00footnotetext: Version 2022-08-08_2. See https://shelah.logic.at/papers/E40/
for possible updates.

# A collection of abstracts of Shelah’s Papers
E40

Saharon Shelah  Einstein Institute of Mathematics
Edmond J. Safra Campus, Givat Ram
The Hebrew University of Jerusalem
Jerusalem, 9190401, Israel
and
Department of Mathematics
Hill Center - Busch Campus
Rutgers, The State University of New Jersey
110 Frelinghuysen Road
Piscataway, NJ 08854-8019 USA<EMAIL_ADDRESS>http://shelah.logic.at

(Date: August 8, 2022)

###### Abstract.

There are here abstracts of most of the papers up to 143 (and 217), mostly
written in 1980/81 with Grossberg. Also more details than in the originals
were added in B5, B8, B217 and C2, C3 were added (the Cxx are representations
of the authors works).

###### Key words and phrases:

model theory, set theory

###### 2010 Mathematics Subject Classification:

Primary: 03Cxx, 03Exx; Secondary: 03-00

The author thanks Alice Leonhardt for the beautiful typing. References like
[She, Th0.2=Ly5] means the label of Th.0.2 is y5. The reader should note that
the version in my website is usually more updated than the one in the
mathematical archive. First typed May 1987

##  1\. Notation

  1. (A)

    1. (a)

$\alpha,\beta,\gamma,\delta,\zeta,\xi,i,j$ denote ordinals; usually $\delta$
denotes a limit ordinal, $\omega$ the first infinite ordinal

    2. (b)

$\lambda,\chi,\mu,\kappa$ stands for infinite cardinals

    3. (c)

$n,m,k,l$ natural numbers

    4. (d)

$\lambda^{\kappa}$ is cardinal exponentiation,
$\lambda^{<\kappa}=\sum\limits_{\mu<\kappa}\lambda^{\mu}$,
$\beth(\lambda,\alpha)=\sum\limits_{\beta<\alpha}2^{\beth(\lambda,\beta)}+\lambda$,
$\beth_{\alpha}=\beth(\aleph_{0},\alpha)$

    5. (e)

${}^{\alpha}\lambda=\\{f:if:\alpha\rightarrow\lambda\\}$ sometimes we call the
member sequences

    6. (f)

${}^{\alpha>}\lambda=\bigcup\limits_{\beta<\alpha}{}^{\beta}\lambda$

    7. (g)

$\eta,\nu$ stands for sequences (or ordinals, usually)

    8. (h)

GCH is the Generalized Continuum Hypothesis

  2. (B)

    1. (a)

$M,N$ (perhaps with index) are models

    2. (b)

${\rm Th}(M)$ \- the collection of first order sentences which are true in
$M$. All languages are with equality.

    3. (c)

a model is $\lambda$-like if it has cardinality $\lambda$ and one of the
relations of the model is an order $<$ such that every initial segment has
cardinality less than $\lambda$, but not the whole.

    4. (d)

vocabularies are denoted by $\tau$.

##  2\. abstracts

(B1) Stable Theories, IJM 7(1969), 186-202.

The stability spectrum (i.e. the set of cardinals in which a theory $T$ is
stable) is characterized for countable theories and investigated in the
general case; ranks for one formula are investigated, as well as existence of
indiscernibles and prime among e.g. $|T|^{+}$-saturated models. Also
categoricity of elementary and of pseudo-elementary classes is investigated
and lower bound on the number of non-isomorphic models of power
$\aleph_{\alpha}$ is given for unstable and unsuperstable $T$ (they are
$|\alpha-\beta|$, $|(\alpha-\beta)/\omega|$ respectively, where
$|T|=\aleph_{\beta}$).

(B2) Note on a Min-Max Problem of Leo Moser, J. Comb. Th. A 6(1969), 187-202

Moser asks how a pair of $(n+1)$-sided dice should be loaded (identically) so
that on throwing the dice the frequency of the most frequently occurring sum
is as small as possible. G.F. Clements finds a relative minimum, conjecturing
that it is always the solution. This conjecture is disproved for $n=3$.

(B3) Finite diagrams stable in power, AML 2(1970), 69-118.

Let $T$ be a complete first order theory, $D_{m}(T)$ the set of complete
$m$-type (consistent with $T$), $D(T)=\bigcup\limits_{m}D_{m}(T)$ and let
$D\subseteq D(T)$. Let

$\begin{array}[]{clcr}K_{D}=\\{M:&M\text{ is a model of }T\text{ such that if
}m<\omega\text{ and }\bar{a}\in{}^{m}M,\\\ &\text{ then the type of
}\bar{a}\text{ (in }M\text{ of course) belongs to }D\\}.\end{array}$

Assume $\mathfrak{C}_{D}$ is a $D$-monster, that is: it is a model of $T$, in
$K_{D}$ and is a $\bar{\kappa}$-sequence homogeneous. We develop stability
theory for it.

Anotated content

§1

  1. [Let $T$ be a complete first order theory, ${\mathfrak{C}}$ its monster models $M,N$ are $\prec\mathfrak{C}$, $A,B$ are $\subseteq\mathfrak{C}$. Let $D(T)$ be the set of complete $L(T)$-types realized in models of $T$ and let $D$ be a subset of $D(T)$. We call $A$ a $D$-set if every finite sequence from $A$ realizes (in $\mathfrak{C}$) a type from $D$, $D(A)$ the minimal such $D$. For a $D$-set let $\mathbf{S}^{n}_{D}(A)$ be the set of $p\in\mathbf{S}^{n}(A)$ such that if $\bar{a}=\langle a_{\ell}:\ell<n\rangle\in{}^{n}{\mathfrak{C}}$ realizes $p$ then $A\cup\bar{a}$ is a $D$-set. $M$ is called a $D$-model if $(M\prec{\mathfrak{C}}$ and) $D(M)\subseteq D$; if $n=1$ we may omit. We say $M$ is $(D,\lambda)$-homogeneous if (it is a $D$-model and) $A\subseteq M$ and $|A|<\lambda$ and $p\in\mathbf{S}_{D}(A)\Rightarrow p$ realized in $M$ (if $D=D(T)$ this means $M$ is $\lambda$-saturated). Let $D$ be fixed, all sets are $D$-sets, all models are $D$-models.]

