# Generalized Von Neumann Universe and Non-Well-Founded Sets

Eugene Zhang

###### Abstract

In this paper, a generalized version of the von Neumann universe known as the
total universe is proposed to formally introduce non-well-founded sets that
include infinitons, semi-infinitons and quasi-infinitons in Russell’s paradox.
All three infinitons are part of infinitely generated sets that are generators
of non-well-founded sets. The total universe combines the well-founded sets
with the non-well-founded ones and turns out to be a model of ZF minus the
axiom of regularity, which can be shown invalid in defining the well-founded
sets. Also, the total universe is shown to be free of Russell’s paradox.

Key Word: Non-well-founded sets; Infiniton; Semi-infiniton; Quasi-infiniton;
Infinitely generated set; Limit of formulas; Rank; Axiom of regularity;
Nullity; Russell’s paradox; Total universe.

###### Contents

  1. 1 Preliminaries
    1. 1.1 Introduction
    2. 1.2 Problems in Von Neumann Universe
    3. 1.3 Correct Rank in Von Neumann Universe
    4. 1.4 Unpacking Operator and Nullity
    5. 1.5 Membership Dimension
  2. 2 Limit of Structures and Formulas
    1. 2.1 Limit in General Language
    2. 2.2 Limit in Language of Set Theory
  3. 3 Non-Well-Founded Sets
    1. 3.1 Introduction
    2. 3.2 Infinitely Generated Set
    3. 3.3 Infiniton
    4. 3.4 Semi-Infiniton
    5. 3.5 Quasi-Infiniton
  4. 4 Total Universe
    1. 4.1 Definitions
    2. 4.2 Rank in Total Universe
    3. 4.3 Set Theory for Total Universe
    4. 4.4 Solution to Russell’s paradox
    5. 4.5 Spectrum of Power Set Operations
  5. 5 Conclusion

###### List of Figures

  1. 1 Diagrams of infinitons and a set of infinitons.
  2. 2 Diagram of a semi-infiniton.
  3. 3 Diagram of a quasi-infiniton.
  4. 4 Diagram of the total universe.

##  1 Preliminaries

###  1.1 Introduction

The investigation of non-well-founded sets began with the work of Mirimanoff
in 1917 [12]. A number of axiomatic systems of non-well-founded sets such as
AFA (by Aczel, Forti and Honsell [1]), SAFA (by Scott), FAFA (by Finsler), and
BAFA (by Boffa), have been proposed thereafter. These systems introduce non-
well-founded sets by replacing the axiom of regularity with separate anti-
foundation axioms. The main problem with these systems is that they lack
precise mathematical descriptions for non-well-founded sets. As the result,
non-well-founded sets are not rigorously defined and the exact process to
generate them is unclear.

In this paper, we will present a model for precisely defining the non-well-
founded sets based on the notion of limit of formulas and generating them by
enlarging the von Neumann universe. First, we show that $V$ is incomplete
because it does not have limit ordinal ranks, a fact that is of fundamental
importance because it implies that non-well-founded sets are necessarily
existent and should be added to $V$ as the limit ordinal ranks. Furthermore,
limits of finite structures and formulas along with an algebraic system to
handle the limit operations are given to provide enough mathematical power for
describing non-well-founded sets. Consequently, the new complete universe of
sets known as the total universe (4.1) can be shown to be a model of ZF minus
the axiom of regularity and free of Russell’s paradox. The axiom of regularity
is invalid even in defining the well-founded sets (section 5). Finally, we
show that the total universe can be generated from the full spectrum of power
set operations.

###  1.2 Problems in Von Neumann Universe

The von Neumann universe (also known as the cumulative hierarchy) is well
known as the class of hereditary well-founded sets and is defined as follows:

$\displaystyle V_{0}\,$ $\displaystyle=\,\varnothing\mathfrak{;}$
$\displaystyle V_{\alpha}\,$
$\displaystyle=\,\mathcal{P}(V_{\alpha-1})\text{,\qquad\qquad\ }\alpha\text{
is any successor ordinal;}$ $\displaystyle V_{\alpha}\,$
$\displaystyle=\underset{\beta<\alpha}{\bigcup}V_{\beta},\text{\qquad\qquad\quad\
}\alpha\text{ is any limit ordinal;}$ $\displaystyle V\,$
$\displaystyle=\underset{\alpha\in Ord}{\bigcup}V_{\alpha}.$ (1.1)

The structure of any set $S$ can be represented by a tree, in which $S$ can be
regarded as the root and all the objects in the transitive closure of $S$ form
the nodes of the tree [6]. A branch (or path) of the tree is a sequence of
nodes connected by “$\in$” from the root to an end node known as a terminal.
Clearly, the only terminal in $V\,$is $\varnothing$. A finite branch consists
of a finite number of nodes, while an infinite branch contains an infinite
number of nodes.

A transfinite sequence
$\gamma_{\alpha}=\left\langle\gamma_{\xi}\colon\xi\leqslant\alpha\right\rangle$
is a function with an ordinal domain where $\alpha$ is its length [4]. A
$\in-\hskip 1.4457pt$sequence is a transfinite sequence $\gamma_{\alpha}$ that
$\gamma_{0}\in\cdots\in\gamma_{\xi}\in\gamma_{\xi+1}\in\cdots\in\gamma_{\alpha}$.
Obviously, any branch of $S$ in $V$ can be represented by a $\in-\hskip
1.4457pt$sequence like $\varnothing=\gamma_{0}\in\cdots\in\gamma_{\alpha}=S$.
As the result, well-founded and non-well-founded sets can be defined upon
$\in-\hskip 1.4457pt$sequences as follows.

###### Definition 1.1

Suppose $S$ is a set (with $\varnothing$ as the only terminal) and
$\gamma_{\alpha}=\left\langle\gamma_{\xi}\colon\xi\leqslant\alpha\right\rangle$
is a $\in-\hskip 1.4457pt$sequence in $S$. Then $S\,$is well-founded (WF) if
any $\gamma_{\alpha}$ of $S$ has $\alpha<\omega.$ $S\,$is non-well-founded
(NWF) if one $\gamma_{\alpha}$ of $S$ has $\alpha\geqslant\omega.$ If all
$\gamma_{\alpha}$ of $S$ have $\alpha\geqslant\omega$, $S\ $is totally non-
well-founded (TNWF).

From definition 1.1, it follows easily that $V\,$consists of only well-founded
sets.

###### Lemma 1.2

$V$ is well-founded and no set in $V$ is non-well-founded.111In this paper, an
existing theorem in set and model theory is listed as a proposition, while a
(mainly) new result is proved as a theorem. Lemmas, corollaries, conclusions
and axioms can have both new and existing results.

Proof. By definition 1.1, we only need to prove (by transfinite induction)
that any $\in-\hskip 1.4457pt$sequence in $V$ is of finite length. First, any
$\in-\hskip 1.4457pt$sequence in $V_{1}$ is of finite length for
$V_{1}=\\{\varnothing\\}$. Suppose any $\in-\hskip 1.4457pt$sequence
$Z_{\xi}\in V_{\beta}$ has length $\xi<\omega$ for $\beta<\alpha$. If $\alpha$
is a successor ordinal and $Z_{\xi}\in V_{\alpha-1}$ has $\xi<\omega$, then
$Z_{\xi+1}$ is a $\in-\hskip 1.4457pt$sequence in $V_{\alpha}$ and
$\xi+1<\omega$. If $\alpha$ is a limit ordinal, for any $X\in
V_{\alpha}=\underset{\beta<\alpha}{\displaystyle\bigcup}V_{\beta}$, there is a
$\gamma<\alpha$ that $X\in V_{\gamma}$. Since any $\in-\hskip
1.4457pt$sequence $Z_{\xi}\in V_{\gamma}$ has $\xi<\omega$, so is $X.$ Thus
$V$ is WF.

If $X\in V$ is NWF, $X$ has a $\in-\hskip 1.4457pt$sequence $Z_{\xi}$ with
length $\xi\geqslant\omega$, contradicting $V$ being WF.

Rank in $V$ is defined as follows.

###### Definition 1.3

The rank of $X$ in $V$ is the least $\alpha$ that $X\in V_{\alpha+1}$ (or
equivalently $X\subset V_{\alpha}$).222A good survey on rank in set theory is
given in [8].

This definition of rank appears to be originated by Mirimanoff [12], developed
by Bernays [2], and given its current form by Tarski [18]. Nonetheless, it is
inadequate for the following reasons. Rank in a universe of the sets is a
function $R$ mapping each set to a unique ordinal number and satisfies the
property of monotonicity, i.e. for any $Y\in X$, $R(Y)<R(X)$, and also
$R(\\{X\\})=R(X)+1$.

First, it is reasonable to assume that a set with one infinite branch (rather
than infinite splittings) has the rank of a limit ordinal. For example,
suppose
$I_{n}=\underset{n}{\underbrace{\\{\cdots\\{\varnothing\\}\cdots\\}}}.$ Then
we have an infinite $\in-\hskip 1.4457pt$sequence as:
$\varnothing\in\\{\varnothing\\}\in\cdots\in I_{n}\in\cdots\in
I_{\omega}$.333$I_{\omega}$ is called an infiniton as in definition 3.14. By
the monotonicity of $R$, we have

$R(I_{n+1})\,=\,R(\\{I_{n}\\})\,=\,R(I_{n})+1\,=\,R(\varnothing)+n+1\quad\text{and
so}\quad R(I_{\omega})\,=\,\omega$

On the other hand, a WF set with infinite splittings like $\omega$ should not
have a rank of a limit ordinal because all of its branches are of finite
length. More specifically, even if for any $n\in\omega$, $R(n)<R(\omega)$, we
can not conclude that $R(\omega)=\omega$ because $\omega+1$ also qualifies.

Furthermore, it is natural to define the rank of a set as the first layer in
$V$ to contain the set. For example, $\\{\\{\varnothing\\}\\}$ should have the
rank of $3$ for it is in $V_{3}$ and not in $V_{2}$. As the result, definition
1.3 is fallacious because it enforces all sets to be contained only in layers
of successor ordinals and never actually involves layers of limit ordinals.

###  1.3 Correct Rank in Von Neumann Universe

From the previous discussion, we adopt a more natural and correct version of
rank in $V$ as follows.

###### Definition 1.4

The rank of $X$ in $V$ is defined as the least $\alpha$ that $X\in V_{\alpha}$
and denoted as $R_{V}(X)$.

###### Lemma 1.5

No set in $V$ has a rank of a limit ordinal.

Proof. Suppose $\alpha$ is a limit ordinal and $R_{V}(X)=\alpha$. Then there
is a $\gamma<\alpha$ that $X\in V_{\gamma}$ or $R_{V}(X)<\alpha$,
contradiction.

###### Corollary 1.6

For any von Neumann ordinal $\alpha\in V,\,R_{V}(\alpha)\,=\,\alpha+1$.

Proof. We prove by transfinite induction. First, since $\varnothing\in
V_{1},\,R_{V}(\mathbf{0})=1$. If $\alpha$ is a successor ordinal, suppose
$R_{V}(\alpha)=\alpha+1$, i.e. $\alpha\in V_{\alpha+1}$ and $\alpha\notin
V_{\alpha}$. Then
$\alpha+1=\alpha\cup\\{\alpha\\}\in\mathcal{P(}V_{\alpha+1})=V_{\alpha+2}$.
Also, $\alpha+1\notin V_{\alpha+1}$ for otherwise $\alpha+1\subset
V_{\alpha}$, which means $\alpha\in V_{\alpha}$, contradiction. So
$R_{V}(\alpha+1)\,=\,\alpha+2.$

If $\alpha$ is a limit ordinal, for any $\gamma<\alpha,\,\gamma\in
V_{\gamma+1}\subset V_{\alpha}$, i.e. $\alpha\subset V_{\alpha}$ and
$\alpha\in V_{\alpha+1}.$ If $\alpha\in
V_{\alpha}=\displaystyle\bigcup\limits_{\beta<\alpha}V_{\beta}$, there is a
$\gamma<\alpha,\,\alpha\in V_{\gamma}$, contradiction. Thus
$R_{V}(\alpha)\,=\,\alpha+1$.

###### Remark 1.7

Corollary 1.6 shows that the rank of any ordinal in $V$ is a successor
ordinal, which is consistent with the fact that $V$ contains only well-founded
sets and no set in $V$ has a rank of a limit ordinal.

Next, we will introduce the notion of the unpacking operator that is important
for the rest discussion in this paper.

###  1.4 Unpacking Operator and Nullity

###### Definition 1.8

Suppose $G=\\{a_{1},a_{2},\cdots\\}.$ The unpacking operator $\ast G$ of $G$
is defined as $\\{\ast G\\}=G,$ i.e. $\ast G=a_{1},a_{2},\cdots$.

###### Remark 1.9

Intuitively, the unpacking operator can be considered as removing the curly
brackets of a set, and $\ast G$ as the collection of $a_{i}$ without the curly
brackets.

###### Example 1.10

Let $S=\\{a_{1},a_{2},\cdots,b_{1},b_{2},\cdots\\}$,
$G_{1}=\\{a_{1},a_{2},\cdots\\}$ and $G_{2}=\\{b_{1},b_{2},\cdots\\}.$ Then
$S=\\{\ast G_{1},\ast G_{2}\\}.$

The unpacking of the empty set $\ast\varnothing$ is of particular importance
because it represents “nothing” or “nullity”, a philosophical term that
denotes the general state of void or nonexistence. The empty set $\varnothing$
is not “nothing” because it is a set with nothing inside it. This can be
understood by viewing a set as a bag — an empty bag is still a bag. Unpacking
the empty set, nonetheless, removes the empty bag, and thus there is nothing
left, or only nullity exists. Since $\varnothing=\\{\ast\varnothing\\}$ and
$\varnothing\subset S,\,\ast\varnothing$ is a member of every set. So we have
the following axiom.

###### Axiom 1.11

$\ast\varnothing$ is known as nullity. Suppose $S$ is any set. Then

  1. (i)

$\varnothing\>\Longleftrightarrow\,\left(\forall
X\in\varnothing\right)\left(X=\ast\varnothing\right)$

  2. (ii)

$\forall S\left(\ast\varnothing\in S\right)$

  3. (iii)

$\forall X(X\in\ast\varnothing\,\,\Longrightarrow\,X=\ast\varnothing)$

  4. (iv)

$\forall
S\left(\left(S,\ast\varnothing\right)\,=\,\left(\ast\varnothing,S\right)\,=\,\ast\varnothing\right)$

###### Remark 1.12

The definition of $\varnothing$ is changed from containing nothing to
containing nullity as its only member. In general, if nothing satisfies a
sentence in the axiom of comprehension, then the solution is nullity.

###### Remark 1.13

$\ast\varnothing$ is a special object that permeates in every set, but is not
involved in the general set operations. Since every set contains
$\ast\varnothing$, most conclusions in set theory remain unchanged.

###### Lemma 1.14

$\\{S,\ast\varnothing\\}\,=\,\\{S\\}$

Proof. By axiom 1.11, for any $X$

$X\in\\{S\\}\,\,\Longleftrightarrow\,X=S\,\vee
X=\ast\varnothing\,\,\Longleftrightarrow\,X\in\\{S,\ast\varnothing\\}$
---

Lemma 1.14 means that $\ast\varnothing$ can be omitted in any set.

###### Corollary 1.15

$S\times\varnothing\,=\,\varnothing\times S\,=\,\varnothing$

Proof. For any $\left(x,y\right)\in S\times\varnothing$, by axiom 1.11(iv),
$\left(x,y\right)\,=\,\left(x,\ast\varnothing\right)\,=\,\ast\varnothing$. So
$S\times\varnothing\,=\,\varnothing$. The second part is proved similarly.

###  1.5 Membership Dimension

###### Definition 1.16

The membership dimension of $S$ (with $\varnothing$ as the only terminal) is
the measure that is defined by the recursive function from $S$ to a cardinal
number, i.e. $D\colon S\rightarrow\mathcal{K}$ and

$D(S)\,=\,\sup\\{D(X)\colon X\in S\\}\,+\,1$ (1.2)

Where $D(\varnothing)=1$ for $\varnothing$ is the terminal and
$\varnothing=\\{\ast\varnothing\\}$ (axiom 1.11).

###### Remark 1.17

Membership dimension is based on cardinal numbers rather than ordinal ones
because it measures the maximum number of curly brackets in a set.

###### Example 1.18

Suppose $\mathbf{n}$ is a finite von Neumann ordinal.

  1. (i)

$D(\mathbf{2})\,=\,D(\\{\varnothing,\\{\varnothing\\}\\})=3$
($D(\\{\varnothing\\})=2\,$ and $D(\varnothing)=1$).

  2. (ii)

$D(\mathbf{n})\,=\,n+1.$

  3. (iii)

$D(\omega)\,=\,\aleph_{0}$ ($D(\omega)>n$ for any $n\in\mathbb{N}$).

The notion of membership dimension gives a necessary condition on when a set
can be a member of itself.

###### Theorem 1.19

Suppose $S_{n}$ are sets of finite membership dimension.

  1. (i)

$S_{1}\in S_{2}\,\,\Longrightarrow\,D(S_{1})<D(S_{2})$

  2. (ii)

$S_{1}\in S_{2}\,\wedge\,\cdots\,\wedge\,S_{n-1}\in
S_{n}\,\,\Longrightarrow\,D(S_{1})<D(S_{n})$

  3. (iii)

$\urcorner\left(S_{1}\in S_{2}\,\wedge\,\cdots\,\wedge\,S_{n-1}\in
S_{n}\,\wedge\,S_{n}\in S_{1}\right)$

Proof. (i) By (1.2), $D(S_{2})\geqslant D(S_{1})+1>D(S_{1})$.

(ii) By (i) and induction.

(iii) If it is true, then by (i) and (ii), $D(S_{n})<D(S_{1})$ and
$D(S_{1})<D(S_{n}),$ which is contradiction. $n=1$ reduces to the case that
there is no $S_{1}$ that $S_{1}\in S_{1}$.

###### Corollary 1.20

That a set is a member of itself or contains a vicious cycle happens only if
it has the infinite membership dimension.

Proof. By theorem 1.19(iii). $n=1$ reduces to the case that $S_{1}\in S_{1}$
means $D(S_{1})=\aleph_{0}.$

Note that the converse is not true. For example, $D(\omega)=\aleph_{0}$, but
$\omega\notin\omega$ for $\omega$ is WF.

Membership dimension allows non-well-founded sets like infinitons to be
defined intuitively as follows.444In later sections, we will give a rigorous
treatment of these sets based on the notion of limit for finite structures. An
infiniton is a set that contains itself as the only member,555In some
literature, it is known as a Quine atom. i.e.

$I=\underset{\aleph_{0}}{\,\underbrace{\\{\cdots\\{\varnothing\\}\cdots\\}}\,}=\underset{\aleph_{0}+1}{\,\underbrace{\\{\\{\cdots\\{\varnothing\\}\cdots\\}\\}}\,}=\left\\{\underset{\aleph_{0}}{\,\underbrace{\\{\cdots\\{\varnothing\\}\cdots\\}}\,}\right\\}=\,\\{I\\}$
(1.3)

Generally, a set that is a member of itself is known as a semi-infiniton that
takes on the following form. Suppose $G\,=\,\\{a_{1},a_{2},\cdots\\}$ and
$a_{k}\in V.$ Then

$Z\,=\,\\{a_{1},a_{2},\cdots,Z\\}\,=\,\\{\ast G,Z\\}$

By replacing $Z$ with itself infinite times, we have

$Z\,=\,\underset{\aleph_{0}}{\underbrace{\\{\ast G,\\{\ast G,\cdots\\{\ast
G,\varnothing\\}\cdots\\}}}$ (1.4)

Then (1.4) is the solution to $Z=\\{\ast G,Z\\}$ for

$Z\,=\,\underset{\aleph_{0}+1}{\underbrace{\\{\ast G,\\{\ast G,\cdots\\{\ast
G,\varnothing\\}\cdots\\}}}\,=\,\\{\ast
G,\underset{\aleph_{0}}{\underbrace{\\{\ast G,\cdots\\{\ast
G,\varnothing\\}\cdots\\}}}\\}\,=\,\\{\ast G,Z\\}$

An infiniton is a special case of a semi-infiniton because $Z=\\{\ast G,Z\\}$
reduces to $Z=\\{Z\\}$ if $G=\varnothing$ (lemma 1.14).