§2

  1. [We call $D$ $\lambda$-good if for every $\lambda$ there is $(D,\lambda)$-homogeneous model of cardinality $\geq\lambda$. We call $D$ $\lambda$-stable if it is $\lambda^{+}$-good and for every $D$-set $A$ of cardinality $\leq\lambda$, $|A|\leq\lambda\Rightarrow|\mathbf{S}_{D}(A)|\leq\lambda$. We say $D$ is stable if it is stable in some $\lambda$. We say $p\in\mathbf{S}^{n}_{D}(A)$ split over $B\subseteq A$ if $0$ for $\bar{a},\bar{b}\subseteq A$ realizing the same type over $B$ and $\varphi$, $\varphi(\bar{x},\bar{a})\equiv\varphi(\bar{x},\bar{b})\in p$. Basic properties of splitting are proved. E.g. if $|\mathbf{S}_{D}(A)|>\mu_{0}=|A|^{<\lambda}+\sum\limits_{\mu<\lambda}2^{|D|^{\mu}}$ then we an find $p\in\mathbf{S}_{D}(A)$ and an increasing sequence $\langle A_{i}:i<\lambda\rangle$, $|A_{i}|<\aleph_{0}+|i|^{+}$ such that $p\restriction A_{i+1}$ split over $A_{i}$, this in turn implies unstability in every $\mu<2^{\lambda}$. Hence, if $D$ is unstable in every $\lambda<\beth\big{(}(2^{|T|})^{+}\big{)}$ then $D$ is unstable (in fact have an indiscernible sequence (which forms a $D$-set and) which ordered by some formula).]

§3

  1. [If $D$ is stable, then $D$ is good, and if $B\subseteq A$, $p\in\mathbf{S}^{n}_{D}(B)$ then $p$ can be extended to some $q\in\mathbf{S}^{n}_{D}(A)$. We also prove that if $D$ is stable in $\lambda$, $A\cup\bigcup\\{\bar{a}_{i}:i<\lambda^{+}\\}$ is a $D$-set, each $\bar{a}_{i}$ a finite sequence then for some $X\subseteq\lambda^{+}$, $|X|=\lambda^{+}$, we have $\langle\bar{a}_{i}:i\in X\rangle$ is an indiscernible set over $A$.]

§4

  1. [We say $p\in\mathbf{S}_{D}(A)$ splits strongly over $B\subseteq A$ if for some $C$, $\langle\bar{a}_{i}:i<\omega\rangle$ we have $A\cup\bigcup\limits_{i<\omega}\bar{a}_{i}\subseteq C$ (the point is that $C$ is a $D$-set) $\langle\bar{a}_{i}:i<\omega\rangle$ is an indiscernible sequence (equivalently set by §3) over $A$ and for some $\psi$, $\psi(\bar{x},\bar{a}_{0}),\neg\psi(\bar{x},\bar{a}_{i})\in p$. We investigate it and using it characterize the stability spectrum. I.e. if $D$ is stable, then for some cardinal $\lambda=\lambda(D)$, $\kappa=\kappa(D)$ which are $<\beth\big{(}2^{|T|})^{+}\big{)}$ we have: $D$ is $\mu$-stable iff $\mu\geq\lambda(D)$ and $\mu=\mu^{<\kappa(D)}$. Also, a formula cannot divide an indiscernible set to two large pieces (i.e. of size $\geq\kappa(D)$).]

§5

  1. [Some variants of $\lambda$-isolation are defined and so $\lambda$-prime models over $(D)$-sets are proved to exist using what is called in [She78, Ch.IV] primary models are defined; in fact, the framework of it applies here (including uniqueness of primary).]

§6

  1. [We prove that there is a $(D,\lambda)$-homogeneous non-$(D,\lambda^{+}$)-homogeneous models in all but “degenerate” cases (parallel to unidimensional $T$).]

§7

  1. [We prove the parallel of “characterizing the categoricity spectrum replacing categorical by all models are homogeneous, for class of the form $\\{M:M$ a model of $T$ omitting every type in a set $\Gamma\\}$.]

We prove: e.g. if $T$ is first order complete countable not categorical in
$\aleph_{2}$ then it has $\geq|\alpha+1|$ non-isomorphic models in
$\aleph_{0}$.

See [She75e] on the spectrum of $\\{\lambda$ : there is a
$(D,\lambda)$-homogeneous model of cardinality $\lambda\\}$.

(B4) On Theories $T$ categorical in $|T|$, JSL 35(1970),73-82.

We prove that, if $T$ is categorical in $|T|$, $|T|=|T|^{\aleph_{0}}$, then
$T$ has a model of power $<|T|$; moreover, $T$ is a definitional extension of
a theory of smaller power. We first note that $T$ has a $\aleph_{1}$-compact
model, and if $T$ is unsuperstable also a non $\aleph_{1}$-compact model of
power $\aleph_{1}$. So $T$ is superstable. We construct a model which is
“almost prime” over a countable indiscernible set, and prove it is maximal in
it, but we can easily build a model without such set.

Superseded by [She78, Ch.IX].

(B5) On languages111Based on part of the author’s M.Sc.Thesis and the summary
here says somewhat more on it. with non-homogeneous strings of quantifiers,
IJM 8(1970),75-79.

In infinitary logic we can introduce a string of quantifiers of the form

$(\ldots Q_{i}x_{i}\ldots Q_{j}x_{j}\ldots)_{i\in I}\varphi(\bar{x})$

$I$ a linear order, $Q_{i}$ is $\exists$ or $\forall$. For $I$ well ordered
the interpretation is by a game, generally by Skolem functions. The theorem is
that we can express such quantifiers by a well ordered sequence of quantifiers
(not so short; and propositial connectives). In fact, only the case when $I$
an inverse of a well ordered set is presented in [She70]; the full proof
appears in the author M. Sc. thesis.

(In somewhat revised formulation this was sent to a student of Takeuli and
included in his thesis, Urbana 1973).

More formally

######  Definition 2.1.

(Mostowski). The meaning of

$M\models(\ldots,(Q_{t}\bar{x}_{t}),\ldots)_{t\in
I}\varphi(\ldots,\bar{x}_{t},\ldots)$

where $I$ is a linear order, $\bar{x}_{t}=\langle
x_{t,\alpha}:\alpha<\alpha_{t}\rangle$, $Q_{t}\in\\{\forall,\exists\\}$,
letting

$I_{\exists}=\\{t:Q_{t}=\exists\\},\ I_{\forall}=\\{t\in I:Q_{t}=\forall\\}$

is that there are functions
$F_{t,\alpha}=F_{t,\alpha}(\ldots,x_{s,\beta},\ldots)_{s\in I_{\forall},\
s<t,\ \beta<\alpha_{s}}$ for $t\in I_{\exists}$, $\alpha<\alpha_{t}$ such that
for every $a_{s,\beta}\in M$ (for $s\in I_{\forall}$, $\beta<\alpha_{s}$);
letting $a_{t,\alpha}=F_{t,\alpha}(\ldots,a_{s,\beta},\ldots)_{s\in
I_{\forall},\ s<t,\ \beta<\alpha_{s}}$ we have $M\models\varphi[\ldots,\langle
a_{t,\gamma}:\gamma<\alpha\rangle,\ldots]_{t\in I}$.

######  Theorem 2.2.

1) The formula $(\ldots,(Q_{t}\bar{x}_{t})\ldots)_{t\in
I}\varphi(\ldots,\bar{x}_{t},\ldots)$ is equivalent to the formula $\psi^{*}$
defined below.

Let

  1. (A)

${\mathscr{T}}=\\{\bar{J}:\bar{J}=\langle J_{i}:i\leq j\rangle$ is a strictly
$\subseteq$-increasing continuous sequence of proper initial segments of $I$,
$J_{0}=\varnothing\\}$

  2. (B)

For $\bar{J}=\langle J_{i}:i\leq j\rangle$ let $t\in\bar{J}$ mean $t\in
J_{j}\setminus\bigcup\limits_{i<j}J_{i}$ and let $\bar{J}_{1}\leq\bar{J}_{2}$
mean $\bar{J}_{1}$ is an initial segment of $\bar{J}_{2}$

  3. (C)

For $\bar{J}=\langle J_{i}:i\leq j\rangle\in\mathscr{T}$ let

$\bar{y}_{\bar{J}}=\langle\ldots,x_{s,\beta},\ldots\rangle_{s\in\bar{J},\beta<\alpha_{s},Q_{s}=\forall}$

$\bar{z}_{\bar{J}}=\langle\ldots,x_{s,\beta},\ldots\rangle_{s\in\bar{J},\beta<\alpha_{s},Q_{s}=\exists}.$

  4. (D)

Lastly, define the formula
$\psi^{*}=(\ldots,(\forall\bar{y}_{\bar{J}})(\exists\bar{z}_{\bar{J}}),\ldots)_{\bar{J}\in\mathscr{T}}\langle\varphi(\ldots,\bar{y}_{\bar{J}_{1}},\bar{z}_{\bar{J}_{1}},\ldots)_{\bar{J}_{1}\leq\bar{J}_{2}}:\bar{J}\in\mathscr{T}\rangle$
which means: as explained below

  5. (E)

in the following game between the player $\exists$ and $\forall$, the player
$\exists$ has a winning strategy;

    1. (a)

in stage $i$ of a play an initial segment $J_{i}$ of $I$, strictly increasing
with $i$ is constructed and a sequence
$\langle\ldots,a^{i}_{s,\beta},\ldots\rangle_{s\in I,\beta<\alpha_{s}}$ are
constructed such that

      1. $(*)$

$s\in J_{i_{0}},i_{0}<i_{1}\Rightarrow
a^{i_{0}}_{s,\beta}=a^{i_{1}}_{s,\beta}$

    2. (b)

in stage $i$, the player $\forall$ chooses $a^{i}_{s,\beta}(s\in
I,\beta<\alpha_{s},Q_{s}=\forall)$ but satisfying $(*)$, then player $\exists$
chooses $a^{i}_{s,\beta}(s\in I,\beta<\alpha_{s},Q_{s}=\exists)$ but
satisfying $(*)$, and then player $\forall$ chooses $J_{j}$ as required. If in
stage $\beta$ $M\models\neg\varphi(\ldots,a^{i}_{s,\beta},\ldots)$, player
$\exists$ loses immediately. If $\exists$ never loses, he wins.

2) Formulas as in (1) (with tree quantifiers indexed by ${\mathscr{T}}$ a well
founded tree) can be translated to one of the form
$(\forall\bar{y}^{0})(\exists\bar{z}^{0})\ldots(\forall\bar{y}^{\alpha})(\exists\bar{z}^{\alpha})\ldots)_{\alpha<\gamma}\bigwedge\limits_{\Upsilon\in\mathscr{T}}\varphi_{\Upsilon}(\bar{x}^{\Upsilon})$
where $\gamma$ is the number of levels of the tree (so in our case, every
increasing sequence of initial segments has length $<\gamma$) and
$\varphi_{\Upsilon}$ the formula “sitting” in the node $\Upsilon$ (we need
$\bar{x}^{\Upsilon}$ not to repeat variables appearing in incomparable nodes
of the tree, in our case all are the same $\varphi$). In fact
$\gamma^{*}\geq{\rm otp}(X)$ for every branch of the tree ${\mathscr{T}}$,
then $\gamma^{*}$ can serve and if $\alpha<\gamma^{*},\ell
g(\bar{y}^{\alpha})=\Sigma\\{|\ell g(\bar{y}_{t})|:t\in{\mathscr{T}}\text{ of
level }\alpha\\},\ell g(\bar{z}^{\alpha})=\Sigma|\ell
g(\bar{z}_{t})|:t\in{\mathscr{T}}\text{ of level }\alpha\\}$ are O.K.

###### Proof..

Straightforward, we may wonder: in the new formulas the
“$(\exists\bar{z}^{\alpha})$ may depend on additional variables but because fo
the structure of
$\bigwedge\limits_{\Upsilon\in\mathscr{T}}\varphi_{\beta}(\bar{x}^{\Upsilon})$
this is not a problem. ∎

Another result from the M.Sc. thesis is:

######  Theorem 2.3.

[Hanf number of $\mathbb{L}_{\lambda^{+},\lambda^{+}}$ with unary function
only.]

Let the vocabulary $\tau$ be $\\{F\\}\cup\\{P_{i}:i<\lambda\\}$ where $F$ is a
unary function and the $P_{i}$ are unary predicates.