A set that contains a vicious cycle is called a quasi-infiniton and is
illustrated as follows. Suppose

$S_{1}=\\{\ast G_{1},S\\},\text{ }S_{2}=\\{\ast G_{2},S_{1}\\},\text{
}\cdots,\text{ }S_{n-1}=\\{\ast G_{n-1},S_{n-2}\\},\text{ }S=\\{\ast
G_{n},S_{n-1}\\}$

Then $S\in S_{1},\,\cdots,\,S_{n-2}\in S_{n-1},\,S_{n-1}\in S$ form a vicious
cycle. Let

$S\,=\,\underset{\aleph_{0}}{\underbrace{\\{\underset{n}{\underbrace{\ast
G_{n},\\{\ast G_{n-1},\cdots,\\{\ast
G_{1}}},\cdots\\{\underset{n}{\underbrace{\ast G_{n},\\{\ast
G_{n-1},\cdots,\\{\ast G_{1}}},\varnothing\\}\cdots\\}}}$ (1.5)

Then $S$ is the solution to $\,Q\,=\,\\{\ast G_{n},\\{\ast
G_{n-1},\cdots,\\{\ast G_{1},Q\\}\cdots\\}$ for

$\displaystyle S\,$
$\displaystyle=\,\underset{\aleph_{0}+n}{\underbrace{\\{\underset{n}{\underbrace{\ast
G_{n},\\{\ast G_{n-1},\cdots,\\{\ast
G_{1}}},\cdots\\{\underset{n}{\underbrace{\ast G_{n},\\{\ast
G_{n-1},\cdots,\\{\ast G_{1}}},\varnothing\\}\cdots\\}}}$
$\displaystyle=\,\\{\underset{n}{\underbrace{\ast G_{n},\\{\ast
G_{n-1},\cdots,\\{\ast
G_{1}}},\underset{\aleph_{0}}{\underbrace{\\{\underset{n}{\underbrace{\ast
G_{n},\\{\ast G_{n-1},\cdots,\\{\ast
G_{1}}},\cdots\\{\underset{n}{\underbrace{\ast G_{n},\\{\ast
G_{n-1},\cdots,\\{\ast G_{1}}},\varnothing\\}\cdots\\}}}\cdots\\}$
$\displaystyle=\,\\{\underset{n}{\underbrace{\ast G_{n},\\{\ast
G_{n-1},\cdots,\\{\ast G_{1}}},S\\}\cdots\\}$

Obviously, generators of a quasi-infiniton form a finite cycle, and a quasi-
infiniton (1.5) is reduced to a semi-infiniton (1.4) if all $G_{k}$
$\left(1\leqslant k\leqslant n\right)$ are identical.

##  2 Limit of Structures and Formulas

In this section, we will investigate limits of (finitely generated) structures
and formulas which provide a rigorous mathematical foundation for
understanding non-well-founded sets (3.2). The limit of finite structures is
an infinite structure that can be described by an infinitely long formula of
$\mathcal{L}_{\omega_{1},\omega_{1}}$ involving countably many conjunctions,
disjunctions and quantifiers. An infinitely long formula involving countably
many quantifiers may be undecidable [5] but is always decidable in
$\mathcal{L}_{\omega_{1},\omega}$ with only a finite number of quantifiers
[16]. First, we discuss the limit of structures in a general language of
$\mathcal{L}_{\omega_{1},\omega}$.

###  2.1 Limit in General Language

We begin with a brief review of some background knowledge in model theory. A
(finitary) formula is a finite well-formed sequence of symbols from a given
alphabet that is part of a formal language. A sentence is a formula that
contains no free variables. A theory is a set of sentences in a first-order
language $\mathcal{L}$ that is closed under logical implication. A model of a
theory $T$ is a structure (a set along with relations, functions and
constants) that satisfies the sentences of $T$. A consistent theory $T$ is a
theory in which there is no sentence $\varphi$ that $T\vdash\varphi$ and
$T\vdash\urcorner\varphi$. A complete theory $T$ is a theory in which for any
sentence $\varphi,$ either $\varphi\in T$ or $\urcorner\varphi\in T$.

Furthermore, given a $\mathcal{L}$-structure $\mathfrak{M}$ of a consistent
theory $T$ and a $n-$tuple $a_{1},\cdots,a_{n}\in\mathfrak{M}$, a set of
formulas $\gamma(x_{1},\cdots,x_{n})$ of $\mathcal{L}$ satisfied by
$a_{1},\cdots,a_{n}$ is a $\mathbf{n-}$type of $T$. A complete type in $T$ is
a maximal consistent set $\Gamma(x_{1},\cdots,x_{n})$ of formulas in the
variables $x_{1},\cdots,x_{n}$. Any non-complete type of $T$ is called a
partial type. A type of $T$ refers to either a $n-$type, a complete type or a
partial type in $T$. The theory of $\mathfrak{M}$ (denoted as
Th$(\mathfrak{M})$) is the set of the sentences satisfied by $\mathfrak{M}$.
In a complete theory $T$, a formula $\varphi$ is called complete in $T$ if for
every formula $\phi,\,T\models\varphi\rightarrow\phi$ or
$T\models\varphi\rightarrow\urcorner\phi$. A type $\Gamma(x_{1},\cdots,x_{n})$
is called an isolated type in $T$ if, for any $\gamma\in\Gamma$, there is a
complete formula $\varphi$ that $T\models\varphi\rightarrow\gamma$ ([3] and
[11]).

An atomic theory $T$ is a theory in which every formula $\gamma$ that is
consistent with $T$ can be derived from a complete formula $\varphi$ in $T$,
i.e. $T\models\varphi\rightarrow\gamma$. A $\mathcal{L}$-structure
$\mathfrak{A}$ is an atomic structure if every $n-$tuple
$a_{1},\cdots,a_{n}\in\mathfrak{A}$ satisfies a complete formula in
Th$(\mathfrak{A})$. Obviously, every type in an atomic theory is isolated. A
$\mathcal{L}$-structure $\mathfrak{M}$ is called $\mathbf{\kappa-}$saturated
if for all subsets $A\subset\mathfrak{M}$ of cardinality less than
$\kappa,\,\mathfrak{M}$ realizes all complete types over $A$. $\mathfrak{M}$
is called countably saturated if it is $\aleph_{0}$-saturated. A countable
$\mathcal{L}$-structure $\mathfrak{M}$ is homogeneous if any isomorphism
between its two finite substructures can be extended to an automorphism of
$\mathfrak{M}$. More precisely, for any two tuples
$\vec{a}=(a_{1},\cdots,a_{n})$ and $\vec{b}=(b_{1},\cdots,b_{n})$ in
$\mathfrak{M}$ that realize the same types, there is an automorphism of
$\mathfrak{M}$ taking $\vec{a}$ to $\vec{b}$. An atomic or saturated structure
is always homogeneous. ([3] and [19])

A $\aleph_{0}$-categorical theory in $\mathcal{L}$ has exactly one countable
structure up to isomorphism. A $\aleph_{0}$-categorical structure is a
countable structure whose theory is $\aleph_{0}$-categorical. We list the
following theorems without proof. Proposition 2.3 states that a
$\aleph_{0}$-categorical structure is made up of only finitely many countable
atomic structures. It is central to $\aleph_{0}$-categorical theories and is
essentially due to Engeler, Ryll-Nardzewski, Svenonius and Vaught.

###### Proposition 2.1

Any atomic structure is homogeneous.

###### Proposition 2.2

Two countable homogeneous structures that realize the same types are
isomorphic.

###### Proposition 2.3

[3, Theorem 2.3.13] Let $T$ be a complete theory. Then the following are
equivalent:

  1. (i)

$T$ is $\aleph_{0}$-categorical.

  2. (ii)

$T$ has a structure that is both atomic and countably saturated.

  3. (iii)

For each $n<\omega$, each type $\Gamma(x_{1},\cdots,x_{n})$ of $T$ has a
complete formula.

  4. (iv)

For each $n<\omega,\>T$ has only finitely many types in $x_{1},\cdots,x_{n}$.

  5. (v)

For each $n<\omega$, there are only finitely many formulas
$\varphi(x_{1},\cdots,x_{n})$ up to equivalence with respect to $T$.

  6. (vi)

All structures of $T$ are atomic.

Now we discuss the limit of structures and formulas in an infinitary language
of $\mathcal{L}_{\omega_{1},\omega}$ for a $\aleph_{0}$-categorical theory.
First, we define the neighborhood of $\omega$ based on the cofinite topology.

###### Definition 2.4

The cofinite topology on $\omega$ is defined as
$\,\mathfrak{T}\,=\,\\{y\subset\omega\colon y=\varnothing\,\vee\,\omega-y$ is
finite$\,\\}$. A neighborhood of
$\mathbf{\omega}\,(\omega-$neighborhood$)\,\mathfrak{H}$ is a member of
$\mathfrak{T}$, i.e. $\mathfrak{H}\in\mathfrak{T}.$

###### Lemma 2.5

$\mathfrak{H}$ is a neighborhood of $\omega$ if and only if $\exists
N\in\omega$ such that $\forall n>N,n\in\mathfrak{H}.$

Proof. Suppose $\mathfrak{H}$ is a neighborhood of $\omega$ and for any
$N\in\omega,$ there is $n>N,\,n\notin\mathfrak{H}.$ Then $\omega-\mathfrak{H}$
is not finite, contradicting definition 2.4. On the other hand, if there is a
$N\in\omega$ such that for any $n>N,n\in\mathfrak{H},$ then
$\omega-\mathfrak{H}$ is finite and $\mathfrak{H}\in\mathfrak{T}$.

###### Definition 2.6

Suppose $\mathcal{L}$ is an infinitary language of
$\mathcal{L}_{\omega_{1},\omega}$ and $T$ is a $\aleph_{0}$-categorical theory
of $\mathcal{L}$. Let $\phi_{n}$ be $n-$types in $T$ and $\,\mathfrak{M}_{n}$
be $\mathcal{L}$-structures that $\,\mathfrak{M}_{n}\models\phi_{n}$. If there
exists a $\omega-$neighborhood $\mathfrak{H}$ that for any
$k,n\in\mathfrak{H}\>(k>n)$, $\mathfrak{M}_{k}\models\phi_{n}$, then
$\\{(\mathfrak{M}_{n},\phi_{n})\colon\mathfrak{M}_{n}\models\phi_{n}\land
n<\omega\\}$ is known as a homogeneous sequence of structures described by
$\phi_{n}$ in $T$.777In the rest discussion, we assume $\mathcal{L}$ is an
infinitary language of $\mathcal{L}_{\omega_{1},\omega}$, $T$ is a
$\aleph_{0}$-categorical theory of $\mathcal{L}$, $\phi_{n}$ and $\varphi_{n}$
are formulas in $T$, $\mathfrak{M}_{n}$ and $\mathfrak{N}_{n}$ are
$\mathcal{L}$-structures, and $\mathfrak{H}$ is a neighborhood of $\omega$
unless further specified.

###### Theorem 2.7

Suppose $T$ is a $\aleph_{0}$-categorical theory of $\mathcal{L}$ and
$\\{(\mathfrak{M}_{n},\phi_{n})\colon\mathfrak{M}_{n}\models\phi_{n}\land
n<\omega\\}$ is a homogeneous sequence of structures in $T$. Then there is a
unique formula $\phi$ in $\mathcal{L}$ (up to equivalence) and a unique
countable atomic structure $\mathfrak{M}$ (up to isomorphism) for
$\\{(\mathfrak{M}_{n},\phi_{n})\\}$ such that $\mathfrak{M}\models\phi$. In
addition, there is a $N\in\omega$ such that
$\,\phi\,=\>\smashoperator[]{\bigwedge_{N<n<\omega}^{}}\phi_{n}$ which is the
complete formula of $\mathfrak{M}$.

Proof. By lemma 2.5, $\exists N\in\omega$ such that $\forall
k>n>N,\,\mathfrak{M}_{k}\models\phi_{n}$. Let $\Sigma=\\{\phi_{n}\colon
n>N\\}$. Clearly, $\Sigma$ is consistent and any finite subset of $\Sigma$ has
a model. So by the compactness theorem, $\Sigma$ has a model. Since arbitrary
large number of $\phi_{n}$ can be realized by $\mathfrak{M}_{k}$, $\Sigma$ has
an infinite model. Let $\mathfrak{M}$ be a countable structure that satisfies
$\Sigma$. Then
$\mathfrak{M}\models\>\smashoperator[]{\bigwedge_{N<n<\omega}^{}}\phi_{n}=\phi$.
Since $\Sigma$ is in a type of $T$, by proposition 2.3(vi), $\mathfrak{M}$ is
a countable atomic structure. Since any $\phi_{n}$ can be derived from $\phi$,
$\phi$ is the complete formula of $\mathfrak{M}$.

Suppose $\mathfrak{N}$ is another countable atomic structure satisfying
$\Sigma$. By proposition 2.1, both $\mathfrak{M}$ and $\mathfrak{N}$ are
homogeneous. And by proposition 2.2, $\mathfrak{M}\simeq\mathfrak{N}$. So
$\mathfrak{M}$ is unique.

###### Corollary 2.8

Suppose in a sequence of structures $\\{(\mathfrak{M}_{n},\phi_{n})\colon\
\mathfrak{M}_{n}\models\phi_{n}\land n<\omega\\}$ of a
$\aleph_{0}$-categorical theory $T$, there are finitely many homogeneous
subsequences of structures
$\\{(\mathfrak{M}_{n_{i}},\phi_{n_{i}})\colon\allowbreak\mathfrak{M}_{n_{i}}\models\phi_{n_{i}}\land
n_{i}<\omega\\}$. Then there is a unique formula $\widetilde{\phi_{i}}$ in
$\mathcal{L}$ (up to equivalence) and a unique countable atomic structure
$\widetilde{\mathfrak{M}_{i}}$ (up to isomorphism) for each
$\\{(\mathfrak{M}_{n_{i}},\phi_{n_{i}})\\}$ such that
$\widetilde{\mathfrak{M}_{i}}\models\widetilde{\phi_{i}}$. In addition, there
is a $N_{i}\in\omega$ such that
$\,\widetilde{\phi_{i}}\,=\>\smashoperator[]{\bigwedge_{N_{i}<n_{i}<\omega}^{}}\phi_{n_{i}}$
which is the complete formula of $\widetilde{\mathfrak{M}_{i}}$.

Proof. Let $\Sigma_{i}=\\{\phi_{n_{i}}\colon n_{i}>N_{i}\\}$. By the proof in
theorem 2.7, $\Sigma_{i}$ is consistent and belongs to a type of $T$. So by
proposition 2.3, it is satisfied by a unique countable atomic structure
$\widetilde{\mathfrak{M}_{i}}$ described by a complete formula
$\widetilde{\phi_{i}}$.

From definition 2.6, theorem 2.7 and corollary 2.8, we have the following
definitions.

###### Definition 2.9

Suppose
$\\{(\mathfrak{M}_{n},\phi_{n})\colon\mathfrak{M}_{n}\models\phi_{n}\land
n<\omega\\}$ is a homogeneous sequence of structures in a
$\aleph_{0}$-categorical theory. The unique countable atomic structure
$\mathfrak{M}$ (up to isomorphism) in $\\{(\mathfrak{M}_{n},\phi_{n})\\}$ is
known as the limit of $\mathfrak{M}_{n}$ and is denoted as
$\underset{n\rightarrow\omega}{\lim}\mathfrak{M}_{n}=\mathfrak{M}$. The unique
formula $\phi$ (up to equivalence) is known as the limit of $\phi_{n}$ and is
denoted as $\underset{n\rightarrow\omega}{\lim}\phi_{n}=\phi$. In both cases,
we also say that the limit of $\phi_{n}$ or the limit of $\mathfrak{M}_{n}$ is
unique.

###### Definition 2.10

Suppose in a sequence of structures
$\\{(\mathfrak{M}_{n},\phi_{n})\colon\mathfrak{M}_{n}\models\phi_{n}\land
n<\omega\\}$ in a $\aleph_{0}$-categorical theory, there are finitely many
homogeneous subsequences of structures
$\\{(\mathfrak{M}_{n_{i}},\phi_{n_{i}})\colon\allowbreak\mathfrak{M}_{n_{i}}\models\phi_{n_{i}}\land
n_{i}<\omega\\}$. Then each
$\underset{i\rightarrow\omega}{\lim}\,\mathfrak{M}_{n_{i}}$ is known as a
sublimit of $\mathfrak{M}_{n}$, and each
$\underset{i\rightarrow\omega}{\lim}\,\phi_{n_{i}}$ is known as a sublimit of
$\phi_{n}$. If some sublimits of $\mathfrak{M}_{n}/\phi_{n}$ are different, we
say
$\underset{n\rightarrow\omega}{\lim}\mathfrak{M}_{n}/\underset{n\rightarrow\omega}{\lim}\phi_{n}$
exist (but not unique).

In the rest discussion, we will not distinguish “$=$” and “$\Leftrightarrow$”
for formulas. So we have

###### Lemma 2.11

$\left(\underset{n\rightarrow\omega}{\lim}\phi_{n}\,=\,\underset{n\rightarrow\omega}{\lim}\,\varphi_{n}\right)\,\Longleftrightarrow\,\left(\underset{n\rightarrow\omega}{\lim}\phi_{n}\,\Longleftrightarrow\,\underset{n\rightarrow\omega}{\lim}\,\varphi_{n}\right)$

###### Corollary 2.12

Suppose $\underset{n\rightarrow\omega}{\lim}\phi_{n}$ and
$\underset{n\rightarrow\omega}{\lim}\varphi_{n}$ are unique. Then

  1. (i)

$\underset{n\rightarrow\omega}{\lim}\phi\,=\,\phi$

  2. (ii)

$\underset{n\rightarrow\omega}{\lim}\phi_{n-p}\,=\,\underset{n\rightarrow\omega}{\lim}\phi_{n}\,\,\left(p<\omega\right)$

  3. (iii)

$\left(\forall
n\in\mathfrak{H}\right)\left(\phi_{n}\,=\,\varphi_{n}\right)\,\Longrightarrow\,\left(\underset{n\rightarrow\omega}{\lim}\phi_{n}\,=\,\underset{n\rightarrow\omega}{\lim}\,\varphi_{n}\right)$

Proof. By theorem 2.7 and definition 2.9.

###### Corollary 2.13

Suppose $\underset{n\rightarrow\omega}{\lim}\mathfrak{M}_{n}$ and
$\underset{n\rightarrow\omega}{\lim}\mathfrak{N}_{n}$ are unique. Then

  1. (i)

$\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}\,=\,\mathfrak{M}$

  2. (ii)

$\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n-p}\,=\,\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\,\,\left(p<\omega\right)$

  3. (iii)

$\left(\forall
n\in\mathfrak{H}\right)\left(\mathfrak{M}_{n}\,=\,\mathfrak{N}_{n}\right)\,\Longrightarrow\,\left(\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\,=\,\underset{n\rightarrow\omega}{\lim}\mathfrak{N}_{n}\right)$

###### Corollary 2.14

Suppose $\mathfrak{M}_{n}\models\phi_{n}$ and for any
$n\in\omega,\,\models\phi_{n+1}\rightarrow\phi_{n}$. Then

$\underset{n\rightarrow\omega}{\lim}\phi_{n}\,=\,\underset{n\rightarrow\omega}{\lim}\,\underset{m\leqslant
n}{\displaystyle\bigwedge}\phi_{m}\,=\underset{n<\omega}{\displaystyle\bigwedge}\phi_{n}$

Proof. Let $\mathfrak{H}=\omega$. By lemma 2.5, for any $1\leqslant
n<k<\omega,\,\models\phi_{k}\rightarrow\phi_{n}$. Since
$\mathfrak{M}_{k}\models\phi_{n}$ for any $k>n\geqslant 1$,
$\\{(\mathfrak{M}_{n},\phi_{n})\\}$ is a homogeneous sequence. Thus by theorem
2.7,
$\underset{n\rightarrow\omega}{\lim}\phi_{n}=\underset{n<\omega}{\displaystyle\bigwedge}\phi_{n}$.
Since for any $1\leqslant m<n,\>\models\phi_{n}\rightarrow\phi_{m}$,
$\underset{m\leqslant
n}{\displaystyle\bigwedge}\phi_{m}\Leftrightarrow\phi_{n}$. So by theorem 2.7
again, $\underset{n\rightarrow\omega}{\lim}\,\underset{m\leqslant
n}{\displaystyle\bigwedge}\phi_{m}\,=\underset{n<\omega}{\displaystyle\bigwedge}\phi_{n}.$

In addition, the following axiom holds for the limit operations.

###### Axiom 2.15

Suppose $\phi_{n}$ and $\varphi_{n}$ are consistent,
$\underset{n\rightarrow\omega}{\lim}\phi_{n}$ and
$\underset{n\rightarrow\omega}{\lim}\,\varphi_{n}$ are unique in a
$\aleph_{0}$-categorical theory. Then

  1. (i)

$\underset{n\rightarrow\omega}{\lim}\left(\phi_{n}\wedge\varphi_{n}\right)$ is
unique and

$\underset{n\rightarrow\omega}{\lim}\left(\phi_{n}\wedge\varphi_{n}\right)\,=\,\underset{n\rightarrow\omega}{\lim}\phi_{n}\wedge\underset{n\rightarrow\omega}{\lim}\,\varphi_{n}$

  2. (ii)

$\underset{n\rightarrow\omega}{\lim}\urcorner\phi_{n}$ is unique and

$\underset{n\rightarrow\omega}{\lim}\urcorner\phi_{n}\,=\,\urcorner\,\underset{n\rightarrow\omega}{\lim}\phi_{n}$

  3. (iii)

$\underset{n\rightarrow\omega}{\lim}\exists x\,\phi_{n}$ is unique $(x$ is a
variable in $\phi_{n})$, and

$\underset{n\rightarrow\omega}{\lim}\exists x\,\phi_{n}\,=\,\exists
x\,\underset{n\rightarrow\omega}{\lim}\phi_{n}$

###### Lemma 2.16

Suppose $\phi_{n}$ and $\varphi_{n}$ are consistent,
$\underset{n\rightarrow\omega}{\lim}\phi_{n}$ and
$\underset{n\rightarrow\omega}{\lim}\,\varphi_{n}$ are unique in a
$\aleph_{0}$-categorical theory. Then

  1. (i)

$\underset{n\rightarrow\omega}{\lim}\left(\phi_{n}\vee\varphi_{n}\right)\,=\,\underset{n\rightarrow\omega}{\lim}\phi_{n}\vee\underset{n\rightarrow\omega}{\lim}\,\varphi_{n}$

  2. (ii)

$\underset{n\rightarrow\omega}{\lim}\left(\phi_{n}\Longrightarrow\varphi_{n}\right)\,=\,\left(\underset{n\rightarrow\omega}{\lim}\phi_{n}\,\Longrightarrow\,\underset{n\rightarrow\omega}{\lim}\,\varphi_{n}\right)$

  3. (iii)

$\underset{n\rightarrow\omega}{\lim}\left(\phi_{n}\Longleftrightarrow\varphi_{n}\right)\,=\,\left(\underset{n\rightarrow\omega}{\lim}\phi_{n}\,\Longleftrightarrow\,\underset{n\rightarrow\omega}{\lim}\,\varphi_{n}\right)$

  4. (iv)

$\underset{n\rightarrow\omega}{\lim}\,\forall x\,\phi_{n}\,=\,\forall
x\,\underset{n\rightarrow\omega}{\lim}\phi_{n}$

Proof. (i) By axiom 2.15

$\displaystyle\underset{n\rightarrow\omega}{\lim}\left(\phi_{n}\vee\varphi_{n}\right)\,\,$
$\displaystyle=\,\,\underset{n\rightarrow\omega}{\lim}\urcorner\left(\urcorner\phi_{n}\,\wedge\,\urcorner\varphi_{n}\right)$
$\displaystyle=\,\,\urcorner\left(\urcorner\,\underset{n\rightarrow\omega}{\lim}\phi_{n}\,\wedge\,\urcorner\,\underset{n\rightarrow\omega}{\lim}\,\varphi_{n}\right)$
$\displaystyle=\,\,\underset{n\rightarrow\omega}{\lim}\phi_{n}\,\vee\,\underset{n\rightarrow\omega}{\lim}\,\varphi_{n}$

(ii) and (iii) follow from (i).