  1. (A)

if $\psi\in\mathbb{L}_{\lambda^{+},\lambda^{+}}$ has models in arbitrarily
large cardinality $<\beth_{\lambda^{+}}$ (in fact $>\beth_{{\rm
q.d.}}(\psi)+1(\lambda)$ suffice), then it has models in arbitrarily large
cardinalities (in fact, any one above $\beth_{{\rm q.d.}}(\psi)+1(\lambda)$)

  2. (B)

if $T\subseteq\mathbb{L}_{\lambda^{+},\lambda^{+}}$ has a model of cardinality
$>\beth_{\lambda^{+}}$, then it has models in arbitrarily large cardinalities.

###### Proof..

For an $L$-model $M$ and $x\in M$ let

$M^{[x]}=(M\restriction\\{y:\text{for some }n<\omega,F^{n}(y)=x\\},x)$

(so $F^{M[x]}$ may be partial and $x$ serve as an individual constant) where
$F^{0}(x)=x$, $F^{n+1}(x)=F(F^{n}(x))$ and let $M^{\langle
x\rangle}=M\restriction\\{y:y\in\bigcup\limits_{n<\omega}M^{F^{n}(x)}\\}$. Let
$\mathrm{Pre}_{M}(x)=\\{y:F^{M}(y)=x\\}$.

Why (A)? Assume for simplicity that q.d.$(\psi)$ is a limit ordinal. If $M$ is
a model of $T$ of cardinality $>\beth_{\text{q.d.}(\psi)}(\lambda)$ then one
of the following occurs:

  1. $(\alpha)$

the number of components (i.e. $M^{\langle x\rangle},x\in M$) is
$>\beth_{\text{q.d.}(\psi)+1}$. So for some $x$, for each $\beta<\text{
q.d.}(\psi)$ the set $\\{M^{\langle y\rangle}:M^{\langle y\rangle},M^{\langle
x\rangle}$ satisfies the same $L_{\infty,\lambda^{+}}$-formulas of quantifier
depth $\leq\beta\\}$ has cardinality $>\lambda$. So we can add to $M$ any
number of isomorphic copies of $M^{\langle x\rangle}$

  2. $(\beta)$

for some $x\in M,\beth_{\text{q.d.}(\psi)+1}<|\text{Pre}_{M}(x)|$ so for some
$y\in\text{ Pre}_{M}(x)$, for every $\beta<\text{ q.d.}(\psi)$, we have
$\\{z\in\text{ Pre}_{M}(x):M^{[z]},M^{[x]}$ satisfies the same
$L_{\infty,\lambda^{+}}$ formulas of quantifier depth $\leq\text{
q.d.}(\psi)\\}$ has cardinality $>\beth_{\text{q.d.}(\psi)}$. So we can just
“increase” $\text{Pre}_{M}(x)$, giving more copies of $M^{\langle x\rangle}$.

Why (B)? Similarly. ∎

(B6) A note on Hanf numbers, PJM, 34(1970),539-543.

We show that for every $\xi<(2^{\kappa})^{+}$, there is a theory $T$ and set
of type $\Gamma$ in a language of power $\kappa$, such that there is a model
of $T$ which omits every $p\in\Gamma$ of power $\lambda$ if and only if
$\lambda\leq\beth_{\xi}$. This completes the computation of an appropriate
Hanf number. We also disprove a conjecture of Morley on the existence of
algebraic elements in such examples.

(B7) On the cardinality of ultraproduct of finite sets, JSL, 35(1970), 83-84.

We prove that if $D$ is an ultrafilter and $\aleph_{0}\leq\lambda=\Pi
n_{i}/D$, then $\lambda^{\aleph_{0}}=\lambda$. The method is to use the amount
of number theory $N=\Pi n_{i}/D$ inherit and $\aleph_{1}$-saturation of $N$.

(B8) Two cardinal compactness, IJM 9(1971), 103-198.

Let $K$ be the class of $(\lambda,\mu)$-models, i.e. models such that
$\|M\|=\lambda$, $|P^{M}|=\mu$. We prove that when $\mu^{\aleph_{0}}=\mu$, $K$
is $\mu$-compact (i.e. a theory $T$ of cardinality $\leq\mu$ has a
$(\lambda,\mu)$-model iff every finite subtheory has a $(\lambda,\mu)$-model).

In fact $\aleph_{0}$-compactness implies $\mu$-compactness, see the abstract
in the Notices of the AMS, [She71b]. We also prove that then any $T$,
$|T|\leq\mu$ which has a $(\lambda,\mu)$-model, has a $(\lambda,\mu)$-model in
which only $\leq 2^{\aleph_{0}}+|T|$ types are realized (e.g. if
$|T|=\aleph_{0}$, and the class of $(\lambda,\mu,\aleph_{0})$-models is
compact, we can get $\aleph_{0}$ types). Also we can get models with many
automorphisms (by a sequence of $\mu$ indiscernibles).

Similar theorems hold for $\mu$-like models, hence cardinality quantifiers,
and even for several such demands.

In the cases that $\aleph_{0}$-compactness holds, transfer theorems are
reduced to partition theorems with finite conclusions.

The models are like incomplete Ehrenfeucht-Mostowski ones, but the
indiscernibility is not “full”, just as much as corresponding to the
appropriate partition theorems.

(B9) Remark to “local definability theory of Reyes”, AML 2(1971), 441-448.

Let $\tau\subseteq\tau_{1}$, $P\in\tau_{1}\setminus\tau$ and $T$ a theory in
$\tau_{1}$. For a model $M$ let

${\rm df}(M)=:\big{|}\\{P:(M,P)\text{ is a reduct of a model of }T\\}\big{|}$

and let ${\rm Df}(\lambda):=\sup\\{{\rm df}(M)^{+}:\|M\|=\lambda\\}$.

Then the following are equivalent:

  1. (i)

for no formula $\theta(\bar{x},\bar{y})\in\tau$,
$T\vdash(\exists\bar{y})(\forall\bar{x})\big{[}P(\bar{x})\equiv\theta(\bar{x},\bar{y})\big{]}$

  2. (ii)

for some $\lambda\geq|\tau_{1}|$, ${\rm Df}(\lambda)>\lambda^{+}$,

  3. (iii)

for every $\lambda\geq|\tau_{1}|$, ${\rm Df}(\lambda)\geq{\rm
Ded}^{*}(\lambda)$ (which is the first $\mu$ such that any tree of power
$\lambda$, has $<\mu$ branches of a fixed height).