(iv) By axiom 2.15

$\displaystyle\underset{n\rightarrow\omega}{\lim}\,\forall x\,\phi_{n}\,$
$\displaystyle=\,\,\urcorner\,\underset{n\rightarrow\omega}{\lim}\,\exists
x\urcorner\phi_{n}$ $\displaystyle=\,\,\urcorner\,\exists
x\urcorner\,\underset{n\rightarrow\omega}{\lim}\phi_{n}$
$\displaystyle=\,\,\forall x\,\underset{n\rightarrow\omega}{\lim}\phi_{n}$

###### Corollary 2.17

Suppose $\mathfrak{M}_{n}\models\phi_{n}$ and for any
$n\in\omega,\,\models\phi_{n-1}\rightarrow\phi_{n}$. Then

$\underset{n\rightarrow\omega}{\lim}\,\underset{m\leqslant
n}{\displaystyle\bigvee}\phi_{m}\,=\underset{n<\omega}{\displaystyle\bigvee}\phi_{n}$

Proof. Let $\mathfrak{H}=\omega$. Then for any
$n\in\mathfrak{H},\,\models\,\urcorner\phi_{n}\rightarrow\,\urcorner\phi_{n-1}$.
So by corollary 2.14

$\underset{n\rightarrow\omega}{\lim}\urcorner\phi_{n}\,=\,\underset{n\rightarrow\omega}{\lim}\,\underset{m\leqslant
n}{\displaystyle\bigwedge}\urcorner\phi_{m}\,=\underset{n<\omega}{\displaystyle\bigwedge}\urcorner\phi_{n}$

Thus by axiom 2.15

$\displaystyle\underset{n\rightarrow\omega}{\lim}\,\underset{m\leqslant
n}{\displaystyle\bigvee}\phi_{m}\,$
$\displaystyle=\,\,\urcorner\,\underset{n\rightarrow\omega}{\lim}\,\underset{m\leqslant
n}{\displaystyle\bigwedge}\urcorner\phi_{m}$
$\displaystyle=\,\,\urcorner\,\underset{n<\omega}{\displaystyle\bigwedge}\urcorner\phi_{n}$
$\displaystyle=\,\,\underset{n<\omega}{\displaystyle\bigvee}\phi_{n}$

###### Corollary 2.18

Suppose $\mathfrak{M}_{n}$ are ascending $\mathcal{L}$-structures that
$\,\mathfrak{M}_{n}\models\phi_{n}$. Then
$\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\,=\underset{n<\omega}{\displaystyle\bigcup}\mathfrak{M}_{n}$.

Proof. Since $\mathfrak{M}_{k}\supset\mathfrak{M}_{n}$ for any
$k>n,\,\mathfrak{M}_{k}\models\phi_{n}$, $\\{(\mathfrak{M}_{n},\phi_{n})\\}$
is a homogeneous sequence. So by theorem 2.7,
$\underset{n\rightarrow\omega}{\lim}\phi_{n}$ is unique. Also, since for any
$1\leqslant m<n,\>\models\phi_{m}\rightarrow\phi_{n}$, $\underset{m\leqslant
n}{\displaystyle\bigvee}\phi_{m}\Leftrightarrow\phi_{n}$. Thus by corollary
2.17

$\underset{n\rightarrow\omega}{\lim}\phi_{n}\,=\,\underset{n\rightarrow\omega}{\lim}\,\underset{m\leqslant
n}{\displaystyle\bigvee}\phi_{m}\,=\underset{n<\omega}{\displaystyle\bigvee}\phi_{n}$

Since
$\underset{n<\omega}{\displaystyle\bigcup}\mathfrak{M}_{n}\models\underset{n<\omega}{\displaystyle\bigvee}\phi_{n}$,
$\,\,\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\,=\underset{n<\omega}{\displaystyle\bigcup}\mathfrak{M}_{n}$.

###### Proposition 2.19

Suppose $\phi_{n}\,=\,\exists x\underset{m\leqslant
n}{\displaystyle\bigwedge}\left(x>m\right)$. Then
$\underset{n\rightarrow\omega}{\lim}\phi_{n}\,=\,\exists
x\underset{m<\omega}{\displaystyle\bigwedge}\left(x>m\right)$.

Proof. Let $\varphi_{n}=(x>n)$. Then $\forall
n\in\omega,\>\models\varphi_{n+1}\rightarrow\varphi_{n}$. So it follows by
corollary 2.14 and axiom 2.15. This confirms that there is an arbitrary large
number in nonstandard number theory.

Now we give the limit of formula for the $\epsilon-N$ formula.

###### Corollary 2.20

Suppose $a_{\omega}=\left\langle a_{m}\colon m<\omega\right\rangle$ is a
sequence in a separable space and

$\varphi_{n}\,=\,\exists x\exists N_{n}\,\forall
m\left(m>N_{n}\,\Longrightarrow|a_{m}-x|<1/n\right)$

Then the limit of formula for $\underset{m\rightarrow\omega}{\lim}a_{m}=x$ is:

$\underset{n\rightarrow\omega}{\lim}\,\underset{p\leqslant
n}{\displaystyle\bigwedge}\varphi_{p}\,=\,\exists
x\underset{n<\omega}{\displaystyle\bigwedge}\exists N_{n}\,\forall
m\left(m>N_{n}\,\Longrightarrow|a_{m}-x|<1/n\right)$

Proof. For any $n\in\omega$ and any $m>N_{n}$, since
$|a_{m}-x|<1/\left(n+1\right)\Longrightarrow|a_{m}-x|<1/n$,
$\models\varphi_{n+1}\rightarrow\varphi_{n}$ for any $n<\omega$. So it follows
by corollary 2.14 and axiom 2.15.

Note that a $\aleph_{0}$-categorical theory (or atomic theory) is absolutely
necessary in the above definitions of limit and axiom 2.15 because of the
following example.888This example is suggested by Martin Goldstern. Suppose
$I_{0}=G_{0}$ and $I_{n+1}=\\{I_{n}\\}$ where $G_{0}\neq\\{G_{0}\\}$. Let
$\chi=\forall x\left(x\neq\\{x\\}\right)$ and $\varphi_{n}=\phi_{n}\wedge\chi$
where $\phi_{n}$ is given in theorem 3.13(iii). Since $I_{n}\models\phi_{n}$
and $\underset{n\rightarrow\omega}{\lim}I_{n}$ is unique, by theorem 3.16(i),
$I_{\omega}=\\{I_{\omega}\\}$. So
$\underset{n\rightarrow\omega}{\lim}\,\varphi_{n}$ is unique but it must also
satisfy $\chi$, which is a contradiction. This can be avoided by the fact that
Th$(I_{n})$ is $\aleph_{0}$-categorical (theorem 3.13(ii)). So by theorem
3.16(iv), $I_{\omega}$ is atomic and there is a complete formula
$\varphi_{\omega}$ for $\operatorname{Th}(I_{\omega})$ such that
$\chi\notin\operatorname{Th}(I_{\omega})$ for
$\varphi_{\omega}\Rightarrow\urcorner\chi$.

###  2.2 Limit in Language of Set Theory

Next, we will study the limit of structures in the (infinitary) language of
set theory $\mathcal{L}^{\prime}=\\{\in\\}$.

###### Axiom 2.21

Suppose $\mathfrak{M}_{n}$ are $\mathcal{L}^{\prime}$-structures in a
$\aleph_{0}$-categorical theory and
$\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}$ is unique. Then the
atomic formula of $\mathcal{L}^{\prime}$ is:

$\underset{n\rightarrow\omega}{\lim}\left(\mathfrak{A}\in\mathfrak{M}_{n}\right)\,\Longleftrightarrow\,\left(\mathfrak{A}\in\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\right)$

All formulas of $\mathcal{L}^{\prime}$ involving the limit operations are
formed recursively based upon the atomic formula above satisfying axiom 2.15.

###### Theorem 2.22

Suppose $\mathfrak{M}_{n}$ and $\mathfrak{N}_{n}$ are
$\mathcal{L}^{\prime}$-structures in a $\aleph_{0}$-categorical theory,
$\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}$ and
$\underset{n\rightarrow\omega}{\lim}\,\mathfrak{N}_{n}$ are unique.999In the
rest of discussion of this section, we assume $\mathfrak{M}_{n}$ and
$\mathfrak{N}_{n}$ are $\mathcal{L}^{\prime}$-structures in a
$\aleph_{0}$-categorical theory,
$\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}$ and
$\underset{n\rightarrow\omega}{\lim}\,\mathfrak{N}_{n}$ are unique unless
further specified.

  1. (i)

$\underset{n\rightarrow\omega}{\lim}\left(\mathfrak{M}_{n}\cup\mathfrak{N}_{n}\right)\,=\,\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\,\cup\underset{n\rightarrow\omega}{\lim}\mathfrak{N}_{n}$

  2. (ii)

$\underset{n\rightarrow\omega}{\lim}\left(\mathfrak{M}_{n}\cap\mathfrak{N}_{n}\right)\,=\,\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\,\cap\underset{n\rightarrow\omega}{\lim}\mathfrak{N}_{n}$

  3. (iii)

$\underset{n\rightarrow\omega}{\lim}\left(\mathfrak{M}_{n}-\mathfrak{N}_{n}\right)\,=\,\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\,-\underset{n\rightarrow\omega}{\lim}\mathfrak{N}_{n}$

Proof. We only prove (i). By axiom 2.21 and lemma 2.16

$\displaystyle\forall\mathfrak{B}\in\underset{n\rightarrow\omega}{\lim}\left(\mathfrak{M}_{n}\cup\mathfrak{N}_{n}\right)\,\,$
$\displaystyle\Longleftrightarrow\,\,\underset{n\rightarrow\omega}{\lim}\left(\forall\mathfrak{B}\in\left(\mathfrak{M}_{n}\cup\mathfrak{N}_{n}\right)\right)$
$\displaystyle\Longleftrightarrow\,\,\underset{n\rightarrow\omega}{\lim}\,\forall\mathfrak{B}\left(\mathfrak{B}\in\mathfrak{M}_{n}\vee\mathfrak{B}\in\mathfrak{N}_{n}\right)$
$\displaystyle\Longleftrightarrow\,\,\forall\mathfrak{B}\left(\mathfrak{B}\in\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\vee\mathfrak{B}\in\underset{n\rightarrow\omega}{\lim}\mathfrak{N}_{n}\right)$
$\displaystyle\Longleftrightarrow\,\,\forall\mathfrak{B}\in\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\cup\underset{n\rightarrow\omega}{\lim}\mathfrak{N}_{n}$

###### Theorem 2.23

$\underset{n\rightarrow\omega}{\lim}\left(\mathfrak{M}_{n}=\mathfrak{A}\right)\,\Longleftrightarrow\,\left(\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}=\mathfrak{A}\right)$

Proof. By axioms 2.21 and lemma 2.16

$\displaystyle\underset{n\rightarrow\omega}{\lim}\left(\mathfrak{M}_{n}=\mathfrak{A}\right)\,\,$
$\displaystyle\Longleftrightarrow\,\,\underset{n\rightarrow\omega}{\lim}\,\forall\mathfrak{B}\left(\mathfrak{B}\in\mathfrak{M}_{n}\Longleftrightarrow\mathfrak{B}\in\mathfrak{A}\right)$
$\displaystyle\Longleftrightarrow\,\,\forall\mathfrak{B}\left(\mathfrak{B}\in\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\Longleftrightarrow\mathfrak{B}\in\mathfrak{A}\right)$
$\displaystyle\Longleftrightarrow\,\,\left(\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}=\mathfrak{A}\right)$

###### Corollary 2.24

  1. (i)

$\underset{n\rightarrow\omega}{\lim}\left(\mathfrak{M}_{n}=\mathfrak{N}_{n}\right)\,\Longleftrightarrow\,\left(\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}=\underset{n\rightarrow\omega}{\lim}\mathfrak{N}_{n}\right)$

  2. (ii)

$\underset{n\rightarrow\omega}{\lim}\left(\mathfrak{M}_{n}\in\mathfrak{N}_{n}\right)\,\Longleftrightarrow\,\left(\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\in\underset{n\rightarrow\omega}{\lim}\mathfrak{N}_{n}\right)$

Proof. (i) is proved similar to theorem 2.23.

(ii) By theorem 2.23

$\displaystyle\underset{n\rightarrow\omega}{\lim}\left(\mathfrak{M}_{n}\in\mathfrak{N}_{n}\right)\,\,$
$\displaystyle\Longleftrightarrow\,\,\underset{n\rightarrow\omega}{\lim}\,\exists\mathfrak{Z}\left(\mathfrak{M}_{n}=\mathfrak{Z}\,\wedge\,\mathfrak{Z}\in\mathfrak{N}_{n}\right)$
$\displaystyle\Longleftrightarrow\,\,\exists\mathfrak{Z}\left(\left(\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}=\mathfrak{Z}\right)\wedge\left(\mathfrak{Z}\in\underset{n\rightarrow\omega}{\lim}\mathfrak{N}_{n}\right)\right)$
$\displaystyle\Longleftrightarrow\,\,\left(\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\in\underset{n\rightarrow\omega}{\lim}\mathfrak{N}_{n}\right)$

###### Corollary 2.25

$\underset{n\rightarrow\omega}{\lim}\\{\mathfrak{M}_{n}\\}\,=\,\\{\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\\}$

Proof. By theorem 2.23

$\displaystyle\forall\mathfrak{B}\left(\mathfrak{B}\in\underset{n\rightarrow\omega}{\lim}\\{\mathfrak{M}_{n}\\}\right)\,\,$
$\displaystyle\Longleftrightarrow\,\,\underset{n\rightarrow\omega}{\lim}\,\forall\mathfrak{B}\left(\mathfrak{B}\in\\{\mathfrak{M}_{n}\\}\right)$
$\displaystyle\Longleftrightarrow\,\,\underset{n\rightarrow\omega}{\lim}\,\forall\mathfrak{B}\left(\mathfrak{B}\,=\,\mathfrak{M}_{n}\right)$
$\displaystyle\Longleftrightarrow\,\,\forall\mathfrak{B}\left(\mathfrak{B}\,=\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\right)$
$\displaystyle\Longleftrightarrow\,\,\forall\mathfrak{B}\left(\mathfrak{B}\in\\{\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\\}\right)$

###### Corollary 2.26

Suppose $\phi_{n}$ are $\mathcal{L}^{\prime}$-formulas in a
$\aleph_{0}$-categorical theory.

  1. (i)

$\underset{n\rightarrow\omega}{\lim}\,\left(\forall\mathfrak{B}\in\mathfrak{M}_{n}\right)\phi_{n}\,\Longleftrightarrow\,\left(\forall\mathfrak{B}\in\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\right)\underset{n\rightarrow\omega}{\lim}\phi_{n}$

  2. (ii)

$\underset{n\rightarrow\omega}{\lim}\,\left(\exists\mathfrak{B}\in\mathfrak{M}_{n}\right)\phi_{n}\,\Longleftrightarrow\,\left(\exists\mathfrak{B}\in\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\right)\underset{n\rightarrow\omega}{\lim}\phi_{n}$

Proof. We only prove (i) and (ii) follows easily.

(i) By lemma 2.16

$\displaystyle\underset{n\rightarrow\omega}{\lim}\left(\forall\mathfrak{B}\in\mathfrak{M}_{n}\right)\phi_{n}\,\,$
$\displaystyle\Longleftrightarrow\,\,\underset{n\rightarrow\omega}{\lim}\,\forall\mathfrak{B}\left(\mathfrak{B}\in\mathfrak{M}_{n}\,\Longrightarrow\,\phi_{n}\right)$
$\displaystyle\Longleftrightarrow\,\,\forall\mathfrak{B}\left(\mathfrak{B}\in\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\,\Longrightarrow\,\underset{n\rightarrow\omega}{\lim}\phi_{n}\right)$
$\displaystyle\Longleftrightarrow\,\,\left(\forall\mathfrak{B}\in\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\right)\underset{n\rightarrow\omega}{\lim}\phi_{n}$

###### Corollary 2.27

$\underset{n\rightarrow\omega}{\lim}\,\left(\exists\,\mathfrak{M}_{n}\in\mathfrak{N}_{n}\right)\phi_{n}\,\Longleftrightarrow\,\left(\exists\,\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\in\underset{n\rightarrow\omega}{\lim}\mathfrak{N}_{n}\right)\underset{n\rightarrow\omega}{\lim}\phi_{n}$

Proof. By corollary 2.24 and theorem 2.23

$\displaystyle\underset{n\rightarrow\omega}{\lim}\,\left(\exists\,\mathfrak{M}_{n}\in\mathfrak{N}_{n}\right)\phi_{n}\,\,$
$\displaystyle\Longleftrightarrow\,\,\underset{n\rightarrow\omega}{\lim}\,\exists\mathfrak{Z}\left(\mathfrak{M}_{n}=\mathfrak{Z}\,\wedge\,\mathfrak{Z}\in\mathfrak{N}_{n}\,\wedge\,\phi_{n}\right)$
$\displaystyle\Longleftrightarrow\,\,\exists\mathfrak{Z}\left(\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}=\mathfrak{Z}\,\wedge\,\mathfrak{Z}\in\underset{n\rightarrow\omega}{\lim}\mathfrak{N}_{n}\,\wedge\,\underset{n\rightarrow\omega}{\lim}\phi_{n}\right)$
$\displaystyle\Longleftrightarrow\,\,\left(\exists\,\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\in\underset{n\rightarrow\omega}{\lim}\mathfrak{N}_{n}\right)\underset{n\rightarrow\omega}{\lim}\phi_{n}$

Lastly, we show that the theory of dense linear order without endpoints can be
obtained through the limit of formulas.

###### Corollary 2.28

Suppose $T$ is the theory of DLO without endpoints and
$\,\mathfrak{M}_{n}\,=\underset{1\leqslant
j<n}{\displaystyle\bigcup}\left(\mathbb{Z}+\frac{j}{n}\right)$. Then
$\underset{n\rightarrow\omega}{\lim}\,\underset{n<\omega}{\displaystyle\bigcup}\mathfrak{M}_{n}=\mathbb{Q}$
and $T=\,$Th$(\mathbb{Q})$.

Proof. Suppose $\varphi_{n},\phi_{n},\delta_{n}$ are sentences specifying the
properties of linear ordering, a dense subset and set without endpoints for
$\mathfrak{M}_{n}$. Then

$\displaystyle\varphi_{n}$ $\displaystyle\,\Longleftrightarrow\,\left(\forall
x,y,z\in\mathfrak{M}_{n}\right)\left(x\leqslant x\wedge\left(x\leqslant
y\wedge y\leqslant x\,\Longrightarrow x=y\right)\wedge\left(x\leqslant y\wedge
y\leqslant z\,\Longrightarrow x\leqslant z\right)\right)$
$\displaystyle\phi_{n}$ $\displaystyle\,\Longleftrightarrow\,\left(\forall
x,y\in\mathfrak{M}_{n}\right)\left(x<y\,\Longrightarrow\left(\exists
z\in\mathfrak{M}_{l}\right)\left(l>n\,\wedge\,x<z<y\right)\right)$
$\displaystyle\delta_{n}$ $\displaystyle\,\Longleftrightarrow\,\left(\forall
x\in\mathfrak{M}_{n}\right)\left(\left(\exists
y\in\mathfrak{M}_{n}\right)\left(y<x\right)\wedge\left(\exists
y\in\mathfrak{M}_{n}\right)\left(x<y\right)\right)$

For any $x,y\in\mathfrak{M}_{n}\left(x<y\right)$, set $N_{n}=2n$. Then
$\forall k>N_{n},\,\exists z\in\mathfrak{M}_{k}$ that $x<z<y$, i.e.
$\mathfrak{M}_{k}\models\phi_{n}$. Since
$\mathbb{Z}\models\varphi_{n}\wedge\delta_{n}$,
$\mathfrak{M}_{k}\models\varphi_{n}\,\wedge\,\phi_{n}\,\wedge\,\delta_{n}$. So
$\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}$ is unique. By
corollary 2.26

$\displaystyle\underset{n\rightarrow\omega}{\lim}\,\varphi_{n}$
$\displaystyle\,\Longleftrightarrow\left(\forall
x,y,z\in\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\right)\left(x\leqslant
x\wedge\left(x\leqslant y\wedge y\leqslant x\,\Longrightarrow
x=y\right)\wedge\left(x\leqslant y\wedge y\leqslant z\,\Longrightarrow
x\leqslant z\right)\right)$
$\displaystyle\underset{n\rightarrow\omega}{\lim}\phi_{n}$
$\displaystyle\,\Longleftrightarrow\left(\forall
x,y\in\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\right)\left(x<y\,\Longrightarrow\left(\exists
z\in\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\right)\left(x<z<y\right)\right)$
$\displaystyle\underset{n\rightarrow\omega}{\lim}\delta_{n}$
$\displaystyle\,\Longleftrightarrow\left(\forall
x\in\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\right)\left(\left(\exists
y\in\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\right)(y<x)\wedge\left(\exists
y\in\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\right)\left(x<y\right)\right)$

Since $T$ is $\aleph_{0}$-categorical and
$\underset{n\rightarrow\omega}{\lim}\,\varphi_{n},\underset{n\rightarrow\omega}{\lim}\phi_{n}$,
$\underset{n\rightarrow\omega}{\lim}\delta_{n}$ are axioms of
Th$(\mathbb{Q})$,
$\underset{n\rightarrow\omega}{\lim}\,\mathfrak{M}_{n}\,=\,\mathbb{Q}$ and
$T\,=\,$Th$(\mathbb{Q})$.

##  3 Non-Well-Founded Sets

###  3.1 Introduction

In this section, we will study infinitely generated sets and three of their
types known as infinitons, semi-infinitons and quasi-infinitons. First, let’s
review existing theories of non-well-founded sets.

The investigation of non-well-founded sets was initiated by Mirimanoff in 1917
[12], in which he formulated the distinction between well-founded and non-
well-founded sets. A number of axiomatic systems of non-well-founded sets have
been proposed thereafter. Since the Zermelo–Fraenkel set theory bans
$\in-\hskip 1.4457pt$sequences of infinite length by the axiom of
regularity101010This is actually fallacious because the axiom of regularity
can also hold for many non-well-founded sets. See section 5. (also known as
the axiom of foundation), most of these systems incorporate non-well-founded
sets by replacing the axiom of regularity with distinct anti-foundation axioms
and are essentially models of ZF minus the axiom of regularity. A notable
exception is New Foundations by Quine [13] that allows non-well-founded sets
without a specific axiom and avoids Russell’s paradox by permitting only
stratified formulas.

There are mainly four anti-foundation axioms by far — AFA (by Aczel, Forti and
Honsell [1]), SAFA (by Scott), FAFA (by Finsler), and BAFA (by Boffa). Each of
them defines a different notion of equality for non-well-founded sets. For
example, AFA bases hypersets (including non-well-founded sets) on accessible
pointed graphs (APG) that two hypersets are equal if and only if they can be
pictured by the same APG. In the universe of AFA, a Quine atom (called
infiniton in this paper) is shown to be existent and unique. The anti-
foundation axioms of AFA, SAFA, FAFA and BAFA specify an increasing sequence
of universes over the von Neumann universe, i.e. $V\subset A\subset S\subset
F\subset B$. In the universe of BAFA which is the largest of the four, the
Quine atoms form a proper class.

The main problem of the above axiomatic systems, however, is that they lack
precise mathematical descriptions for non-well-founded sets. For instance, a
non-well-founded set such as a Quine atom in AFA is an APG that can be
unfolded into an infinite tree. As we learn later, a tree with an infinite
branch is a countable structure that must be handled by the limit of finite
structures and formulas. Consequently, AFA only describes countable structures
intuitively and does not provide enough mathematical rigor for depicting their
structures and operations.

Furthermore, there have been efforts to introduce non-well-founded sets by
enlarging the von Neumann universe (through removing the axiom of regularity).
For examples, in [15], $V$ is modified through the iterative conception of a
set that includes some non-well-founded sets; in [20], $V$ is expanded through
the process of bisimulation. However, these attempts are inadequate because
they fail to identify exactly why $V$ needs be enlarged, as well as precisely
how the non-well-founded sets are generated.

In this paper, we will present a new way to generate non-well-founded sets by
enlarging the von Neumann universe along with the precise reason why $V$ needs
be enlarged as well as the exact process to generate these sets. First, we can
see that $V$ is incomplete because it does not have the limit ordinal ranks
(lemma 1.5). This fact is of fundamental importance because it implies that
non-well-founded sets are necessarily existent and should take on the limit
ordinal ranks in a complete universe of sets.111111More precisely, only
generators of non-well-founded sets known as infinitely generated sets take on
the rank of limit ordinals in the total universe. However, (many) non-well-
founded sets can have the rank of successor ordinals. Then non-well-founded
sets are added to $V$ as infinitely generated sets with limit ordinal ranks to
form the total universe (4.1). Furthermore, limits of finite structures and
formulas discussed in the previous section can provide rigorous analysis for
three types of infinitely generated sets as infinitons, semi-infinitons and
quasi-infinitons that appear in Russell’s paradox. Consequently, the total
universe is a model of ZF minus the axiom of regularity and free of Russell’s
paradox.

###  3.2 Infinitely Generated Set

An infinitely generated set is a generator of non-well-founded sets and
contains only one infinite branch. It is the limit of well-found sets known as
finitely generated sets. As discussed in a previous section, an infinitely
generated set is an infinite structure that is (generally) described by an
infinitely long formula of $\mathcal{L}_{\omega_{1},\omega_{1}}$ involving
countably many quantifiers. A $\mathcal{L}_{\omega_{1},\omega_{1}}$ formula
may be undecidable but is always decidable in a language of finitely many
quantifiers $\mathcal{L}_{\omega_{1},\omega}$ ([5] and [16]). First, we
introduce the notion of a finitely generated set.

###### Definition 3.1

Suppose $G_{k}\in V_{\omega}\,\left(0\leqslant k\leqslant n\right)$. A
finitely generated set is a finite $\mathcal{L}^{\prime}$-structure that is
defined as:

$H_{n}(G_{n},\cdots,G_{0})\,=\,\\{\ast G_{n},\\{\ast G_{n-1},\cdots\\{\ast
G_{1},G_{0}\\}\cdots\\}$ (3.1)

Where $G_{n}\left(1\leqslant k\leqslant n\right)$ are principal generators and
$G_{0}$ is a base generator of $H_{n}$. It can also be defined recursively as:

$H_{n}(G_{n},\cdots,G_{0})\,=\,\\{\ast G_{n},H_{n-1}(G_{n-1},\cdots,G_{0})\\}$

An infinitely generated set is defined as the limit (definition 2.9) of
finitely generated sets in $V_{\omega}$. In a later section, we will extend it
to well-founded sets of higher ranks in the total universe.

###### Definition 3.2

Suppose $H_{n}$ is defined in (3.1) and $\mathcal{G}=\\{G_{n}\colon G_{n}\in
V_{\omega},\,n<\omega\\}$. An infinitely generated set (IGS) (at $\omega$) is
defined as:

$H_{\omega}(\mathcal{G})\,=\,\underset{n\rightarrow\omega}{\lim}\,H_{n}(G_{n},\cdots,G_{0})$
(3.2)

Where $G_{n}\left(n\geqslant 1\right)$ are principal generators and $G_{0}$ is
a base generator of $H_{\omega}$.

###### Remark 3.3

Each IGS has one infinite branch. A non-well-founded set with multiple
infinite branches can be formed from IGS through power set operations. So IGS
are generators of the non-well-founded sets.

From definition 2.9, we can see that the limit of finitely generated sets
exists if $H_{n}$ is a homogeneous sequence in a $\aleph_{0}$-categorical
theory. As the result, it is essential to find out conditions for Th$(H_{n})$
to be $\aleph_{0}$-categorical and homogeneous. First, let’s review more
background knowledge in model theory.

A countable $\mathcal{L}$-structure $\mathfrak{M}$ is homogeneous if any
isomorphism between its two finite substructures can be extended to an
automorphism of $\mathfrak{M}$. A first-order theory $T$ has quantifier
elimination if, for every formula $\varphi(x_{1},\cdots,x_{n})$ there is a
quantifier-free formula $\phi(x_{1},\cdots,x_{n})$ such that $T\vdash\forall
x_{1}\cdots\forall
x_{n}(\varphi(\overline{x})\leftrightarrow\phi(\overline{x}))$ [10]. The
skeleton (or age) of a countable $\mathcal{L}$-structure $\mathfrak{M}$ is the
class of all finite $\mathcal{L}$-structures, each of which is isomorphic to a
substructure of $\mathfrak{M}$. An amalgamation class (or Fraisse class) is a
class of finite $\mathcal{L}$-structures with the hereditary, the joint
embedding and the amalgamation properties ([10]). We summarize the above in
the following propositions without proof.

###### Proposition 3.4

[10, Corollary 3.1.3] Let $\mathfrak{M}$ be a countable
$\mathcal{L}$-structure which is homogeneous in a finite relational language.
Then Th$(\mathfrak{M})$ is $\aleph_{0}$-categorical.

###### Proposition 3.5

[10, Proposition 3.1.6] Let $\mathfrak{M}$ be a $\aleph_{0}$-categorical
structure in a relational language. Then $\mathfrak{M}$ is homogeneous if and
only if Th$(\mathfrak{M})$ has quantifier elimination.

###### Proposition 3.6

A countable class of $\mathcal{L}$-structures is an amalgamation class if and
only if it is the skeleton of a countable homogeneous $\mathcal{L}$-structure
$\mathfrak{M}$. The amalgamation class is unique and is the Fraisse limit of
$\mathfrak{M}$.

Now we apply the above results to study the limit of $H_{n}$ in the language
of set theory $\mathcal{L}^{\prime}=\\{\in\\}$. First, we need the notion of
an amalgamation class in $V_{\omega}$.

###### Definition 3.7

An amalgamation class $\mathcal{K}$ of $V_{\omega}$ is a collection of
finitely generated sets in $V_{\omega}$ that satisfies the following
properties.

  1. (i)

(Heredity) If $H\in\mathcal{K}$, then any $J$ that is isomorphic to a finitely
generated subset of $H$ is in $\mathcal{K}$.

  2. (ii)

(Joint embedding) If $H_{1},H_{2}\in\mathcal{K}$, then there is a
$J\in\mathcal{K}$ and embeddings $f\colon H_{1}\to J$ and $g\colon H_{2}\to
J$.

  3. (iii)

(Amalgamation) If $H_{1},H_{2},J_{1}\in\mathcal{K}$ and embeddings
$f_{0}\colon H_{1}\to H_{2}$ and $f_{1}\colon H_{1}\to J_{1}$, then there is a
$J_{2}\in\mathcal{K}$ and embeddings $g_{0}\colon H_{2}\to J_{2}$ and
$g_{1}\colon J_{1}\to J_{2}$ with $g_{0}\circ f_{0}=g_{1}\circ f_{1}$.

###### Lemma 3.8

Suppose $H_{n}$ is defined in (3.1) and $\mathcal{G}=\\{G_{n}\colon G_{n}\in
V_{\omega},\,n<\omega\\}$. If $\\{H_{n}\colon n<\omega\\}$ is an amalgamation
class, then Th$(H_{n})$ is $\aleph_{0}$-categorical.

Proof. Since $H_{n}$ is the only (isomorphic) copy of finite
$\mathcal{L}^{\prime}$-structures (finitely generated sets) of rank $n$, its
skeleton is $\\{H_{n}\colon n<\omega\\}$. By proposition 3.6, $\\{H_{n}\\}$ is
homogeneous. So by proposition 3.4, Th$(H_{n})$ is $\aleph_{0}$-categorical.

###### Corollary 3.9

Suppose $H_{n}$ is an amalgamation class.

  1. (i)

If $H_{n}$ is a homogeneous sequence, then
$\underset{n\rightarrow\omega}{\lim}H_{n}$ is unique.

  2. (ii)

If $H_{n}$ consists only of finitely many homogeneous subsequences, then there
are finitely many sublimits for $\underset{n\rightarrow\omega}{\lim}H_{n}$.

Proof. By lemma 3.8, definition 2.9 and 2.10.

The axiom of extensionality for infinitely generated sets is as follows.

###### Axiom 3.10

Suppose $H_{\omega}(\mathcal{G}_{1})$ and $H_{\omega}(\mathcal{G}_{2})$ are
unique where $\,\mathcal{G}_{1}\,=\,\\{G_{n}^{1}\colon G_{n}^{1}\in
V_{\omega},\,n<\omega\\}$ and $\,\mathcal{G}_{2}\,=\,\\{G_{n}^{2}\colon
G_{n}^{2}\in V_{\omega},\,n<\omega\\}$. Then

$\left(\forall
n<\omega\right)\left(G_{n}^{1}=G_{n}^{2}\right)\,\Longrightarrow\,H_{\omega}(\mathcal{G}_{1})=H_{\omega}(\mathcal{G}_{2})$

The following concept is significant for the rest discussion.

###### Definition 3.11

Suppose $H_{\omega+\gamma}=\\{\ast G_{\omega+\gamma},\,H_{\omega+\gamma-1}\\}$
where $G_{\omega+\gamma}\in V_{\omega}\left(\gamma\geqslant 1\right)$. If for
any $\alpha>\omega$, there is a (successor ordinal) $\beta>\alpha$ that
$H_{\beta}=H_{\omega}$, then $H_{\omega}$ is called
$\mathbf{\omega-}$invariant.

###### Remark 3.12

Generally, an IGS does not have an immediate member, i.e. there is no $z\in
H_{\omega}$. The significance of $\omega-$invariance is that a
$\omega-$invariant set always has an immediate member. For example, if
$H_{\omega}$ is $\omega-$invariant, i.e. there is a $\beta>\omega$ that
$H_{\beta}=H_{\omega}$, then $H_{\beta-1}\in H_{\beta}=H_{\omega}$.

Next, we will study three types of infinitely generated sets that are
fundamental in Russell’s paradox.

###  3.3 Infiniton

An infiniton is a set that contains itself as the only member, i.e.
$I=\\{I\\}$, a fact that will be proved rigorously next.

###### Theorem 3.13

Suppose for each $n<\omega,$ $I_{n}$ is a finitely generated set and
$I_{n}=\\{I_{n-1}\\}$ with $I_{0}\,=\,G_{0}\in V_{\omega}.$ Then

  1. (i)

$I_{n}\,=\underset{n}{\,\,\underbrace{\\{\cdots\\{G_{0}\\}\cdots\\}}}$.

  2. (ii)

Th$(I_{n})$ is $\aleph_{0}$-categorical and has quantifier elimination.

  3. (iii)

$\underset{n\rightarrow\omega}{\lim}I_{n}$ is unique.

Proof. (i) follows easily by replacing $I_{n}$ recursively $n$ times.

(ii) For any $i,j<\omega$, suppose $I_{i}$ is mapped to $I_{j}$. Since this
can be extended to an automorphism of $I_{n}$, $\\{I_{n}\colon n<\omega\\}$ is
homogeneous. So by proposition 3.4 and 3.5, Th$(I_{n})$ is
$\aleph_{0}$-categorical and has quantifier elimination.

(iii) Suppose $\phi_{n}(S)$ means there exists a unique $\in-\hskip
1.4457pt$sequence of length $n$ in $S$, i.e.

$\phi_{n}(S)\,\Longleftrightarrow\,\left(\left(Y_{n}=S\right)\,\wedge\,\,\smashoperator[]{\bigwedge_{1\leqslant
j\leqslant n}^{}}\,\,\exists!\,Y_{j}\,\exists!\,Y_{j-1}\left(Y_{j-1}\in
Y_{j}\right)\right)$

Clearly, for any $n<\omega$, $I_{n}\models\phi_{n}$. Furthermore, for any
$n<\omega$ and any $k>n,\,I_{k}\models\phi_{n}$. So $I_{n}$ is a homogeneous
sequence. By (ii) and definition 2.9,
$\underset{n\rightarrow\omega}{\lim}I_{n}$ is unique.

###### Definition 3.14

In theorem 3.13, $\underset{n\rightarrow\omega}{\lim}I_{n}$ is known as the
infiniton generated by $G_{0}$ and is denoted as:

$\underset{n\rightarrow\omega}{\lim}I_{n}\,=\,I_{\omega}\,=\,\underset{\aleph_{0}}{\underbrace{\\{\cdots\\{G_{0}\\}\cdots\\}}}\,=\,\\{G_{0}\\}_{\mathcal{I}}$

Where $G_{0}$ is a base generator of $I_{\omega}$.

###### Definition 3.15

$S|_{\mathcal{I}}\,=\,\\{\\{G_{0}\\}_{\mathcal{I}}\colon G_{0}\in S\\}$ is
known as the set of the infinitons from $S$.

###### Theorem 3.16

Suppose for each $n<\omega,\,I_{n}=\\{I_{n-1}\\}$ and $I_{0}=G_{0}\in
V_{\omega}$. Then141414In the rest discussion, $I$ or $I_{\omega}$ is presumed
to be an infiniton generated from $G_{0}$ unless further specified.

  1. (i)

$I_{\omega}\,=\,\\{I_{\omega}\\}$.

  2. (ii)

$I_{\omega}$ is $\omega-$invariant.

  3. (iii)

A type of $I_{\omega}$ is that there exists a unique $\in-\hskip
1.4457pt$sequence of length $\omega$ in $I_{\omega}$.

  4. (iv)

$I_{\omega}$ is atomic and there is a complete formula for Th$(I_{\omega})$.

Proof. (i) By theorem 3.13, $I_{\omega}$ is unique. So by corollary 2.13 and
2.25

$I_{\omega}\,=\underset{n\rightarrow\omega}{\lim}I_{n}\,=\underset{n\rightarrow\omega}{\lim}\\{I_{n-1}\\}\,=\,\\{\underset{n\rightarrow\omega}{\lim}I_{n-1}\\}\,=\,\\{I_{\omega}\\}$

(ii) Obviously, for any $\alpha$ that $\omega<\alpha<\omega 2$

$I_{\alpha}\,=\,\\{I_{\alpha-1}\\}\,=\cdots=\,\underset{\alpha-\omega}{\underbrace{\\{\cdots\\{I_{\omega}\\}\cdots\\}}}\,=\,I_{\omega}$

Then it follows by transfinite induction that for any
$\alpha,\,I_{\alpha}\,=\,I_{\omega}$.

(iii) Let $\gamma_{n}=\left\langle I_{j}\colon j\leqslant n\right\rangle$ be
the unique $\in-\hskip 1.4457pt$sequence of length $n$ in $I_{n}$. Then
$I_{n}$ satisfies

$\varphi_{n}\,\Longleftrightarrow\,\smashoperator[]{\bigwedge_{1\leqslant
j\leqslant n}^{}}\,\exists!\,I_{j}\,\exists!\,I_{j-1}\left(I_{j-1}\in
I_{j}\right)\,\Longleftrightarrow\,\,\smashoperator[]{\bigwedge_{1\leqslant
j\leqslant n-1}^{}}\,\,\exists!\,I_{j}\,\exists!\,I_{j-1}\left(I_{j-1}\in
I_{j}\right)\,\wedge\,\exists!\,I_{n}\,\exists!\,I_{n-1}\left(I_{n-1}\in
I_{n}\right)$

Since $\underset{n\rightarrow\omega}{\lim}I_{n}\,=\,I_{\omega}$, by axiom
2.15, corollary 2.14 and 2.27

$\varphi_{\omega}\,\Longleftrightarrow\underset{n\rightarrow\omega}{\lim}\,\varphi_{n}\,\Longleftrightarrow\underset{n<\omega}{\displaystyle\bigwedge}\exists!\,I_{n}\,\exists!\,I_{n-1}\left(I_{n-1}\in
I_{n}\right)\,\wedge\,\exists!\,I_{\omega}\left(I_{\omega}\in
I_{\omega}\right)$

By theorem 2.7, $I_{\omega}\models\varphi_{\omega}$ where $\varphi_{\omega}$
describes $\gamma_{\omega}=\left\langle I_{n}\colon
n\leqslant\omega\right\rangle$, the unique $\in-\hskip 1.4457pt$sequence of
length $\omega$ in $I_{\omega}$.

(iv) Since Th$(I_{\omega})$ is $\aleph_{0}$-categorical, by proposition 2.3,
$I_{\omega}$ is atomic and $\varphi_{\omega}$ is a complete formula since any
tuple of $I_{0},\cdots,I_{n}$ satisfies $\varphi_{\omega}$. Furthermore, any
(finite) type of Th$(I_{\omega})$ can be derived from $\varphi_{\omega}.$

###### Corollary 3.17

Suppose
$I_{0}^{\prime}\,=\underset{k}{\,\underbrace{\\{\cdots\\{G_{0}\\}\cdots\\}}}$
and $I_{n}^{\prime}=\\{I_{n-1}^{\prime}\\}$ for each $n<\omega$. Then
$\underset{n\rightarrow\omega}{\lim}I_{n}^{\prime}\,=\underset{n\rightarrow\omega}{\lim}I_{n}$.

Proof. Since
$I_{n}\,=\underset{n}{\,\underbrace{\\{\cdots\\{G_{0}\\}\cdots\\}}}$,
$I_{n}^{\prime}=I_{n+k}$. So it follows by corollary 2.13.