######  Conjecture 2.4.

If for some $\lambda\geq|\tau_{1}|$, ${\rm Df}(\lambda)>{\rm Ded}(\lambda)$
then for every $\lambda$, ${\rm Df}(\lambda)=(2^{\lambda})^{+}$.

(B10) Stability, the f.c.p. and superstability, AML 3(1971), 271-362.

We investigate in detail stable formulas, ranks of types and their
definability, the f.c.p., some syntactical properties of unstable formulas,
indiscernible sets and degrees of types in superstable theories. There is a
list of all results connected with those properties, or whose proof use them.

This list may still be of value. Superseded by [She78, Ch.II].

(B11) On the number of non-almost isomorphic models of $T$ in a power,
PJM, 36(1971), 811-818.

Let $T$ be a first order theory. Two models are almost isomorphic if they are
elementarily equivalent in the language $\mathbb{L}_{\infty,\omega}$. We
investigate the number of non almost-isomorphic models of $T$ of power
$\lambda$ as a function of $\lambda$, $I(T,\lambda)$. We prove
$\mu>\lambda\geq|T|$ and $I(T,\lambda)\leq\lambda$ implies $I(T,\mu)\leq
I(T,\lambda)$.

We also get downward Lowenheim Skolem theorem for the corresponding variant of
a rigid model.

(B12) The number of non-isomorphic models of an unstable first-order theory,
IJM 4(1971), 473-487.

It is proved that if $T$ is an unstable (first-order) theory,
$\lambda>|T|+\aleph_{0}$, then $T$ has exactly $2^{\lambda}$ non-isomorphic
models of cardinality $\lambda$. In fact we have stronger results: this is
true for pseudo-elementary classes, and for almost every
$\lambda\geq|T|+\aleph_{1}$.

The method is contradictory orders. Not too many Ehrenfeucht Mostowski models
built on pairwise contradictory order are isomorphic. Many pairwise
contradictory orders are constructed, by using stationary sets.

In passing we deal with the existence of a family of $2^{\lambda}$ subsets of
$\lambda$, each of power $\lambda$, the intersection of any two finite, if it
does not exist, our proof works for $\lambda=|T|+\aleph_{1}$, if it does
exist, for some $\aleph_{\alpha}<\lambda$,
$2^{\lambda}=2^{|\alpha|}+2^{\aleph_{0}}$ (in fact it follows that
$\lambda<2^{\aleph_{0}}\vee(\exists\mu,\alpha)$,
$\mu<\alpha=\aleph_{\alpha}\leq\lambda<\mu^{\aleph_{0}}$) and then we use a
different proof.

Superceded by [She87c, Ch.III,§3], even better [Shear, Ch.III,§3].

(B13) Every two elementary equivalent models have isomorphic ultrapowers,
IJM, 10(1971), 224-233.

We prove that every two elementarily equivalent models have isomorphic
ultrapower. In fact if $M,N$ are of cardinality $\leq\lambda$, then there is
such an ultrafilter on $2^{\lambda}$.

Later the author notes that (see Stern, [Ste76]) the parallel theorem for
Banach spaces, and so get Craig interpolation theorem for the suitable logic.

(B14) Saturation of Ultrapowers and Keisler’s Order

Superseded by [She78, Ch.VI], except §1, which gives examples of uncountable
theories.

(B15) Uniqueness and characterization of prime models over sets for totally
transcendental first-order theories, JSL (1972), 107-113.

If $T$ is a complete first-order totally transcendental theory then, over
every $T$-structure $A$, the prime model is unique up to isomorphism over $A$.
Moreover, $M$ is a prime model over $A$ iff:

  1. (a)

every finite sequence for $M$ realizes an isolated type over $A$, and

  2. (b)

there is no uncountable indiscernible set of $A$ in $M$. Hence the uniqueness
of the differential closure of a differential field of characteristic zero
follows.

The proof is by a suitable induction on rank.

(B16) A combinatorial problem: stability and order for models and theories in
infinitary languages PJM 41(1972).

Some infinite combinatorial problems are solved. Their model-theoretic
representation is: if the model $M$ is unstable in $\lambda$,
$\lambda=\lambda^{<\mu}+\sum\limits_{\kappa<\mu}2^{2^{\kappa}}$, then in $M$
there is a set of sequences from $M$, of fixed finite length ordered by one
formula. We then prove that if $M$ is stable in $\lambda$ and has no “large”
ordered set, every set of $>\lambda$ elements contains an indiscernible
subset. We also prove that if $\psi\in\mathbb{L}_{\lambda^{+},\omega}$,
$\varphi(\bar{x},\bar{y})\in\mathbb{L}_{\infty,\omega}$ and $\psi$ has a model
in which $\psi$ orders a set of power $\mu$ for arbitrarily large $\mu$ (or
just $\mu<\beth_{(2^{\kappa})^{+}}$ where
$\psi,\varphi\in\mathbb{L}_{\kappa^{+},\omega}$) then $\psi$ has $2^{\mu}$
non-isomorphic models of power $\mu$ for each $\mu$.

Most model theory is superseded by [She87c, Ch.I] and Theorem 2.5 by [GS86].

(B17) For what filters is every reduced product saturated IJM 12(1972), 23-31.

In this paper we characterize the filters $D$ such that for every sequence
$\langle M_{i}:i<I\rangle$ we have $\prod\limits_{i\in I}M_{i}/D$ is
$\lambda$-saturated (where $\lambda>\aleph_{0}$). The characterization is: $D$
is $\lambda$-good, $D$ is $\aleph_{0}$-incomplete and ${\mathscr{P}}(I)/D$ is
a $\lambda$-saturated Boolean Algebra. (Note that when the Boolean algebra
${\mathscr{P}}(I)/D$ is $\lambda$-saturated it can be characterized directly;
we can also restrict ourselves e.g. to $\Pi_{n}$-formulas only).

(B18) On Models with power-like orderings JSL 37(1972), 247-267.

We prove here theorems of the form: if $T$ has a model $M$ in which $P_{1}(M)$
is $\kappa_{1}$-like ordered, $P_{2}(M)$ is $\kappa_{2}$-like ordered…, and
$Q_{1}(M)$ is of power $\lambda_{1},\dotsc,$ then $T$ has a model $N$ in which
$P_{1}(M)$ is $\kappa^{\prime}_{1}$-like ordered…, $Q_{1}(N)$ is of power
$\lambda^{\prime}_{1},\ldots$ (In this article $\kappa$ is a strong limit
singular cardinal, and $\kappa^{\prime}$ is a singular cardinal). If the
language has power $\chi$, this is written
$\chi:\langle\kappa_{1},\kappa_{2},\dotsc,|\lambda_{1},\ldots\rangle\rightarrow\langle\kappa^{\prime}_{1},\kappa^{\prime}_{2},\dotsc,|\lambda^{\prime}_{1},\ldots\rangle$.

We also sometimes add the condition that $M,N$ omits some types. The results
are seemingly the best possible, i.e. according to our knowledge about
$n$-cardinal problems (or, more precisely, a certain variant of them, speaking
on cofinality, too).

Our method will enable us to reduce such problems to transfer problems which
do not mention $\kappa$-like orderings. In many cases we can translate
problems of the form
$\chi:\langle\kappa_{1},\ldots|\lambda_{1},\ldots\rangle\rightarrow\langle\kappa^{\prime}_{1},\ldots|\lambda^{\prime}_{1},\ldots\rangle$
to problems of the form

$\chi:\langle{\rm
cf}(\kappa_{1}),\dotsc,\lambda_{1},\ldots\rangle\rightarrow\langle{\rm
cf}(\kappa^{\prime}_{1}),\dotsc,\lambda^{\prime}_{1},\ldots\rangle.$

As an example of our results, concerning transfer between pairs of cardinals
we have: if $\chi\leq\lambda^{\prime}<\kappa^{\prime},\chi\leq\lambda<\kappa$,
then
$\chi:\langle\kappa|\lambda\rangle\rightarrow\langle\kappa^{\prime}|\lambda^{\prime}\rangle$
when at least one of the following conditions is satisfied:

  1. (A)

${\rm cf}(\kappa^{\prime})\leq\lambda^{\prime}$,

  2. (B)

${\rm cf}(\kappa)>\beth_{\omega}(\lambda)$,

  3. (C)

${\rm cf}(\kappa)>\lambda$, ${\rm
cf}(\kappa^{\prime})=(\lambda^{\prime})^{+}$,
$\lambda^{\prime}=\sum\limits_{\mu<\lambda}(\lambda^{\prime})^{\mu}$.

In general, the transfers are affected by using incomplete Ehrenfeucht-
Mostowski-like types obtained from certain polarized partition theorems.

In addition we prove that:

######  Theorem 2.5.

There is a nice logic which is (fully) compact; stronger than first order. We
got it adding the quantifier “the cofinality of an order is $\kappa$” to first
order logic gives a compact logic (provably in ZFC of course; subsequently
more such logics were found).

We also suggest transfer theorems starting with finite cardinals, some
problems on them and the easily answered cases. (Those corresponding to the
gaps $(\aleph_{0}),(\beth_{\omega},\aleph_{0})$), see p.250 and see more in
[She75b].

In proving the theorems or omitting types we use passing to non-well ordered
models of fragments of set theory (§5).

(B19) Separability properties of almost disjoint families of sets, IJM
12(1972), 207-214 (with P. Erdős).

We deal with familiies of subsets of $\omega$. It is proved (in ZFC) that:
there is a strongly $n$-separable non $(n+1)$-separable maximal almost
disjoint family (of infinite subset of $\omega$), any for $n=2$, but (under
MA) not necessarily for $n=3$. Also we have some similar results, also
independent families and related finitary problems are discussed.

Saharon: read.

(B20) On power-like models for hyperinaccessible cardinals JSL 37(1972),
531-537 (with J.H. Schmerl).

The main result of this paper is the following transfer theorem: if $T$ is an
elementary theory which has $\kappa$-like model where $\kappa$ is Mahlo of
order $\omega$, then $T$ has a $\lambda$-like model for each $\lambda>\text{
card}(T)$. This can be expressed as a transfer theorem for generalized
quantifiers (we get compactness and omitting-type results, too).

The partition theorem originally used by the second author is:

######  Theorem 2.6.

If $f_{\ell}$ is an $\ell$-place function on $\kappa$, where $\kappa$ is
$(m+n)$-Mahlo then there is an $m$-Mahlo $\lambda<\kappa$ and
$A\subseteq\lambda$ unbounded, such that if
$\alpha_{1}<\ldots<\alpha_{\ell}\in A$, $\ell\leq n$,
$f(\alpha_{1},\dotsc,\alpha_{\ell})<\alpha_{i}$ then the value of $f$ does not
depend on $\alpha_{i+1},\dotsc,\alpha_{n}$.

What is sufficient (and proved there) is the version with having
$A=\\{\alpha_{1}<\alpha_{2}<\ldots<\alpha_{n}\\}$ with $m=0$.

(B21) On problems of Moser and Hanson Proc. Symp. in Graph Theory, Springer
Lecture, No.303, 75-80 (with P. Erdős).

The following problem is due to L. Moser. Let $A_{1},\dotsc,A_{n}$ be any $n$
sets. Take the largest subfamily $A_{i_{1}},\dotsc,A_{i_{r}}$ which is union-
free; i.e. $A_{i_{j_{1}}}\cup A_{i_{j_{2}}}\neq A_{i_{j_{3}}}$, $1\leq
j_{i}\leq r$, $1\leq j_{2}\leq r$, $1\leq j_{3}\leq r$, for every triple of
distinct sets $A_{j_{1}},A_{j_{2}},A_{j_{3}}$. Put $f(n)=\min(r)$, where the
minimum is taken over all families of $n$ distinct sets.

Determine or estimate $f(n)$. Improving previous bounds we show

$\sqrt{2n}-1<f(n)<2\sqrt{n+1}.$

(B22) A note on model complete models and generic models, Proc. AMS 34(1972),
509-514.

We prove that there are many maximal model complete ($=$ generic) models, and
that there exists an (uncountable) theory with no generic models.