###### Remark 3.18

Corollary 3.17 shows that different base generators could generate the same
infiniton. By choosing the one with the least rank, an infiniton is unique to
its base generator.

###### Definition 3.19

The base generator $G_{0}$ of an infiniton $I_{\omega}$ is the one with the
least rank (in $V_{\omega}$).

###### Corollary 3.20

$2$ infinitons are identical if and only if their base generators are the
same.

Proof. Suppose
$I_{n}\,=\underset{n}{\,\underbrace{\\{\cdots\\{G_{0}\\}\cdots\\}}}$ and
$I_{n}^{\prime}\,=\underset{n}{\,\underbrace{\\{\cdots\\{G_{0}^{\prime}\\}\cdots\\}}}$.
If $G_{0}=G_{0}^{\prime}$, then $I_{n}=I_{n}^{\prime}$ for any $n<\omega$. So
by corollary 2.13, $I_{\omega}=I_{\omega}^{\prime}$. Conversely, by theorem
3.16, if $I_{\omega}=I_{\omega}^{\prime}$, then $I_{\omega}$ and
$I_{\omega}^{\prime}$ have the same $\in-\hskip 1.4457pt$sequence of length
$\omega$. Thus by definition 3.19, $G_{0}=G_{0}^{\prime}$.

###### Corollary 3.21

Suppose $I$ is an infiniton and $S|_{\mathcal{I}}$ a set of infinitons.

  1. (i)

$I\neq\varnothing$.

  2. (ii)

$D(I)\,=\,D(S|_{\mathcal{I}})\,=\,\aleph_{0}$.

  3. (iii)

Any infiniton and set of infinitons are TNWF.

  4. (iv)

$I\notin V$.

Proof. (i) Since $I\in I$ and
$\varnothing\notin\varnothing,\,I\neq\varnothing$.

(ii) If $D(I)<\aleph_{0}$, then by (1.2), $D(I)<D\left(\\{I\\}\right)=D(I)$,
contradiction. And by definition 3.15, $D(S|_{\mathcal{I}})\geqslant
D(I)\,=\,\aleph_{0}$.

(iii) By definition 1.1, 3.14 and 3.15.

(iv) By lemma 1.2 and (iii).

The tree structures for infinitons and a set of infinitons are shown in Figure
1. Intuitively, any infiniton consists of one infinite (broken) branch, and
all branches of a set of infinitons are infinite.

Figure 1: Diagrams of infinitons and a set of infinitons.

###### Theorem 3.22

  1. (i)

$S_{1}\subset S_{2}\,\Longrightarrow\,S_{1}|_{\mathcal{I}}\subset
S_{2}|_{\mathcal{I}}$

  2. (ii)

$\left(S_{1}\cup S_{2}\right)|_{\mathcal{I}}\,=\,S_{1}|_{\mathcal{I}}\cup
S_{2}|_{\mathcal{I}}$

  3. (iii)

$\left(S_{1}\cap S_{2}\right)|_{\mathcal{I}}\,\subset S_{1}|_{\mathcal{I}}\cap
S_{2}|_{\mathcal{I}}$

  4. (iv)

$S_{1}|_{\mathcal{I}}-S_{2}|_{\mathcal{I}}\,\subset\left(S_{1}-S_{2}\right)|_{\mathcal{I}}$

  5. (v)

$\left.\left(\underset{\alpha\in
D}{\displaystyle\bigcup}S_{\alpha}\right)\right|_{\mathcal{I}}=\underset{\alpha\in
D}{\displaystyle\bigcup}\left.S_{\alpha}\right|_{\mathcal{I}}$

Proof. (i) is obvious.

(ii) $``\supset\textquotedblright$ follows from (i) for $\left(S_{1}\cup
S_{2}\right)|_{\mathcal{I}}\supset S_{1}|_{\mathcal{I}}$ and $\left(S_{1}\cup
S_{2}\right)|_{\mathcal{I}}\supset S_{2}|_{\mathcal{I}}$. For any
$\\{G_{0}\\}|_{\mathcal{I}}\in\left(S_{1}\cup S_{2}\right)|_{\mathcal{I}}$,
$G_{0}\in S_{1}\cup S_{2}$. So $\\{G_{0}\\}|_{\mathcal{I}}\in
S_{1}|_{\mathcal{I}}$ or $\\{G_{0}\\}|_{\mathcal{I}}\in S_{2}|_{\mathcal{I}}$.
This proves $``\subset\textquotedblright$.

(iii) By (i).

(iv) For any $\\{G_{0}\\}|_{\mathcal{I}}\in
S_{1}|_{\mathcal{I}}-S_{2}|_{\mathcal{I}}$, $G_{0}\in S_{1}$ and $G_{0}\notin
S_{2}$. So
$\\{G_{0}\\}|_{\mathcal{I}}\in\left(S_{1}-S_{2}\right)|_{\mathcal{I}}$.

(v) By (i), for any $\alpha\in D$

$\left.S_{\alpha}\right|_{\mathcal{I}}\,\subset\left.\left(\underset{\alpha\in
D}{\displaystyle\bigcup}S_{\alpha}\right)\right|_{\mathcal{I}}\quad\text{and
so}\quad\underset{\alpha\in
D}{\displaystyle\bigcup}\left.S_{\alpha}\right|_{\mathcal{I}}\,\subset\left.\left(\underset{\alpha\in
D}{\displaystyle\bigcup}S_{\alpha}\right)\right|_{\mathcal{I}}$

For any $\\{G_{0}\\}|_{\mathcal{I}}\,\in\left.\left(\underset{\alpha\in
D}{\displaystyle\bigcup}S_{\alpha}\right)\right|_{\mathcal{I}}$, there is a
$\beta\in D$ that $G_{0}\in S_{\beta}$. So

$\\{G_{0}\\}|_{\mathcal{I}}\in\left.S_{\beta}\right|_{\mathcal{I}}\,\subset\underset{\alpha\in
D}{\displaystyle\bigcup}\left.S_{\alpha}\right|_{\mathcal{I}}$

Thus (v) follows.

###### Corollary 3.23

The axiom of regularity fails for any infiniton and set of infinitons.

Proof. Suppose $I=\\{I\\}$ and $S|_{\mathcal{I}}=\\{I_{k}\colon
I_{k}=\\{I_{k}\\}\,\wedge\,k\in\mathbb{N}\\}$. AR fails for $I$ since $I\in
I$. For each $I_{k}\in S|_{\mathcal{I}}$, $I_{k}\cap
S|_{\mathcal{I}}\,=\,I_{k}$ for $I_{k}\in I_{k}$. Since no $y\in
S|_{\mathcal{I}}$ satisfies $y\cap S|_{\mathcal{I}}=\varnothing$, AR fails for
$S|_{\mathcal{I}}$ too.

###### Corollary 3.24

Any set of infinitons is not a member of itself.

Proof. Let $S=\\{I_{1},I_{2},\cdots\\}$ where $I_{k}$ are distinct infinitons.
If $S\in S$, then $S=I_{j}=\\{I_{j}\\}$. This leads to
$I_{1}=I_{2}=\cdots=I_{j}$, contradiction.

Corollary 3.24 can be extended to any set like
$\\{\\{\\{I_{1},I_{2}\\},I_{3}\\},I_{4}\\},$ i.e. all sets of infinitons, all
subset of sets of infinitons and so on (corollary 4.36).

###  3.4 Semi-Infiniton

A semi-infiniton is a set that is a member of itself, i.e. $X\in X$. From
(1.4) and (3.2), we can see that an infinitely generated set with only one
principal generator is a semi-infiniton.

###### Theorem 3.25

Suppose for each $n<\omega,\,Z_{n}$ is a finitely generated set and
$Z_{n}=\\{\ast G,Z_{n-1}\\}$ with $Z_{0}\,=\,G_{0}\in V_{\omega}$ and $G\in
V_{\omega}$. Then

  1. (i)

$Z_{n}\,=\,\underset{n}{\underbrace{\\{\ast G,\\{\ast G,\cdots\\{\ast
G,G_{0}\\}\cdots\\}}}$.

  2. (ii)

Th$(Z_{n})$ is $\aleph_{0}$-categorical and has quantifier elimination.

  3. (iii)

$\underset{n\rightarrow\omega}{\lim}Z_{n}$ is unique.

Proof. (i) follows easily by replacing $Z_{n}$ recursively $n$ times.

(ii) For any $i,j<\omega$, suppose $Z_{i}$ is mapped to $Z_{j}$. Since this
can be extended to an automorphism of $Z_{n}$, $\\{Z_{n}\colon n<\omega\\}$ is
homogeneous. So by proposition 3.4 and 3.5, Th$(Z_{n})$ is
$\aleph_{0}$-categorical and has quantifier elimination.

(iii) Suppose $\phi_{n}(S)$ means there exists a unique $\in-\hskip
1.4457pt$sequence of length $n$ in $S$ and with each sublevel of $S$
containing the members of $G$, i.e.

$\phi_{n}(S)\,\Longleftrightarrow\,\left(\left(Y_{n}=S\right)\,\wedge\,\,\smashoperator[]{\bigwedge_{1\leqslant
j\leqslant n}^{}}\,\,\exists!\,Y_{j}\left(\exists!\,Y_{j-1}\left(Y_{j-1}\in
Y_{j}\right)\,\wedge\,\left(\forall z\in G\right)\left(z\in
Y_{j}\right)\right)\right)$

Clearly, for any $n<\omega$, $Z_{n}\models\phi_{n}$. Furthermore, for any
$n<\omega$ and any $k>n,\,Z_{k}\models\phi_{n}$. So $Z_{n}$ is a homogeneous
sequence. By (ii) and definition 2.9,
$\underset{n\rightarrow\omega}{\lim}Z_{n}$ is unique.

###### Definition 3.26

In theorem 3.25, $\underset{n\rightarrow\omega}{\lim}Z_{n}$ is known as the
semi-infiniton generated by $G\,(G\neq\varnothing)$ and $G_{0}$, and is
denoted as:

$\underset{n\rightarrow\omega}{\lim}Z_{n}\,=\,Z_{\omega}\,=\,\underset{\aleph_{0}}{\underbrace{\\{\cdots\\{\ast
G,\\{\ast G,G_{0}\\}\cdots\\}}}\,=\,\\{G,G_{0}\\}|_{\mathfrak{S}}$

Where $G$ is the principal generator and $G_{0}$ is a base generator of
$Z_{\omega}$.

###### Theorem 3.27

Suppose for each $n<\omega,\,Z_{n}\,=\,\\{\ast G,Z_{n-1}\\}$ with
$Z_{0}\,=\,G_{0}\in V_{\omega}$ and $G\in V_{\omega}$. Then151515In the rest
discussion, $Z_{\omega}$ is presumed to be a semi-infiniton generated from $G$
and $G_{0}$ unless further specified.

  1. (i)

$Z_{\omega}\,=\,\\{\ast G,Z_{\omega}\\}$.

  2. (ii)

$Z_{\omega}$ is $\omega-$invariant.

  3. (iii)

A type of $Z_{\omega}$ is that there exists a unique $\in-\hskip
1.4457pt$sequence of length $\omega$ in $Z_{\omega}$ with each sublevel
$n\,(n\leqslant\omega)$ of $Z_{\omega}$ containing the members of $G$.

  4. (iv)

$Z_{\omega}$ is atomic and there is a complete formula for Th$(Z_{\omega})$.

Proof. (i) By corollary 2.13, 2.25 and theorem 2.22

$\displaystyle Z_{\omega}\>$
$\displaystyle=\>\underset{n\rightarrow\omega}{\lim}\\{\ast G,Z_{n-1}\\}$
$\displaystyle=\>\underset{n\rightarrow\omega}{\lim}\left(G\,\cup\\{Z_{n-1}\\}\right)$
$\displaystyle=\>G\,\cup\\{\underset{n\rightarrow\omega}{\lim}Z_{n-1}\\}$
$\displaystyle=\>\\{\ast G,Z_{\omega}\\}$

(ii) Obviously, for any $\alpha$ that $\omega<\alpha<\omega 2$

$Z_{\alpha}\,=\,\\{\ast
G,Z_{\alpha-1}\\}\,=\cdots=\,\underset{\alpha-\omega}{\underbrace{\\{\ast
G,\cdots\\{\ast G,\\{\ast G,Z_{\omega}\\}\\}\cdots\\}}}\,=\,Z_{\omega}$

Then it follows by transfinite induction that for any
$\alpha,\,Z_{\alpha}\,=\,Z_{\omega}$. Also for any $z\in G,\,z\in Z_{\omega}$.

(iii) Let $\gamma_{n}=\left\langle Z_{j}\colon j\leqslant n\right\rangle$ be
the unique $\in-\hskip 1.4457pt$sequence of length $n$ in $Z_{n}$ with each
sublevel $j$ $(j<n)$ of $Z_{n}$ containing the members of $G$. Then $Z_{n}$
satisfies

$\delta_{n}\,\Longleftrightarrow\,\,\smashoperator[]{\bigwedge_{1\leqslant
j\leqslant n}^{}}\,\,\exists!\,Z_{j}\left(\exists!\,Z_{j-1}\left(Z_{j-1}\in
Z_{j}\right)\,\wedge\,\left(\forall z\in G\right)\left(z\in
Z_{j}\right)\right)$

Since $\underset{n\rightarrow\omega}{\lim}Z_{n}\,=\,Z_{\omega}$, by axiom
2.15, corollary 2.14 and 2.27

$\displaystyle\delta_{\omega}\>$
$\displaystyle\Longleftrightarrow\>\underset{n\rightarrow\omega}{\lim}\underset{1\leqslant
j\leqslant
n}{\displaystyle\bigwedge}\exists!\,Z_{j}\left(\exists!\,Z_{j-1}\left(Z_{j-1}\in
Z_{j}\right)\,\wedge\,\left(\forall z\in G\right)\left(z\in
Z_{j}\right)\right)$
$\displaystyle\Longleftrightarrow\>\underset{n\rightarrow\omega}{\lim}\underset{1\leqslant
j\leqslant
n-1}{\displaystyle\bigwedge}\exists!\,Z_{j}\left(\exists!\,Z_{j-1}\left(Z_{j-1}\in
Z_{j}\right)\,\wedge\,\left(\forall z\in G\right)\left(z\in
Z_{j}\right)\right)\,\wedge\vskip 6.0pt plus 2.0pt minus 2.0pt$
$\displaystyle\>\qquad\underset{n\rightarrow\omega}{\lim}\,\exists!\,Z_{n}\left(\exists!\,Z_{n-1}\left(Z_{n-1}\in
Z_{n}\right)\,\wedge\,\left(\forall z\in G\right)\left(z\in
Z_{n}\right)\right)\vskip 6.0pt plus 2.0pt minus 2.0pt$
$\displaystyle\Longleftrightarrow\>\underset{n<\omega}{\displaystyle\bigwedge}\exists!\,Z_{n}\left(\exists!\,Z_{n-1}\left(Z_{n-1}\in
Z_{n}\right)\,\wedge\,\left(\forall z\in G\right)\left(z\in
Z_{n}\right)\right)\,\wedge\,\vskip 6.0pt plus 2.0pt minus 2.0pt$
$\displaystyle\>\qquad\exists!\,Z_{\omega}\left(\left(Z_{\omega}\in
Z_{\omega}\right)\,\wedge\,\left(\forall z\in G\right)\left(z\in
Z_{\omega}\right)\right)$

By theorem 2.7, $Z_{\omega}\models\delta_{\omega}$ where $\delta_{\omega}$
describes $\gamma_{\omega}=\left\langle Z_{n}\colon
n\leqslant\omega\right\rangle$, the unique $\in-\hskip 1.4457pt$sequence of
length $\omega$ in $Z_{\omega}$ with each sublevel $n$ $(n\leqslant\omega)$ of
$Z_{\omega}$ containing the members of $G$.

(iv) Since Th$(Z_{\omega})$ is $\aleph_{0}$-categorical, by proposition 2.3,
$Z_{\omega}$ is atomic and $\delta_{\omega}$ is a complete formula since any
tuple of $Z_{0},\cdots,Z_{n}$ satisfies $\delta_{\omega}$. Furthermore, any
(finite) type of Th$(Z_{\omega})$ can be derived from $\delta_{\omega}$.

Now we give another condition for an IGS to be a semi-infiniton.

###### Corollary 3.28

Suppose $H_{n}$ is a finitely generated set (3.1). If
$\underset{n\rightarrow\omega}{\lim}H_{n}=H_{\omega}$ and
$\underset{n\rightarrow\omega}{\lim}G_{n}=G_{\omega}$, then

  1. (i)

$H_{\omega}\,=\,\\{\ast G_{\omega},H_{\omega}\\}$.

  2. (ii)

$H_{\omega}$ is $\omega-$invariant.

Proof. (i) By corollary 2.25 and theorem 2.22

$\displaystyle H_{\omega}\>$
$\displaystyle=\>\underset{n\rightarrow\omega}{\lim}\\{\ast G_{n},H_{n-1}\\}$
$\displaystyle=\>\underset{n\rightarrow\omega}{\lim}\left(G_{n}\cup\\{H_{n-1}\\}\right)$
$\displaystyle=\>\underset{n\rightarrow\omega}{\lim}G_{n}\cup\\{\underset{n\rightarrow\omega}{\lim}H_{n-1}\\}$
$\displaystyle=\>\\{\ast G_{\omega},H_{\omega}\\}$

(ii) Obviously, for any $\alpha$ that $\omega<\alpha<\omega 2$

$H_{\alpha}\,=\,\\{\ast
G_{\omega},H_{\alpha-1}\\}\,=\cdots=\,\underset{\alpha-\omega}{\underbrace{\\{\ast
G_{\omega},\cdots\\{\ast G_{\omega},\\{\ast
G_{\omega},H_{\omega}\\}\\}\cdots\\}}}\,=\,H_{\omega}$

Then it follows by transfinite induction that for any $\alpha$,
$H_{\alpha}=H_{\omega}$. In addition, for any $z\in G_{\omega}$, $z\in
H_{\omega}$.

The tree structure for a semi-infiniton is shown in Figure 2. Intuitively, a
semi-infiniton has one infinite (broken) branch.

Figure 2: Diagram of a semi-infiniton.

###### Corollary 3.29

Suppose $Z$ is a semi-infiniton.

  1. (i)

$Z\neq\varnothing$

  2. (ii)

$Z\neq\\{Z\\}$

  3. (iii)

$D(Z)\,=\,\aleph_{0}$

  4. (iv)

$Z\notin V$

Proof. (i) Since $\varnothing\notin\varnothing,\,Z\neq\varnothing$.

(ii) If $Z=\\{Z\\},\,Z$ is an infiniton, contradicting definition 3.26.

(iii) If $D(Z)<\aleph_{0}$, then by (1.2), $D(Z)<D(\\{Z\\})=D(Z)$,
contradiction.

(iv) Since $Z$ has an infinite branch, it is NWF. By lemma 1.2, $Z\notin V$.

###### Definition 3.30

$\ S|_{\mathfrak{S}}=\\{\\{G,G_{0}\\}|_{\mathfrak{S}}\colon G_{0},G\in S\\}$
is known as the set of the semi-infinitons from $S$.

###### Corollary 3.31

  1. (i)

$S_{1}\subset S_{2}\,\Longrightarrow\,S_{1}|_{\mathfrak{S}}\subset
S_{2}|_{\mathfrak{S}}$

  2. (ii)

$\left(S_{1}\cup
S_{2}\right)|_{\mathfrak{S}}\,\supset\,S_{1}|_{\mathfrak{S}}\cup
S_{2}|_{\mathfrak{S}}$

  3. (iii)

$\left(S_{1}\cap
S_{2}\right)|_{\mathfrak{S}}\,\subset\,S_{1}|_{\mathfrak{S}}\cap
S_{2}|_{\mathfrak{S}}$

  4. (iv)

$\left.\left(\underset{\alpha\in
D}{\displaystyle\bigcup}S_{\alpha}\right)\right|_{\mathfrak{S}}\supset\underset{\alpha\in
D}{\displaystyle\bigcup}\left.S_{\alpha}\right|_{\mathfrak{S}}$

  5. (v)

$S_{\alpha}\uparrow\,\Longrightarrow\,\left.\left(\underset{\alpha\in
D}{\displaystyle\bigcup}S_{\alpha}\right)\right|_{\mathfrak{S}}=\underset{\alpha\in
D}{\displaystyle\bigcup}\left.S_{\alpha}\right|_{\mathfrak{S}}$

Proof. (i) is obvious. (ii), (iii) and (iv) follow from (i).

(v) For any $\\{G,G_{0}\\}|_{\mathfrak{S}}\in\left.\left(\underset{\alpha\in
D}{\displaystyle\bigcup}S_{\alpha}\right)\right|_{\mathfrak{S}}$, there are
$\gamma_{1},\gamma_{2}\in D$ that $G\in S_{\gamma_{1}}$ and $G_{0}\in
S_{\gamma_{2}}$. Let $\gamma=\max\\{\gamma_{1},\gamma_{2}\\}$. Then
$G,G_{0}\in S_{\gamma}$ and
$\\{G,G_{0}\\}|_{\mathfrak{S}}\in\left.S_{\gamma}\right|_{\mathfrak{S}}\subset\underset{\alpha\in
D}{\displaystyle\bigcup}\left.S_{\alpha}\right|_{\mathfrak{S}}$. This proves
$``\subset\textquotedblright$ and it follows by (iv).

The following shows that an infiniton is a special case of a semi-infiniton.