(B23) Some counterexamples in the partition calculus, J. Comb. Th. A 15(1973),
167-174. (with F. Galvin)

We show that $\aleph_{1}\nrightarrow[\aleph_{1}]^{2}_{4}$, i.e. the pairs
(2-element subsets; edges of the complete graph) of a set of cardinality
$\aleph_{1}$ can be colored with 4 colors so that every uncountable subset
contains pairs of every color, and
$2^{\aleph_{0}}\nrightarrow[2^{\aleph_{0}}]^{2}_{\aleph_{0}}$, i.e. the pairs
of real numbers can be colored with $\aleph_{0}$ colors so that every set of
reals of cardinality $2^{\aleph_{0}}$ contains pairs of every color. These
results are counterexamples to certain transfinite analogs of Ramsey’s
theorem. Results of this kind were obtained previously by Sierpinski and by
Erdős, Hajnal and Rado. The Erdős-Hajnal-Rado result is much stronger than
ours, but they used the continuum hypothesis and we do not. As by-products, we
get an uncountable tournament with no uncountable transitive subtournament,
and an uncountable partially ordered set such that every uncountable subset
contains an infinite antichain and a chain isomorphic to the rationals.

(B24) First order theory of permutation groups, IJM 14(1973), 149-162.

(B25) Errata to first order, theory of permutation groups, IJM 15(1973),
437-441.

We solve the problem of the elementary equivalence (definability) of the
permutation groups over cardinals $\aleph_{\alpha}$, by getting bi-
interpretability results. We show that it suffices to solve the problem of
elementary equivalence (definability) for the ordinals $\alpha$ in certain
second order logic, and this is reduced to the case of
$\alpha<(2^{\aleph_{0}})^{+}$. E.g. if the ordinal
$((2^{\aleph_{0}})^{+})^{\omega}$ (ordinal exponentiation) divides
$\alpha_{1},\alpha_{2}$ and ${\rm cf}(\alpha_{1})$, ${\rm
cf}(\alpha_{2})>2^{\aleph_{0}}$ or ${\rm cf}(\alpha_{1})={\rm
cf}(\alpha_{2})$, then the permutation groups of
$\aleph_{\alpha_{1}},\aleph_{\alpha_{2}}$ are elementarily equivalent. We also
solve a problem of Mycielski and McKenzie on embedding of free groups in
permutation groups, and discuss some weak second-order quantifiers.

(B26) Notes on combinatorial set theory, IJM 14(1973), 262-277.

We shall prove some unconnected theorems:

1) (GCH) $\omega_{\alpha+1}\rightarrow(\omega_{\alpha}+\xi)^{2}_{2}$ when
$\aleph_{\alpha}$ is regular and $|\xi|^{+}<\aleph_{\alpha}$.

2) There is a Jónsson algebra in $\aleph_{\alpha+n}$, and
$\aleph_{\alpha+n}\rightarrow[\aleph_{\alpha+n}]^{n+1}_{\aleph_{\alpha+n}}$ if
$2^{\aleph_{\alpha}}\leq\aleph_{\alpha+n}$.

3) [Universal graph] If $\lambda>\aleph_{0}$ is a strong limit cardinal, then
among the graphs with $\leq\lambda$ vertices each of valence $<\lambda$ there
is a universal one.

4) (GCH) If $f$ is a set mapping on $\omega_{\alpha+1}$ ($\aleph_{\alpha}$
regular) satisfying $\alpha<\beta<\lambda\Rightarrow|f(\alpha)\cap
f(\beta)|<\aleph_{\alpha}$, then there is a free subset of order-type $\xi$
for every $\xi<\omega_{\alpha+1}$.

(B27) Size direction games over the real line. III., IJM, 14(1973), 442-449
(with G. Moran).

(B28) There are just four second-order quantifiers, IJM, 14(1973), 262-277.

Among the second-order quantifiers ranging over relations satisfying a first-
order sentence, there are four for which any other one is bi-interpretable
with one of them: the trivial, monadic, permutational and a full second-order.
(We deal with infinite, one sorted models). The interpretation is in fact
first-order (second-order parameters but not quantifiers are used).

See representation in Handbook of model theoretic logics, Baldwin’s article
and [She86].

(B29) A substitute for Hall’s theorem for families with infinite sets, J.
Comb. Th. A. 16(1974), 199-208.

A sufficient condition for the existence of a system of distinct
representatives for a family $S$ is that $x\in A\in S$ implies the number of
elements of $A$ is not smaller than the number of sets in $S$ to which $x$
belongs.

The method is: by Hall’s theorem we reduce the problem for countable $S$ to
the divergence of an infinite sum, which is then proved to diverge. For $S$
uncountable see [MS74].

(B30) The cardinals of simple models for universal theories, Proc. of the
Symp. in honor of Tarski’s 70th birthday, Proc. Symp. in Pure Math 25(1974),
53-74 (with R. McKenzie)

Some results about spectra of cardinals of simple algebras in varieties and,
more generally, about the cardinals of $T$-simple models where $T$ is a
universal theory are obtained and applications discussed. $M$ is $T$-simple if
every homomorphism from $M$ to a model $N$ of $T$ is an embedding or a
constant. It is shown that if the language of $T$ has power $\kappa$ and if
there exists a $T$-simple model whose power exceeds $2^{\kappa}$, then
$T$-simple models exist in all powers $\lambda\geq\kappa$ (the point is that
2-indiscernibility is enough, even a sequence of approximations). It is
further shown that if the language of $T$ is countable, and if there exists an
uncountable $T$-simple model, then there exists a $T$-simple model with the
power $2^{\aleph_{0}}$. Also counterexamples are given showing the results are
essentially best possible.

There are generalizations.

(B33) The Hanf number of omitting complete types, PJM 50(1974).

It is proved in this paper that the Hanf number $m^{c}$ of omitting complete
types by models of complete countable theories is the same as that of omitting
not necessarily complete type by models of a countable theory. I.e.

######  Theorem 2.7.

For every ordinal $\alpha<\omega_{1}$ there is a countable first-order
vocabulary $\tau_{\alpha}$ and a complete theory $T_{\alpha}$ in
$\mathbb{L}(\tau_{\alpha})$ such that:

  1. (i)

$p=\\{P(\chi_{0})\\}\cup\\{\chi_{0}\neq c_{n}:n<\omega\\}$ is a complete type
for $T_{\alpha}$

  2. (ii)

$T_{\alpha}$ has a model of cardinality $\beth_{\alpha}$ omitting $p$

  3. (iii)

$T_{\alpha}$ has no model of cardinality $>\beth_{\alpha}$ omitting $p$.

(B34) Weak definability in infinitary languages, JSL 38(1973), 339-404.