###### Corollary 3.32

  1. (i)

$\\{G_{0}\\}_{\mathcal{I}}\,=\,\\{\varnothing,G_{0}\\}|_{\mathfrak{S}}$

  2. (ii)

$\left.S\right|_{\mathcal{I}}\,\subset\left.S\right|_{\mathfrak{S}}$

Proof. (i) Suppose for each $n<\omega,\,I_{n}=\\{I_{n-1}\\}$,
$Z_{n}=\\{\ast\varnothing,Z_{n-1}\\}$, and $I_{0}=Z_{0}=G_{0}\in V_{\omega}$.
By lemma 1.14, $I_{n}=Z_{n}$ for all $n$. So by corollary 2.13,
$\underset{n\rightarrow\omega}{\lim}I_{n}\,=\,\underset{n\rightarrow\omega}{\lim}Z_{n}$,
or $\\{G_{0}\\}_{\mathcal{I}}\,=\,\\{\varnothing,G_{0}\\}|_{\mathfrak{S}}$.

(ii) follows from (i).

A semi-infiniton is also related to an infiniton in the following result.

###### Corollary 3.33

$\qquad Z\in Z\,\wedge\,Z$ is transitive $\,\Longleftrightarrow\,Z=\\{Z\\}$

Proof. If $Z=\\{Z\\}$, then $Z\in\\{Z\\}=Z$. Also, $Z\in Z$ means
$\\{Z\\}\subset Z$. So $Z\subset Z$, i.e. $Z$ is transitive. Conversely, $Z\in
Z$ means $\\{Z\\}\subset Z$. Since $Z$ is transitive, $Z\in\\{Z\\}$ means
$Z\subset\\{Z\\}$. So $Z=\\{Z\\}$.

###  3.5 Quasi-Infiniton

A quasi-infiniton is a set that contains a vicious cycle, i.e. $Q\in
X_{1},\,X_{1}\in X_{2}$, $\cdots,\,X_{n-1}\in Q$. From (1.5) and (3.2), we can
see that an infinitely generated set whose principal generators form a finite
cycle is a quasi-infiniton.

###### Theorem 3.34

Suppose for each $n<\omega,\,Q_{n}$ is a finitely generated set and
$Q_{n}=\\{\ast G_{n},Q_{n-1}\\}$ with $Q_{0}\,=\,G_{0}\in V_{\omega}$, and its
principle generators $G_{n}$ form a finite cycle, i.e. there are only $l$
$\left(l>1\right)$ distinct $G_{j}\in V_{\omega}$ $\left(1\leqslant j\leqslant
l\right)$ and $|G_{1}|<|G_{2}|<\cdots<|G_{l}|$. Then there are $l$ sublimits
for $Q_{n}$ which are decided by formulas as:

$\phi_{n}(S)\,\Longleftrightarrow\,\left(\left(Y_{pl+q}=S\right)\,\,\,\,\smashoperator[]{\bigwedge_{q\leqslant
j\leqslant
q+l-1}^{}}\,\,\,\,\exists!\,Y_{\left(p-1\right)l+j+1}\left(\begin{array}[c]{c}\exists!\,Y_{\left(p-1\right)l+j}\left(Y_{\left(p-1\right)l+j}\in
Y_{\left(p-1\right)l+j+1}\right)\\\\[5.16663pt] \wedge\,\left(\forall z\in
G_{j+1}\right)\left(z\in
Y_{\left(p-1\right)l+j+1}\right)\end{array}\right)\right)$

Where $n=pl+q$ $\left(p,l,q<\omega,\,0\leqslant q<l\right)$, $G_{j+1}=G_{r}$,
$j+1\equiv r$ $mod$ $l$ for $r\neq 0$, and $j+1=l$ for $r=0$. The $l$
sublimits of $Q_{n}$ are distinct and are as follows $\left(0\leqslant
q<l\right)$:

$Q_{\omega,q}\,=\,\\{\ast G_{l+q},\\{\ast G_{l+q-1},\cdots\\{\ast
G_{q+1},Q_{\omega,q}\\}\cdots\\}$

Proof. First, we prove the case of $q=0$. Clearly

$\displaystyle Q_{l}\,$ $\displaystyle=\,\\{\ast G_{l},\\{\ast
G_{l-1},\cdots\\{\ast G_{1},G_{0}\\}\cdots\\}$ $\displaystyle Q_{2l}\,$
$\displaystyle=\,\\{\ast G_{l},\\{\ast G_{l-1},\cdots\\{\ast
G_{1},Q_{l}\\}\cdots\\}$ $\displaystyle\,\vdots$ $\displaystyle Q_{pl}\,$
$\displaystyle=\,\\{\ast G_{l},\\{\ast G_{l-1},\cdots\\{\ast
G_{1},Q_{\left(p-1\right)l}\\}\cdots\\}$

Since for any $i,j<\omega$, the mapping from $Q_{il}$ to $Q_{jl}$ can be
extended to an automorphism, $Q_{pl}$ is homogeneous. So by proposition 3.4,
Th$(Q_{pl})$ is $\aleph_{0}$-categorical. In addition, $Q_{pl}$ is satisfied
by the following formula $\left(n=pl\right)$:

$\phi_{n}(S)\,\Longleftrightarrow\,\left(\left(Y_{pl}=S\right)\,\,\,\smashoperator[]{\bigwedge_{0\leqslant
j\leqslant
l-1}^{}}\,\,\,\exists!\,Y_{\left(p-1\right)l+j+1}\left(\begin{array}[c]{c}\exists!\,Y_{\left(p-1\right)l+j}\left(Y_{\left(p-1\right)l+j}\in
Y_{\left(p-1\right)l+j+1}\right)\\\\[5.16663pt] \wedge\,\left(\forall z\in
G_{j+1}\right)\left(z\in
Y_{\left(p-1\right)l+j+1}\right)\end{array}\right)\right)$

Since for any $p<\omega$ and any $k>p$, $Q_{kl}\models\phi_{pl}$, $Q_{pl}$ is
a homogeneous subsequence of $Q_{n}$. So by definition 2.9,
$\underset{p\rightarrow\omega}{\lim}Q_{pl}$ is unique. Consequently, by
corollary 2.13, 2.25 and theorem 2.22

$\displaystyle Q_{\omega,0}\>$
$\displaystyle=\>\underset{p\rightarrow\omega}{\lim}\\{\ast G_{l},\\{\ast
G_{l-1},\cdots\\{\ast G_{1},Q_{\left(p-1\right)l}\\}\cdots\\}$
$\displaystyle=\>\\{\ast G_{l},\\{\ast G_{l-1},\cdots\\{\ast
G_{1},\underset{p\rightarrow\omega}{\lim}Q_{\left(p-1\right)l}\\}\cdots\\}$
$\displaystyle=\>\\{\ast G_{l},\\{\ast G_{l-1},\cdots\\{\ast
G_{1},Q_{\omega,0}\\}\cdots\\}$

For $q=1$

$\displaystyle Q_{l+1}\,$ $\displaystyle=\,\\{\ast G_{1},\\{\ast
G_{l},\cdots\\{\ast G_{2},Q_{1}\\}\cdots\\}$ $\displaystyle\>\vdots$
$\displaystyle Q_{pl+1}\,$ $\displaystyle=\,\\{\ast G_{1},\\{\ast
G_{l},\cdots\\{\ast G_{2},Q_{\left(p-1\right)l+1}\\}\cdots\\}$

So the subsequence $Q_{pl+1}$ of $Q_{n}$ has the unique limit as:

$Q_{\omega,1}\,=\,\underset{p\rightarrow\omega}{\lim}Q_{pl+1}\,=\,\\{\ast
G_{1},\\{\ast G_{l},\cdots\\{\ast G_{2},Q_{\omega,1}\\}\cdots\\}$

Since $G_{1}\neq G_{l}$, $Q_{\omega,0}\neq Q_{\omega,1}$. The above proof can
easily be extended to any $Q_{pl+q}$. Thus there are $l$ distinct sublimits
for $Q_{n}$ in total.

###### Definition 3.35

Suppose for each $n<\omega$, $Q_{n}=\\{\ast G_{n},Q_{n-1}\\}$ with
$Q_{0}=G_{0}\in V_{\omega}$, whose principle generators form a finite cycle,
i.e. there are only $l$ $\left(l>1\right)$ distinct $G_{j}\in V_{\omega}$
$\left(1\leqslant j\leqslant l\right)$, and $|G_{1}|<|G_{2}|<\cdots<|G_{l}|$.
By theorem 3.34, $\underset{n\rightarrow\omega}{\lim}Q_{n}$ has $l$ distinct
values, each of which is known as a quasi-infiniton generated by
$\mathcal{G}=\\{G_{k}\colon G_{k}\in V_{\omega}$, $0\leqslant k\leqslant l$,
$|G_{1}|<\cdots<|G_{l}|\,\\}$ and is denoted as $\left(0\leqslant q<l\right)$:

$Q_{\omega,q}\,=\,\\{\ast G_{l+q},\\{\ast G_{l+q-1},\cdots\\{\ast
G_{q+1},Q_{\omega,q}\\}\cdots\\}\,=\,\\{G_{k},l\\}|_{\mathfrak{Q}}$

Where for $1\leqslant j\leqslant l$, $G_{l+j}\,=\,G_{j}$. The collection of
the limits of $Q_{n}$ is denoted as:

$\underset{n\rightarrow\omega}{\lim}Q_{n}\,=\,Q_{\omega}\,=\,\\{Q_{\omega,q}\colon
0\leqslant q<l\\}$

Where $G_{k}\,\left(1\leqslant k\leqslant l\right)$ are principal generators
and $G_{0}$ is a base generator of $Q_{\omega}$. Each quasi-infiniton contains
a vicious cycle of length $l$ known as the length of $Q_{\omega}$.

###### Theorem 3.36

Suppose $Q_{\omega}$ is defined in definition 3.35. Then161616In the rest
discussion, $Q_{w}$ is presumed to be the collection of quasi-infinitons, and
$Q$ a quasi-infiniton from the base generator of $G_{0}$ and $l$ principal
generators of $G_{k}$ $\left(1\leqslant k\leqslant l\right)$ unless further
specified.

  1. (i)

Each $Q_{\omega,q}$ of $Q_{\omega}$ is $\omega-$invariant.

  2. (ii)

For each $0\leqslant q<l$, $\,Q_{\omega,q+1}\,=\,\\{\ast
G_{q+1},\,Q_{\omega,q}\\}$.

  3. (iii)

The type for each $Q_{\omega,q}$ of $Q_{\omega}$ is that there exists a
vicious cycle of length $l$ with each sublevel $j$ $\left(1\leqslant
j\leqslant l\right)$ containing the members of $G_{j}$.

  4. (iv)

$Q_{\omega}$ is a $\aleph_{0}$-categorical structure with $l$ atomic
substructures.

Proof. (i) WLOG, suppose $\,Q_{\omega,0}\,=\,\\{\ast G_{l},\\{\ast
G_{l-1},\cdots\\{\ast G_{1},Q_{\omega,0}\\}\cdots\\}$. For any $\alpha$ that
$\omega<\alpha<\omega 2$, if $\left(m-1\right)l\leqslant\alpha-\omega<ml$, let
$\beta\,=\,\omega+ml$. Then

$\displaystyle Q_{\beta}\>$ $\displaystyle=\>\\{\ast G_{l},\\{\ast
G_{l-1},\cdots\\{\ast G_{1},Q_{\left(m-1\right)l}\\}\cdots\\}$
$\displaystyle=\>\\{\ast G_{l},\\{\ast G_{l-1},\cdots\\{\ast G_{1},\\{\ast
G_{l},\\{\ast G_{l-1},\cdots\\{\ast
G_{1},Q_{\left(m-2\right)l}\\}\\}\cdots\\}$ $\displaystyle\>\>\vdots$
$\displaystyle=\>\underset{\beta-\omega}{\underbrace{\\{\ast
G_{l},\cdots\\{\ast G_{1},\\{\ast G_{l},\cdots\\{\ast
G_{1},Q_{\omega,0}\\}\\}\cdots\\}}}$ $\displaystyle=\>Q_{\omega,0}$

Then it follows by transfinite induction. Other quasi-infinitons of
$Q_{\omega}$ can be proved similarly.

(ii) We only prove $Q_{\omega,1}\,=\,\\{\ast G_{1},\,Q_{\omega,0}\\}$. By
corollary 2.25 and theorem 3.34

$\displaystyle\\{\ast G_{1},\,Q_{\omega,0}\\}\>$ $\displaystyle=\>\\{\ast
G_{1},\,\underset{p\rightarrow\omega}{\lim}Q_{pl}\\}$
$\displaystyle=\>\underset{p\rightarrow\omega}{\lim}\\{\ast G_{1},\\{\ast
G_{l},\cdots\\{\ast G_{2},Q_{\left(p-1\right)l+1}\\}\cdots\\}$
$\displaystyle=\>\\{\ast G_{1},\\{\ast G_{l},\cdots\\{\ast
G_{2},Q_{\omega,1}\\}\cdots\\}$ $\displaystyle=\>Q_{\omega,1}$

(iii) Since $\left(n=pl\right)$

$Q_{pl}\,=\,\\{\ast G_{l},\\{\ast G_{l-1},\cdots\\{\ast
G_{1},Q_{\left(p-1\right)l}\\}\cdots\\}\>=\>\underset{p}{\underbrace{\\{\ast
G_{l},\cdots\\{\ast G_{1},\\{\ast G_{l},\cdots\\{\ast
G_{1},Q_{0}\\}\\}\cdots\\}}}$

The type for $Q_{pl}$ is:

$\chi_{pl}\,\Longleftrightarrow\underset{1\leqslant i\leqslant
p}{\displaystyle\bigwedge}\text{ }\underset{0\leqslant j\leqslant
l-1}{\displaystyle\bigwedge}\exists!\,Q_{\left(i-1\right)l+j+1}\left(\begin{array}[c]{c}\exists!\,Q_{\left(i-1\right)l+j}\left(Q_{\left(i-1\right)l+j}\in
Q_{\left(i-1\right)l+j+1}\right)\\\\[5.16663pt] \wedge\,\left(\forall z\in
G_{j+1}\right)\left(z\in Q_{\left(i-1\right)l+j+1}\right)\end{array}\right)$

In general, the type for $Q_{pl+q}$ $\left(n=pl+q,\,0\leqslant q<l\right)$ is:

$\chi_{pl+q}\,\Longleftrightarrow\underset{1\leqslant i\leqslant
p}{\displaystyle\bigwedge}\text{ }\underset{q\leqslant j\leqslant
q+l-1}{\displaystyle\bigwedge}\exists!\,Q_{\left(i-1\right)l+j+1}\left(\begin{array}[c]{c}\exists!\,Q_{\left(i-1\right)l+j}\left(Q_{\left(i-1\right)l+j}\in
Q_{\left(i-1\right)l+j+1}\right)\\\\[5.16663pt] \wedge\,\left(\forall z\in
G_{j+1}\right)\left(z\in Q_{\left(i-1\right)l+j+1}\right)\end{array}\right)$

By theorem 3.34, each $Q_{pl+q}$ $\left(0\leqslant q<l\right)$ is homogeneous
and has a unique limit. So by corollary 2.14, 2.27 and theorem 3.34

$\displaystyle\chi_{\omega,q}\,\,$
$\displaystyle\Longleftrightarrow\,\,\underset{p\rightarrow\omega}{\lim}\chi_{pl+q}$
$\displaystyle\Longleftrightarrow\,\,\underset{p\rightarrow\omega}{\lim}\underset{1\leqslant
i\leqslant p-1}{\displaystyle\bigwedge}\text{ }\underset{q\leqslant j\leqslant
q+l-1}{\displaystyle\bigwedge}\exists!\,Q_{\left(i-1\right)l+j+1}\left(\begin{array}[c]{c}\exists!\,Q_{\left(i-1\right)l+j}\left(Q_{\left(i-1\right)l+j}\in
Q_{\left(i-1\right)l+j+1}\right)\\\\[5.16663pt] \wedge\,\left(\forall z\in
G_{j+1}\right)\left(z\in Q_{\left(i-1\right)l+j+1}\right)\end{array}\right)$
$\displaystyle\qquad\wedge\,\underset{p\rightarrow\omega}{\lim}\underset{q\leqslant
j\leqslant
q+l-1}{\displaystyle\bigwedge}\exists!\,Q_{\left(p-1\right)l+j+1}\left(\begin{array}[c]{c}\exists!\,Q_{\left(p-1\right)l+j}\left(Q_{\left(p-1\right)l+j}\in
Q_{\left(p-1\right)l+j+1}\right)\\\\[5.16663pt] \wedge\,\left(\forall z\in
G_{j+1}\right)\left(z\in Q_{\left(p-1\right)l+j+1}\right)\end{array}\right)$
$\displaystyle\Longleftrightarrow\,\,\underset{p<\omega}{\displaystyle\bigwedge}\text{
}\underset{q\leqslant j\leqslant
q+l-1}{\displaystyle\bigwedge}\exists!\,Q_{\left(p-1\right)l+j+1}\left(\begin{array}[c]{c}\exists!\,Q_{\left(p-1\right)l+j}\left(Q_{\left(p-1\right)l+j}\in
Q_{\left(p-1\right)l+j+1}\right)\\\\[5.16663pt] \wedge\,\left(\forall z\in
G_{j+1}\right)\left(z\in Q_{\left(p-1\right)l+j+1}\right)\end{array}\right)$
$\displaystyle\qquad\wedge\underset{q\leqslant j\leqslant
q+l-1}{\displaystyle\bigwedge}\exists!\,Q_{\omega,j+1}\left(\exists!\,Q_{\omega,j}\left(Q_{\omega,j}\in
Q_{\omega,j+1}\right)\,\wedge\,\left(\forall z\in G_{j+1}\right)\left(z\in
Q_{\omega,j+1}\right)\right)$

By theorem 2.7, $Q_{\omega,q}\models\chi_{\omega,q}$ where $\chi_{\omega,q}$
describes a vicious cycle of length $l$ with each sublevel $j$
$\left(1\leqslant j\leqslant l\right)$ containing the members of $G_{j}$.

(iv) From the above results, we can see that each $Q_{n}$ satisfies one and
only one of $\chi_{pl+q}$. So there are only finitely many $(l)$ $n-$types. By
proposition 2.3, Th$(Q_{\omega})$ is $\aleph_{0}$-categorical with $l$ atomic
substructures of $Q_{\omega,q}$, each having a complete formula of
$\chi_{\omega,q}$.

The tree structure for a quasi-infiniton is shown in Figure 3. Intuitively, a
quasi-infiniton has one infinite branch and the nodes of the infinite branch
form a finite cycle.

Figure 3: Diagram of a quasi-infiniton.

###### Corollary 3.37

  1. (i)

$Q\notin Q$ and $Q\neq\varnothing$.

  2. (ii)

$D(Q)\,=\,\aleph_{0}$ and $Q\notin V$.

Proof. (i) If $Q\in Q$, the length of $Q$ is $1$ and $Q$ is a semi-infiniton.
Also, no $Q_{1}\in\varnothing$. So $\varnothing\in Q_{1}$ and
$Q_{1}\in\varnothing$ are impossible.

(ii) By theorem 3.36(i) and (1.2), $D(Q)\,=\,\aleph_{0}$. Since $Q$ has an
infinite branch, it is NWF. So $Q\notin V$.

###### Definition 3.38

$S|_{\mathfrak{Q}}=\\{\\{G_{k},l\\}|_{\mathfrak{Q}}\colon G_{k}\in
S,\,0\leqslant k\leqslant l,\,l>1\\}$ is known as the set of the quasi-
infinitons from $S$.

###### Corollary 3.39

  1. (i)

$S_{1}\subset S_{2}\>\Longrightarrow\>S_{1}|_{\mathfrak{Q}}\subset
S_{2}|_{\mathfrak{Q}}$

  2. (ii)

$\left(S_{1}\cup
S_{2}\right)|_{\mathfrak{Q}}\,\supset\,S_{1}|_{\mathfrak{Q}}\cup
S_{2}|_{\mathfrak{Q}}$

  3. (iii)

$\left(S_{1}\cap
S_{2}\right)|_{\mathfrak{Q}}\,\subset\,S_{1}|_{\mathfrak{Q}}\cap
S_{2}|_{\mathfrak{Q}}$

  4. (iv)

$\left.\left(\underset{\alpha\in
D}{\displaystyle\bigcup}S_{\alpha}\right)\right|_{\mathfrak{Q}}\supset\underset{\alpha\in
D}{\displaystyle\bigcup}\left.S_{\alpha}\right|_{\mathfrak{Q}}$

  5. (v)

$S_{\alpha}\uparrow\hskip
4.30554pt\Longrightarrow\left.\left(\underset{\alpha\in
D}{\displaystyle\bigcup}S_{\alpha}\right)\right|_{\mathfrak{Q}}=\underset{\alpha\in
D}{\displaystyle\bigcup}\left.S_{\alpha}\right|_{\mathfrak{Q}}$

Proof. (i) is obvious. (ii), (iii) and (iv) follow from (i).

(v) For any $\\{G_{k},l\\}|_{\mathfrak{Q}}\in\left.\left(\underset{\alpha\in
D}{\displaystyle\bigcup}S_{\alpha}\right)\right|_{\mathfrak{Q}}$, there is a
$\gamma\in D$ that $G_{k}\in S_{\gamma}$ for $0\leqslant k\leqslant l$. Thus
$\\{G_{k},l\\}|_{\mathfrak{Q}}\in\left.S_{\gamma}\right|_{\mathfrak{Q}}\subset\underset{\alpha\in
D}{\displaystyle\bigcup}\left.S_{\alpha}\right|_{\mathfrak{Q}}$. This proves
$``\subset\textquotedblright$ and it follows by (iv).

##  4 Total Universe

In this section, we will present and investigate a hierarchy for combining the
well-founded sets with the non-well-founded sets known as the total universe.
We will also show that the total universe is free of Russell’s paradox.

###  4.1 Definitions

First, we need to generalize the $\omega-$neighborhood and the limit of
formulas.

###### Definition 4.1

Suppose $\alpha$ is a limit ordinal $\left(\alpha>\omega\right)$ and
$\alpha_{0}$ the limit ordinal immediately below $\alpha$. The cofinite
topology on $\alpha$ is defined as:
$\,\mathfrak{T}\,=\,\\{y\subset\alpha\colon
y=\varnothing\,\vee\,(\alpha_{0}\subset y\,\wedge\,\alpha-y$ is finite$)\\}$.
A neighborhood of $\mathbf{\alpha}$ $(\alpha-$neighborhood$)$ $\mathfrak{H}$
is a member of $\mathfrak{T}$.

###### Lemma 4.2

$\mathfrak{H}$ is a neighborhood of $\alpha$ if and only if
$\exists\beta\in\alpha-\alpha_{0}$ that
$\forall\gamma\left(\beta<\gamma<\alpha\right)\Rightarrow\gamma\in\mathfrak{H}$.

Proof. Suppose $\mathfrak{H}$ is a neighborhood of $\alpha$ and for any
$\beta\in\alpha-\alpha_{0}$, there is a $\gamma$ that $\beta<\gamma<\alpha$
and $\gamma\notin\mathfrak{H}$. Then $\alpha-\mathfrak{H}$ is not finite,
contradicting definition 4.1. On the other hand, if there is a
$\beta\in\alpha-\alpha_{0}$ such that for any $\gamma$ of
$\beta<\gamma<\alpha$, $\gamma\in\mathfrak{H}$, then $\alpha-\mathfrak{H}$ is
finite and $\mathfrak{H}\in\mathfrak{T}$.

###### Definition 4.3

Suppose $\mathcal{L}$ is an infinitary language of
$\mathcal{L}_{\omega_{1},\omega}$, $T$ is a $\aleph_{0}$-categorical theory of
$\mathcal{L}$ and $\alpha$ is a limit ordinal $\left(\alpha>\omega\right)$.
Let $\phi_{\gamma}$ be types in $T$ and $\mathfrak{M}_{\gamma}$ be
$\mathcal{L}$-structures that $\mathfrak{M}_{\gamma}\models\phi_{\gamma}$. If
there exists a $\alpha-$neighborhood $\mathfrak{H}$ that for any
$\beta,\gamma\in\mathfrak{H}\>(\beta>\gamma)$,
$\mathfrak{M}_{\beta}\models\phi_{\gamma}$, then
$\\{(\mathfrak{M}_{\gamma},\phi_{\gamma})\colon\mathfrak{M}_{\gamma}\models\phi_{\gamma}\land\gamma<\alpha\\}$
is known as a homogeneous sequence of structures described by $\phi_{\gamma}$
in $T$.

###### Definition 4.4

Suppose
$\\{(\mathfrak{M}_{\gamma},\phi_{\gamma})\colon\mathfrak{M}_{\gamma}\models\phi_{\gamma}\land\gamma<\alpha\\}$
is a homogeneous sequence of structures in a $\aleph_{0}$-categorical theory.
The unique countable atomic structure $\mathfrak{M}$ (up to isomorphism) in
$\\{(\mathfrak{M}_{\gamma},\phi_{\gamma})\\}$ is known as the limit of
$\mathfrak{M}_{\gamma}$ and is denoted as
$\underset{\gamma\rightarrow\alpha}{\lim}\mathfrak{M}_{\gamma}=\mathfrak{M}$.
The unique formula $\phi$ (up to equivalence) is known as the limit of
$\phi_{\gamma}$ and is denoted as
$\underset{\gamma\rightarrow\alpha}{\lim}\phi_{\gamma}=\phi$. In both cases,
we also say that the limit of $\phi_{\gamma}$ or the limit of
$\mathfrak{M}_{\gamma}$ is unique.

###### Definition 4.5

Suppose in a sequence of structures
$\\{(\mathfrak{M}_{\gamma},\phi_{\gamma})\colon\mathfrak{M}_{\gamma}\models\phi_{\gamma}\land\gamma<\alpha\\}$
in a $\aleph_{0}$-categorical theory, there are finitely many homogeneous
subsequences of structures
$\\{(\mathfrak{M}_{\gamma_{i}},\phi_{\gamma_{i}})\colon\allowbreak\mathfrak{M}_{\gamma_{i}}\models\phi_{\gamma_{i}}\land\gamma_{i}<\alpha\\}$.
Then each
$\underset{\gamma_{i}\rightarrow\alpha}{\lim}\,\mathfrak{M}_{\gamma_{i}}$ is
known as a sublimit of $\mathfrak{M}_{\gamma}$, and each
$\underset{\gamma_{i}\rightarrow\alpha}{\lim}\,\phi_{\gamma_{i}}$ is known as
a sublimit of $\phi_{\gamma}$. If some sublimits of
$\mathfrak{M}_{\gamma}/\phi_{\gamma}$ are different, we say
$\underset{\gamma\rightarrow\alpha}{\lim}\mathfrak{M}_{\gamma}/\underset{\gamma\rightarrow\alpha}{\lim}\phi_{\gamma}$
exist (but not unique).

(Almost) all conclusions in section 2 hold for the limit ordinals as well. We
can simply replace $n$ with $\gamma$ ($\gamma$ is any successor ordinal in a
neighborhood of a limit ordinal $\alpha$ above $\omega$), and
$\underset{n\rightarrow\omega}{\lim}$ with
$\underset{\gamma\rightarrow\alpha}{\lim}$. Now we define the total universe
based upon the von Neumann universe.

###### Definition 4.6

The total universe is:

$\displaystyle T_{0}\,$ $\displaystyle=\,\varnothing;$ $\displaystyle
T_{\alpha}\,$
$\displaystyle=\,\mathcal{P}(T_{\alpha-1})\text{,\qquad\qquad\qquad\qquad}\alpha\operatorname{\>is\>any\>successor\>ordinal;}$
$\displaystyle T_{\alpha}\,$
$\displaystyle=\underset{\beta<\alpha}{\bigcup}T_{\beta}\,\cup\left.\left(\underset{\beta<\alpha}{\bigcup}T_{\beta}\right)\right|_{\mathcal{\aleph}_{0}},\text{\qquad}\alpha\operatorname{\>is\>any\>limit\>ordinal;}$
$\displaystyle T\,$ $\displaystyle=\underset{\alpha\in
Ord}{\bigcup}T_{\alpha}.$ (4.1)

###### Remark 4.7

Note that (4.1) is based on the generalization of definition 3.2 and 4.9.
Since the total universe contains the well-founded sets, it is similar to
(1.1). The key difference is that the infinitely generated sets are created at
each limit ordinal in addition to the von Neumann universe.

###### Definition 4.8

Suppose $\alpha$ is a limit ordinal and $H_{\gamma}$ is the same as $H_{n}$ in
(3.1) and (3.2) except
$G_{\gamma}\in\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}$
$\left(\gamma<\omega\right)$ and $\mathcal{G}=\\{G_{\gamma}\colon
G_{\gamma}\in\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta},\gamma<\omega\\}$.
An infinitely generated set (at $\alpha$) is defined as:

$H_{\alpha}(\mathcal{G)}\,=\,\underset{\gamma\rightarrow\alpha}{\lim}\,H_{\gamma}(G_{\gamma},\cdots,G_{0})$
(4.2)

###### Definition 4.9

$\left.S\right|_{\aleph_{0}}\,=\,\\{H_{\alpha}(\mathcal{G)}\colon\mathcal{G}=\\{G_{\gamma}\colon
G_{\gamma}\in S,\gamma<\omega\\}\\}$ is known as the set of the infinitely
generated sets from $S$.

###### Remark 4.10

All $G_{n}\in V_{\omega}$ in definition 3.14, 3.26 and 3.35 are changed to
$G_{\gamma}\in\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}$
$\left(\gamma<\omega\right)$.

The axiom of extensionality for IGS can be modified from axiom 3.10.

###### Axiom 4.11

Suppose $\mathcal{G}_{1}=\\{G_{\gamma}^{1}\colon
G_{\gamma}^{1}\in\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta},\,\gamma<\omega\\}$
and $\mathcal{G}_{2}=\\{G_{\gamma}^{2}\colon
G_{\gamma}^{2}\in\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta},\,\gamma<\omega\\}$.
Then

$\left(\forall\gamma<\omega\right)\left(G_{\gamma}^{1}\,=\,G_{\gamma}^{2}\right)\>\Longrightarrow\>H_{\alpha}(\mathcal{G}_{1})\,=\,H_{\alpha}(\mathcal{G}_{2})$

###### Corollary 4.12

  1. (i)

$S_{1}\subset S_{2}\>\Longrightarrow\>S_{1}|_{\aleph_{0}}\subset
S_{2}|_{\aleph_{0}}$

  2. (ii)

$S_{1}|_{\aleph_{0}}\cup S_{2}|_{\aleph_{0}}\,\subset\,\left(S_{1}\cup
S_{2}\right)|_{\aleph_{0}}$

  3. (iii)

$\left(S_{1}\cap S_{2}\right)|_{\aleph_{0}}\,\subset\,S_{1}|_{\aleph_{0}}\cap
S_{2}|_{\aleph_{0}}$

  4. (iv)

$\underset{\alpha\in
D}{\displaystyle\bigcup}\left.S_{\alpha}\right|_{\aleph_{0}}\,\subset\left.\left(\underset{\alpha\in
D}{\displaystyle\bigcup}S_{\alpha}\right)\right|_{\aleph_{0}}$

Proof. (i) By definition 4.9, for any
$H_{\alpha}(\mathcal{G)}\in\left.S_{1}\right|_{\aleph_{0}}$ and any
$G_{\gamma}\in S_{1}$, $G_{\gamma}\in S_{2}$. So
$H_{\alpha}(\mathcal{G)}\in\left.S_{2}\right|_{\aleph_{0}}$. The rest follow
from (i).

###### Corollary 4.13

$\left.\left(\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\right)\right|_{\mathfrak{S}}\cup\left.\left(\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\right)\right|_{\mathfrak{Q}}\subset\left.\left(\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\right)\right|_{\mathcal{\aleph}_{0}}$

Proof. By definition 3.30, 3.38, 4.9 and (4.1).

Rank in the total universe is the same as that of the von Neumann universe
(definition 1.4).

###### Definition 4.14

The rank of $S$ in $T$ is defined as the least $\alpha$ that $S\in T_{\alpha}$
and denoted as $R_{T}(S)$.

The notion of $\omega-$invariance (definition 3.11) can be extended to any
limit ordinal.

###### Definition 4.15

Suppose $\alpha$ is a limit ordinal, $H_{\alpha}$ is an IGS and
$H_{\alpha+\xi}=\\{\ast G_{\alpha+\xi},\,H_{\alpha+\xi-1}\\}$, where
$G_{\alpha+\xi}\in\underset{\delta<\alpha}{\displaystyle\bigcup}T_{\delta}$
$\left(\xi\geqslant 1\right)$. If for any $\gamma>\alpha$, there is a
(successor ordinal) $\beta>\gamma$ that $H_{\beta}=H_{\alpha}$, then
$H_{\alpha}$ is called $\,\mathbf{\omega-}$invariant.

###### Lemma 4.16

Suppose $\alpha$ is a limit ordinal, $H_{\alpha}$ is $\,\omega-$invariant and
$G_{\xi}\in\underset{\mathfrak{\delta}<\alpha}{\displaystyle\bigcup}T_{\mathfrak{\delta}}\,\left(\xi\leqslant\beta\right)$.
Then

$H_{\alpha}\mathcal{\,}=\,\\{\ast G_{\beta},\\{\ast G_{\beta-1},\cdots\\{\ast
G_{\alpha+1},H_{\alpha}\\}\cdots\\}$

Proof. Suppose $\alpha^{\prime}$ is the limit ordinal immediately above
$\alpha$. WLOG, by definition 4.15, assume for any $\gamma$
$\left(\alpha<\gamma<\alpha^{\prime}\right)$, there is a $\beta$
$\left(\gamma<\beta<\alpha^{\prime}\right)$ that $H_{\beta}=H_{\alpha}$. Then

$H_{\beta}\,=\,\\{\ast G_{\beta},H_{\beta-1}\\}\,=\,\\{\ast G_{\beta},\\{\ast
G_{\beta-1},\cdots\\{\ast G_{\alpha+1},H_{\alpha}\\}\cdots\\}\,=\,H_{\alpha}$

where
$G_{\xi}\in\underset{\mathfrak{\delta}<\alpha}{\displaystyle\bigcup}T_{\mathfrak{\delta}}$
for $\xi\leqslant\beta$.

###### Remark 4.17

Lemma 4.16 shows that a $\omega-$invariant set always has an immediate member,
while (in general) an IGS does not have one.

Furthermore, the union operator and transitive closure need be extended to the
transfinite case.

###### Definition 4.18

Suppose ${\displaystyle\bigcup}S=\\{z\colon\exists y\left(y\in S\,\wedge\,z\in
y\right)\\}$. The $\mathbf{\alpha}^{th}$ union operator is defined
(recursively) as:

  1. (i)

If $\alpha$ is a successor ordinal, then

${\displaystyle\bigcup}^{\alpha}S\,=\,{\displaystyle\bigcup}\>{\displaystyle\bigcup}^{\alpha-1}S\quad\left({\displaystyle\bigcup}^{0}S\,=\,S\right)$

  2. (ii)

If $\alpha$ is a limit ordinal, then

${\displaystyle\bigcup}^{\alpha}S\,=\,\underset{\beta<\alpha}{\displaystyle\bigcup}{\displaystyle\bigcup}^{\beta}S$

###### Definition 4.19

Suppose $\alpha_{0}$ is the least ordinal $\alpha$ that
${\displaystyle\bigcup}^{\alpha}S\,=\,{\displaystyle\bigcup}^{\alpha+1}S$.
Then the transitive closure of $S$ is:

$TC(S)\,=\underset{\alpha\leqslant\alpha_{0}}{\displaystyle\bigcup}{\displaystyle\bigcup}^{\alpha}S$
(4.3)

###### Corollary 4.20

  1. (i)

For any $\alpha<\omega$, $V_{\alpha}=T_{\alpha}$. Otherwise,
$V_{\alpha}\subsetneq T_{\alpha}$.

  2. (ii)

$V\subsetneq T$.

  3. (iii)

$T_{\alpha}$ contains all ordinals less than $\alpha$. $T$ contains all
ordinals.

Proof. (i) By (1.1) and (4.1), for any $\alpha<\omega$,
$V_{\alpha}=T_{\alpha}$. Clearly

$T_{\omega}\,=\,\underset{\beta<\omega}{\displaystyle\bigcup}T_{\beta}\,\cup\left.\left(\underset{\beta<\omega}{\displaystyle\bigcup}T_{\beta}\right)\right|_{\mathcal{\aleph}_{0}}\,\supsetneq\,\underset{\beta<\omega}{\displaystyle\bigcup}T_{\beta}\,=\,V_{\omega}$

Suppose $V_{\beta}\subsetneq T_{\beta}$ for any $\beta$,
$\omega<\beta<\alpha$. Then if $\alpha$ is a successor ordinal

$T_{\alpha}\,=\,\mathcal{P}(T_{\alpha-1})\,\supsetneq\,\mathcal{P}(V_{\alpha-1})\,=\,V_{\alpha}$

If $\alpha$ is a limit ordinal, then

$T_{\alpha}\,=\,\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\,\cup\left.\left(\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\right)\right|_{\mathcal{\aleph}_{0}}\,\supsetneq\,\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\,\supsetneq\,\underset{\beta<\alpha}{\displaystyle\bigcup}V_{\beta}\,=\,V_{\alpha}$

(ii) By (i), (1.1) and (4.1).

(iii) By definition 4.8,
$\left.\left(\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\right)\right|_{\mathcal{\aleph}_{0}}$
contains no ordinals since any set in it is NWF, but all ordinals are WF. So
$T_{\alpha}$ has the same ordinals as $V_{\alpha}$ and by corollary 1.6,
$T_{\alpha}$ contains all ordinals less than $\alpha$. By (4.1), $T$ contains
all ordinals.

###### Corollary 4.21

  1. (i)

Each $T_{\alpha}$ is transitive.

  2. (ii)

$T$ is transitive.

Proof. (i) We prove by the transfinite induction. First, since
$T_{1}=\\{\varnothing\\}$ and $\varnothing\subset T_{1}$, $T_{1}$ is
transitive. Suppose it is true for any $\beta<\alpha$. Then if $\alpha$ is a
successor ordinal, for any $X\in T_{\alpha}\,=\,\mathcal{P}(T_{\alpha-1})$,
$X\subset T_{\alpha-1}$. So for any $Y\in X$, $Y\in T_{\alpha-1}$. Since
$T_{\alpha-1}$ is transitive, $Y\subset T_{\alpha-1}$ and $Y\in T_{\alpha}$.
Thus $T_{\alpha}$ is transitive.

If $\alpha$ is a limit ordinal, then for any $X\in T_{\alpha}$, if
$X\in\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}$, there is a
$\gamma<\alpha$ that $X\in T_{\gamma}$. Since $T_{\gamma}$ is transitive,
$X\subset T_{\gamma}\subset T_{\alpha}$. If
$X\in\left.\left(\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\right)\right|_{\mathcal{\aleph}_{0}}$
and $X$ is $\omega-$invariant, then by lemma 4.16

$X\,=\,\\{\ast G_{\eta},\\{\ast
G_{\eta-1},\cdots\\}\cdots\\},\>G_{\xi}\in\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\,\subset\,T_{\alpha}\text{
\ for }\xi\leqslant\eta.$

Then there is a $\gamma<\alpha$ that $G_{\eta}\in T_{\gamma}$. Since
$T_{\gamma}$ is transitive, for any $z\in G_{\eta}$, $z\in T_{\gamma}\subset
T_{\alpha}$. Furthermore, $\\{\ast G_{\eta-1},\\{\ast
G_{\eta-2},\cdots\\}\cdots\\}$ is also $\omega-$invariant and

$\\{\ast G_{\eta-1},\\{\ast
G_{\eta-2},\cdots\\}\cdots\\}\in\left.\left(\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\right)\right|_{\mathcal{\aleph}_{0}}\subset\,T_{\alpha}$

So $X\subset T_{\alpha}$. If $X$ is not $\omega-$invariant, then $X$ does not
have immediate members. Both cases show that $T_{\alpha}$ is transitive if
$\alpha$ is a limit ordinal.

(ii) By (i) and (4.1).

###### Corollary 4.22

  1. (i)

For any $\alpha$, $T_{\alpha}\,\subset\,\mathcal{P}(T_{\alpha})$

  2. (ii)

$\alpha<\beta\>\Longrightarrow\>T_{\alpha}\subset T_{\beta}$

Proof. (i) By corollary 4.21, $T_{\alpha}$ is transitive. So for any $X\in
T_{\alpha}$, $X\subset T_{\alpha}$. Thus $X\in\mathcal{P}(T_{\alpha})$ and (i)
follows.

(ii) Suppose it is true for any $\alpha<\gamma<\beta$. If $\beta$ is a
successor ordinal, then by (i)

$T_{\beta}\,=\,\mathcal{P}(T_{\beta-1})\,\supset\,T_{\beta-1}\,\supset\,T_{\alpha}$

If $\beta$ is a limit ordinal, then

$T_{\beta}\,=\,\underset{\gamma<\beta}{\displaystyle\bigcup}T_{\gamma}\,\cup\left.\left(\underset{\gamma<\beta}{\displaystyle\bigcup}T_{\gamma}\right)\right|_{\mathcal{\aleph}_{0}}\,\supset\,\underset{\gamma<\beta}{\displaystyle\bigcup}T_{\gamma}\,\supset\,T_{\alpha}$

###  4.2 Rank in Total Universe

Now we investigate more on rank in the total universe.

###### Lemma 4.23

Suppose $X\in T$. Then

$Y\in X\>\Longrightarrow\>R_{T}(Y)\leqslant R_{T}(X)$

Proof. Suppose $R_{T}(X)=\alpha$. Then by definition 4.14, $X\in T_{\alpha}$.
By corollary 4.21, $T_{\alpha}$ is transitive. Thus for $Y\in X$, $Y\in
T_{\alpha}$, i.e. $R_{T}(Y)\leqslant\alpha$.

###### Lemma 4.24

If $R_{T}(X)$ is a successor ordinal, then

$Y\in X\>\Longrightarrow\>R_{T}(Y)<R_{T}(X)$

Proof. Suppose $R_{T}(X)=\alpha$ is a successor ordinal. Then $X\in
T_{\alpha}\,=\,\mathcal{P}(T_{\alpha-1})$ and $X\subset T_{\alpha-1}$. Thus
$Y\in T_{\alpha-1}$ and $R_{T}(Y)\leqslant\alpha-1<R_{T}(X)$.

###### Corollary 4.25

If $R_{T}(X)$ is a limit ordinal, then $X$ is an infinitely generated set.

Proof. Suppose $R_{T}(X)=\alpha$ is a limit ordinal. If
$X\in\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}$, then there is a
$\gamma<\alpha$ that $X\in T_{\gamma}$, i.e. $R_{T}(X)\leqslant\gamma<\alpha$,
contradiction. So
$X\in\left.\left(\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\right)\right|_{\mathcal{\aleph}_{0}}$.

###### Corollary 4.26

Suppose $\alpha$ is a limit ordinal and $H$ is $\omega-$invariant. Then
$R_{T}(H)=\alpha$.

Proof. By lemma 4.16, $H=\\{\ast G_{\gamma},\\{\ast G_{\gamma-1},\cdots\\{\ast
G_{\alpha+1},H\\}\cdots\\}$, where
$G_{\xi}\in\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}$ for
$\xi\leqslant\gamma$. Let $H_{1}=\\{\ast G_{\gamma-1},\cdots\\{\ast
G_{\alpha+1},H\\}\cdots\\}$. If $R_{T}(H)>\alpha$, then by lemma 4.24,
$R_{T}(H)<R_{T}(H_{1})<R_{T}(H)$, contradiction. Thus $R_{T}(H)=\alpha$.

Note that there are sets that appear to have successor ranks, but actually
have limit ordinal ranks.

###### Example 4.27

Suppose $a,b\in V_{\omega}$ and $Q=\\{a,b,2\\}|_{\mathfrak{Q}}$. Then
$R_{T}(\\{b,Q\\})=\omega$.

At first sight, $\\{b,Q\\}\in T_{\omega+1}$ for $\\{b,Q\\}\subset T_{\omega}$.
But by theorem 3.34, $\\{a,\\{b,Q\\}\\}=Q$. So
$\\{b,\\{a,\\{b,Q\\}\\}\\}=\\{b,Q\\}$, i.e. $\\{b,Q\\}$ is $\omega-$invariant.
By corollary 4.26, $R_{T}(\\{b,Q\\})=\omega$.

The total universe can be partitioned by rank as follows.

######  Theorem 4.28 (Partition Formula by Rank)

171717$SOrd$ is the set of all successor ordinals and $LOrd$ is the set of all
limit ordinals.

Suppose $A_{\alpha}=\\{X\colon R_{T}(X)=\alpha\,\wedge\,X\in T\\}$.

  1. (i)

If $\alpha$ is a successor ordinal, then
$A_{\alpha}\,=\,T_{\alpha}-T_{\alpha-1}$

  2. (ii)

If $\alpha$ is a limit ordinal, then

$A_{\alpha}\,=\,T_{\alpha}\,-\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\,\,=\left.\left(\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\right)\right|_{\mathcal{\aleph}_{0}}-\,\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\,\,=\,\underset{\beta<\alpha}{\displaystyle\bigcap}\left(T_{\alpha}-T_{\beta}\right)$

  3. (iii)

In general, $T$ can be partitioned by rank as:

$T\,\,=\underset{\alpha\in
SOrd}{\displaystyle\bigcup}\left(T_{\alpha}-T_{\alpha-1}\right)\,\,\cup\underset{\alpha\in
LOrd}{\displaystyle\bigcup}\,\,\underset{\beta<\alpha}{\displaystyle\bigcap}\left(T_{\alpha}-T_{\beta}\right)$

Proof. (i) If $\alpha$ is a successor ordinal, then by definition 4.14,
$R_{T}(X)=\alpha$ if and only if $X\in T_{\alpha}$ and $X\notin T_{\alpha-1}$.

(ii) If $\alpha$ is a limit ordinal, by definition 4.14, $R_{T}(X)=\alpha$ if
and only if $X\in T_{\alpha}$ and for any $\beta<\alpha$, $X\notin T_{\beta}$.
So by (4.1)

$A_{\alpha}\,=\,T_{\alpha}\,-\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\,\,=\,\underset{\beta<\alpha}{\bigcup}T_{\beta}\,\cup\left.\left(\underset{\beta<\alpha}{\bigcup}T_{\beta}\right)\right|_{\mathcal{\aleph}_{0}}-\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\,\,=\left.\left(\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\right)\right|_{\mathcal{\aleph}_{0}}-\,\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}$

On the other hand, clearly we have

$T_{\alpha}\,-\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\,\,=\,\underset{\beta<\alpha}{\displaystyle\bigcap}\left(T_{\alpha}-T_{\beta}\right)$

(iii) For any $X\in T$ and $R_{T}(X)=\alpha$, if $\alpha$ is a successor
ordinal, then by (i)

$X\in T_{\alpha}-T_{\alpha-1}\,\subset\underset{\alpha\in
SOrd}{\displaystyle\bigcup}\left(T_{\alpha}-T_{\alpha-1}\right)$

If $\alpha$ is a limit ordinal, then by (ii)

$X\in
T_{\alpha}-\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\,\,\subset\underset{\alpha\in
LOrd}{\displaystyle\bigcup}\,\,\underset{\beta<\alpha}{\displaystyle\bigcap}\left(T_{\alpha}-T_{\beta}\right)$

Obviously, each category is disjoint from others.

We also have the following theorems for rank in the total universe.

###### Theorem 4.29

Suppose $X\in T$.

  1. (i)

If $R_{T}(X)$ is a successor ordinal, then there is either a $Y\in X$ that
$R_{T}(Y)=R_{T}(X)-1$, or a sequence $Y_{n}\in X$ that
$R_{T}(Y_{n})\rightarrow R_{T}(X)-1$ as $n\rightarrow\infty$.

  2. (ii)

If $R_{T}(X)$ is a limit ordinal and $X$ is $\omega-$invariant, then there is
one and only one $Y\in X$ that $R_{T}(Y)=R_{T}(X)$.

Proof. (i) Suppose $R_{T}(X)=\alpha$ is a successor ordinal. By lemma 4.24,
for any $Y\in X$, $R_{T}(Y)\leqslant\alpha-1$. If there is a $Y\in X$ that
$R_{T}(Y)=\alpha-1$, then done. So suppose there is no $Y\in X$ that
$R_{T}(Y)=\alpha-1$.

If $\alpha-1$ is a successor ordinal, then for any $Y\in X$, $Y\in
T_{\alpha-2}$. So $X\subset T_{\alpha-2}$ and $X\in T_{\alpha-1}$, i.e.
$R_{T}(X)\leqslant\alpha-1$, contradiction. If $\alpha-1$ is a limit ordinal,
then there is a sequence $Y_{n}\in X$ that $R_{T}(Y_{n})\rightarrow\alpha-1$.
Otherwise suppose there is a $\beta<\alpha-1$ that for any $Y\in X$,
$R_{T}(Y)\leqslant\beta$, i.e. $Y\in T_{\beta}$. So $X\subset T_{\beta}$ and
$X\in T_{\beta+1}$, i.e. $R_{T}(X)\leqslant\beta+1<\alpha$, contradiction.

(ii) Suppose $R_{T}(X)=\alpha$ is a limit ordinal. Then by lemma 4.16,
$X=\,\\{\ast G_{\gamma},\\{\ast G_{\gamma-1},\cdots\\}\cdots\\}$, where
$G_{\xi}\in\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}$ for
$\xi\leqslant\gamma$. Let $Y=\\{\ast G_{\gamma-1},\\{\ast
G_{\gamma-2},\cdots\\}\cdots\\}$. Then $Y$ is $\omega-$invariant and by
corollary 4.26, $R_{T}(Y)=\alpha$. Furthermore, since
$G_{\gamma}\in\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}$,
$R_{T}(G_{\gamma})<\alpha$. So by lemma 4.23, for any $a\in G_{\gamma}$,
$R_{T}(a)<\alpha$, i.e. $Y$ is the only one in $X$ that $R_{T}(Y)=\alpha$.

###### Corollary 4.30

Suppose $X\in T$.

  1. (i)

If $R_{T}(X)$ is a successor ordinal, then

$R_{T}(X)\,=\,\sup\\{R_{T}(Y)\colon Y\in X\\}\,+\,1$

  2. (ii)

If $R_{T}(X)$ is a limit ordinal and $X$ is $\omega-$invariant, then

$R_{T}(X)\,=\,\sup\\{R_{T}(Y)\colon Y\in X\\}$

Proof. (i) By lemma 4.24, for any $Y\in X$, $R_{T}(Y)\leqslant R_{T}(X)-1$.
Then it follows by theorem 4.29(i).

(ii) By theorem 4.29(ii).

###  4.3 Set Theory for Total Universe

In this section, we will introduce an expanded theory of Zermelo-Fraenkel set
theory known as EZF to handle both the well-founded and non-well-founded sets.
The language of EZF is the language of ZF with the primitive symbol of
“$\in$”. The axioms of EZF are similar to those of ZF [4] with modifications
in the axioms of extensionality and union. The main result is that the total
universe is a model of EZF. First, we list the axioms of EZF as follows.

  1. I.

Extensionality.

  2. (i)

If $X$ and $Y$ are IGS (an IGS $A\left(\mathcal{G}_{A}\right)$ is generated by
$\mathcal{G}_{A}\,=\,\\{G_{\gamma}^{A}\colon G_{\gamma}^{A}\in S$,
$\gamma<\omega\\}$), then181818This is the same as axiom 4.11.

$\left(\forall\gamma<\omega\right)\left(G_{\gamma}^{X}\,=\,G_{\gamma}^{Y}\right)\>\Longrightarrow\>X(\mathcal{G}_{X})\,=\,Y(\mathcal{G}_{Y})$

  3. (ii)

If $X$ and $Y$ are any other sets, then

$\forall z\left(z\in X\Longleftrightarrow z\in Y\right)\,\Longrightarrow\,X=Y$

  4. II.

Non-well-foundedness. Suppose $\alpha\geqslant\omega$ is a limit ordinal and
$\alpha_{0}$ is the limit ordinal immediately below $\alpha$. Then

$\exists
X\left(\begin{array}[c]{c}\underset{\alpha_{0}\leqslant\gamma\leqslant\alpha}{\displaystyle\bigwedge}\exists\,X_{\gamma}\,\exists\,X_{\gamma-1}\left(X_{\gamma-1}\in
X_{\gamma}\right)\,\wedge\underset{\gamma<\alpha_{0}}{\displaystyle\bigwedge}\exists\,X_{\gamma}\,\exists\,X_{\gamma-1}\left(X_{\gamma-1}\in
X_{\gamma}\right)\,\wedge\,X_{\alpha}=X\end{array}\right)$

  5. III.

Union. Suppose ${\displaystyle\bigcup}^{\alpha}X$ is given in definition 4.18.
Then for any $\alpha\geqslant 1$

$\forall X\,\exists Y\left(Y=\,{\displaystyle\bigcup}^{\alpha}X\right)$

  6. IV.

Pairing.

$\forall x\,\forall y\,\exists S\,\forall z\left(z\in
S\,\Longleftrightarrow\,z=x\vee z=y\right)$

  7. V.

Power Set.

$\forall X\,\exists Y\,\forall y\left(y\in Y\Longleftrightarrow\,y\subset
X\right)$

  8. VI.

Infinity.

$\exists S\left(\varnothing\in S\,\wedge\,(\forall x\in
S)\left(x\cup\\{x\\}\in S\right)\right)$

  9. VII.

Replacement. Suppose $\phi$ is a formula and $p$ is a tuple. Then

$\forall x\,\forall y\,\forall
z\left(\phi(x,y,p)\wedge\phi(x,z,p)\Longrightarrow
y=z\right)\>\Longrightarrow\>\forall X\,\exists Y\,\forall y\left(y\in
Y\Longleftrightarrow\left(\exists x\in X\right)\phi(x,y,p)\right)$

  10. VIII.

Separation. Suppose $\phi$ is a formula and $p$ is a tuple. Then

$\forall X\,\forall p\,\exists Y\,\forall y\left(y\in
Y\Longleftrightarrow\,y\in X\wedge\phi(y,p)\right)$

Since infinitely generated sets (generally) do not have immediate members, the
axiom of extensionality for EZF must be modified to handle IGS. As the result,
I(i) is added to decide if two IGS are equal. For all other sets, the axiom of
extensionality in ZF applies. II means that there are sets in EZF that have
infinite $\in-\hskip 1.4457pt$sequences (non-well-founded sets). The axiom of
regularity is dropped in EZF because it can not even define the well-founded
sets (corollary 5.2).

Since non-well-founded sets contain infinite $\in-\hskip 1.4457pt$sequences,
the union axiom in ZF must be modified as well. III is based on definition
4.18 and means that the result of any union operation is a set. The rest
axioms (IV,$\cdots$,VIII) of EZF remain the same as those of ZF. Consequently,
EZF is essentially ZF minus the axiom of regularity. We have the following
conclusion.

###### Corollary 4.31

$T$ is a model of EZF.

Proof. Extensionality. All sets in $T$ are either WF or NWF. For a NWF set, it
either has immediate members or not. I(ii) applies for all WF and NWF sets
with immediate members. For two IGS without immediate members, I(i) can decide
if they are equal by checking the equality of their generators, which can be
either WF or NWF sets at lower ranks. Thus by transfinite recursion, I holds
for all sets in $T$.

Non-well-foundedness. By theorem 3.16, $I_{\omega}$ contains an infinite
$\in-\hskip 1.4457pt$sequence. (By corollary 3.23, $I_{\omega}$ also fails the
axiom of regularity.) Thus II holds in $T$.

Union. Suppose $X\in T$ and $R_{T}(X)=\gamma$. If $\gamma$ is a successor
ordinal, $X\subset T_{\gamma-1}$. By corollary 4.21 and theorem 4.29, for any
$y\in X$ and $z\in y$, $z\in T_{\gamma-1}$ and so ${\displaystyle\bigcup}X\in
T_{\gamma}$. Suppose for any $\beta<\alpha$,
${\displaystyle\bigcup}^{\beta}X\in T_{\gamma}$. By definition 4.18,
${\displaystyle\bigcup}^{\alpha}X\in T_{\gamma}$. If $\gamma$ is a limit
ordinal, suppose $X$ is $\omega-$invariant. By lemma 4.16,
${\displaystyle\bigcup}^{\alpha}X\in T_{\gamma+\omega}$. In both cases, we
have ${\displaystyle\bigcup}^{\alpha}X\in T$. Thus III holds in $T$.

Pairing. For any sets $a,b\in T$, $a,b\in T_{\alpha}$. So $\\{a,b\\}\subset
T_{\alpha}$ and $\\{a,b\\}\in T_{\alpha+1}$. Thus IV holds in $T$.

Power set. For any set $X\in T$, $X\in T_{\alpha}$. By corollary 4.21,
$X\subset T_{\alpha}$. So for any $Y\subset X$, $Y\subset T_{\alpha}$ and
$Y\in\mathcal{P}(T_{\alpha})=T_{\alpha+1}$. Thus $\mathcal{P}(X)\subset
T_{\alpha+1}$ and $\mathcal{P}(X)\in T_{\alpha+2}$, and V holds in $T$.

Infinity. Since $\omega\in T$, VI holds in $T$.

Replacement. Suppose $f=\\{\left(x,y\right)\in T\colon\phi(x,y,p)\\}$. Then
the first part of VII implies $f$ is a function. Let $Y=f(X)$. Then there is a
(least) $\alpha$ that $f\in T_{\alpha}$ and $X\in T_{\alpha}$. So
$f(X)=\\{f(x)\colon x\in X\\}\in T_{\alpha}$. Thus

$\forall y\left(y\in Y\,\Longleftrightarrow\,\left(\exists x\in
X\right)\left(\left(x,y\right)\in f\right)\,\Longleftrightarrow\,\left(\exists
x\in X\right)\left(\phi(x,y,p)\right)\right)$

And VII holds in $T$.

Separation. Let $\varphi(x,y,p)=\left(x=y\wedge\phi(x,p)\right)$. Clearly,
$\varphi$ is a functional formula. So for $X\in T$, by VII,
$Y=\\{y\colon\left(\exists x\in X\right)\varphi(x,y,p)\\}\in T$ and

$\forall y\left(y\in Y\,\Longleftrightarrow\,\left(\exists x\in
X\right)\left(x=y\wedge\phi(x,p)\right)\,\Longleftrightarrow\,\left(y\in
X\wedge\phi(y,p)\right)\right)$

Thus VIII holds in $T$.

###  4.4 Solution to Russell’s paradox

In this section, we will show that the total universe is free of Russell’s
paradox. First, we have the following conclusion.

###### Lemma 4.32

  1. (i)

$\left.T\right|_{\mathfrak{S}}\,=\underset{\alpha\in
Ord}{\displaystyle\bigcup}\left.T_{\alpha}\right|_{\mathfrak{S}}$

  2. (ii)

$\left.T\right|_{\mathfrak{Q}}\,=\underset{\alpha\in
Ord}{\displaystyle\bigcup}\left.T_{\alpha}\right|_{\mathfrak{Q}}$

Proof. By (4.1), corollary 3.31(v), 4.22(ii) and 3.39(v).

###### Definition 4.33

The infiniton class $\mathcal{I}$ of the total universe includes all the semi-
infinitons and quasi-infinitons in $T$. The non-infiniton class $\mathcal{N}$
of $T$ is the complement of $\mathcal{I}$ in $T$, i.e.
$\mathcal{N}=T-\mathcal{I}$.

###### Theorem 4.34

Suppose $X\in T$.

  1. (i)

If $X\in\mathcal{I}$, then $R_{T}(X)$ is a limit ordinal.

  2. (ii)

If $R_{T}(X)$ is a successor ordinal, then $X\in\mathcal{N}$.

Proof. (i) Suppose $X\in\mathcal{I}$ and $R_{T}(X)=\alpha$ is a successor
ordinal. Then $X\in T_{\alpha}=\mathcal{P}(T_{\alpha-1})$ and $X\subset
T_{\alpha-1}$. If $X\in X$, then $X\in T_{\alpha-1}$ and
$R_{T}(X)\leqslant\alpha-1<\alpha$, contradiction.

If $X$ is a quasi-infiniton, suppose $X\in Y_{1},Y_{1}\in
Y_{2},\cdots,Y_{n}\in X$. Then by lemma 4.24, $R_{T}(Y_{n})<R_{T}(X)$. Thus by
lemma 4.23, $R_{T}(X)\leqslant R_{T}(Y_{1})\leqslant\cdots\leqslant
R_{T}(Y_{n})<R_{T}(X)$, contradiction again.

(ii) If $X\notin\mathcal{N}$, then $X\in\mathcal{I}$. By (i), $R_{T}(X)$ is a
limit ordinal.

From theorem 4.34, lemma 1.2 and 1.5, we can see that no well-founded sets are
in the infiniton class, i.e. no well-founded set is a member of itself or
contains a vicious cycle.

###### Corollary 4.35

$V\cap\mathcal{\,I}\,=\,\varnothing$

###### Corollary 4.36

Suppose $X\in T$. Then

  1. (i)

$X$ is not a semi-infiniton except $X\in\left.T\right|_{\mathfrak{S}}$.

  2. (ii)

$X$ is not a quasi-infiniton except $X\in\left.T\right|_{\mathfrak{Q}}$.

Proof. We prove by transfinite induction.

(i) First, for any $X\in V_{\omega}$, $R_{T}(X)<\omega$. So by theorem
4.34(ii), $X\notin X$. Suppose it is true for $R_{T}(X)<\alpha$. Then for
$R_{T}(X)=\alpha$, if $\alpha$ is a successor ordinal, $X\notin X$. If
$\alpha$ is a limit ordinal, by corollary 4.25,
$X\in\left.\left(\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\right)\right|_{\mathcal{\aleph}_{0}}$.
Thus $X\in X$ if and only if
$X\in\left.\left(\underset{\beta<\alpha}{\displaystyle\bigcup}T_{\beta}\right)\right|_{\mathfrak{S}}\subset\left.T\right|_{\mathfrak{S}}$.

(ii) is similar to (i).

###### Remark 4.37

Corollary 4.36 shows that all semi-infinitons and quasi-infinitons in $T$ are
infinitely generated. Thus all semi-infinitons and quasi-infinitons in $T$ are
precisely determined.

###### Corollary 4.38

  1. (i)

$\mathcal{I}\,=\,\left.T\right|_{\mathfrak{S}}\cup\left.T\right|_{\mathfrak{Q}}$

  2. (ii)

$\mathcal{I}\,\subset\underset{\alpha\in
LOrd}{\displaystyle\bigcup}\,\,\underset{\beta<\alpha}{\displaystyle\bigcap}\left(T_{\alpha}-T_{\beta}\right)\,\subset\,T$

  3. (iii)

$T\,=\,\mathcal{N}\cup\,\mathcal{I}$ and
$\mathcal{N}\cap\,\mathcal{I}\,=\,\varnothing$

  4. (iv)

$\mathcal{N}\,=\underset{\alpha\in
Ord}{\displaystyle\bigcup}\left(T_{\alpha}-\mathcal{I}\right)\,=\underset{\alpha\in
Ord}{\displaystyle\bigcup}\mathcal{N}_{\alpha}$

  5. (v)

$\mathcal{I}$ contains no ordinals. $\mathcal{N}_{\alpha}$ contains all
ordinals less than $\alpha$. $\mathcal{N}$ contains all ordinals.

Proof. (i) By definition 4.33 and corollary 4.36.

(ii) By theorem 4.28 and 4.34.

(iii) By definition 4.33.

(iv) By (iii) and (4.1).

(v) Since all ordinals are WF and all sets in $\mathcal{I}$ are NWF,
$\mathcal{I}$ contains no ordinals. So
$\mathcal{N}_{\alpha}=T_{\alpha}-\mathcal{I}$ and $\mathcal{N}$ contain the