We prove that if a model of cardinality $\kappa$ can be expanded to a model of
a sentence $\psi$ from $\mathbb{L}_{\lambda^{+},\omega}$ by adding a suitable
predicate in more than $\kappa$ ways, then it has a submodel of power $\mu$
which can be expanded to a model of $\psi$ in $>\mu$ ways provided that
$\lambda,\kappa,\mu$ satisfy suitable conditions.

######  The Main Theorem 2.8.

1) Let $\psi$ be a sentence in $\mathbb{L}_{\lambda^{+},\omega}(\tau+P)$, $P$
a unary predicate not in $\tau$, $|\tau|\leq\lambda$, $M$ a $\tau$-model of
cardinality $\aleph_{\alpha+\beta}$ such that:

$\big{|}\\{P:P\subseteq|M|,(M,P)\models\psi\\}\big{|}>\aleph_{\alpha+\beta}.$

Assume further that $\beta<\omega_{1}$, $\aleph_{\alpha}$ has cofinality
$\aleph_{0}$, $\mu_{n}\geq\lambda$, $\mu=\sum\limits_{n<\omega}\mu_{n}$,
$\mu_{n}<\mu_{n+1}$ and
$\kappa<\aleph_{\alpha}\Rightarrow\kappa^{\mu_{n}}<\aleph_{\alpha}$ for
$n<\omega$.

Then $M$ has an elementary submodel $N$ of cardinality $\mu$ such that

$\big{|}\big{\\{}P:P\subseteq|N|,\
(N,P)\models\psi\big{\\}}\big{|}\geq\mu^{\aleph_{0}}.$

Another theorem which we shall not prove, as its proof is simpler is

######  Theorem 2.9.

Let $\psi\in\mathbb{L}_{\lambda^{+},\omega}(\tau+P)$, $M$ a $\tau$-model of
cardinality $\kappa$ such that
$|\\{P:P\subseteq|M|,(M,P)\models\psi\\}|>\kappa$. Assume further that
$\mu\geq\lambda$, $\kappa^{\mu}=\kappa$. Then $M$ has an elementary submodel
$N$ of cardinality $\mu$ such that

$\big{|}\big{\\{}P:P\subseteq|N|,\
(N,P)\models\psi\big{\\}}\big{|}>\mu^{\aleph_{0}}.$

Also, (see middle of page 400 there)

######  Theorem 2.10.

If $T$ is a complete (first order) theory, $|T|=\lambda^{+}$, $\lambda$
regular (for simplicity) and every $n$-type of cardinality $<\lambda$ can be
extended to complete $n$-type of cardinality $<\lambda$ (holds if
$|D(T)|<2^{\lambda}$), then $T$ has a model in which every finite sequence
realizes a complete type of cardinality $<\lambda$.

(B35) Sufficiency conditions for the existence of transversals Cand. J. Math
26 (1974), 948-961 (with E.C. Milner).

The main theorem has an interesting formulation in terms of bipartite graphs.
A bipartite graph is a triple $\Gamma=\langle X,\Delta,Y\rangle$ with vertex
set $X\cup Y$ ($X,Y$ disjoint sets) and edge set
$\Delta\subseteq\\{\\{x,y\\}:x\in X,y\in Y\\}$. Let $v(z)=:\big{|}\\{u\in
X\cup Y:\\{u,z\\}\in\Delta\\}\big{|}$ (for $z\in X\cup Y$) be the valency
function of $\Gamma$. Then the main theorem is equivalent to the following
statement: if $\Gamma=\langle X,\Delta,Y\rangle$ is a bipartite graph such
that $v(x)>0$ for $x\in X$ and $v(x)\geq v(y)$ whenever $x\in X,y\in Y$ and
$\\{x,y\\}\in\Delta$, then there is a matching from $X$ into $Y$, i.e. there
is a 1-1 function $\varphi:X\rightarrow Y$ such that
$\\{x,\varphi(x)\\}\in\Delta$ (for $x\in X$).

In §7 we prove even stronger results.

(B36) Remarks on Cardinal invariants in Topology, General Topology and its
Applications 7(1977), 251-259.

In §2 we investigate to what degree singular calibres are preserved by
products (for regular calibres Sanin solved the problem). We also generalize
the concept to pairs of cardinals.

We prove that

######  Theorem 2.11.

If $\lambda$ is a calibre of $X,{\rm cf}(\lambda)>\aleph_{0}$, then $\lambda$
is a calibre of $X^{I}$.

In §1 we construct a $T_{2}$ space $X$ with $d(X)=2^{\aleph_{0}}$ and
$s(x)=\aleph_{0}$; i.e. there is no discrete subspace of power $\aleph_{1}$.
We also show that the upper bound in the definition of $h(X)=\lambda$ is
realized if $X$ is Hausdorff and ${\rm cf}(\lambda)=\aleph_{0}$. The theorem
on $h(X)=\lambda$, for $X$ Hausdorff was proved independently by Hajnal and
Juhasz and the author.

In §3 we show that the class of calibres of a $T_{2}$ space is just about
arbitrary.

(B37) A two-cardinal theorem, Proc. AMS 481(1975), 207-213.

We prove the following theorem and deal with some related questions: if for
all $n<\omega$, $T$ has a model $M$ such that
$n+|Q^{M}|^{n}\leq|P^{M}|<\aleph_{0}$ then for all $\lambda,\mu$ such that
$|T|\leq\mu\leq\lambda<{\rm Ded}^{*}(\mu)$ (e.g.
$\mu=\aleph_{0},\lambda=2^{\aleph_{0}}$), $T$ has a model of type
$(\lambda,\mu)$, i.e. $|Q^{M}|=\mu$, $|P^{M}|=\lambda$. We use the existence
of free subsets.

Question: Is our result the best possible? That is, does there exist a
sentence which for every $n$ has a model $M$,
$\aleph_{0}>|P^{M}|>|Q^{M}|^{n}$, $|Q^{M}|\geq n$, but does not have a
$(2^{\mu},\mu)$-model for some $\mu$, and even: has a $(\lambda,\mu)$-model
iff $\mu\leq\lambda<{\rm Ded}^{*}(\mu)$ (assuming for some $\mu,{\rm
Ded}^{*}(\mu)\leq 2^{\mu}$).

Conjecture: $\\{(m_{i},n_{i},k_{i}):i<\omega\\}\rightarrow(\lambda,\mu,k)$
when $m_{i}\geq n^{i}_{i}$, $n_{i}\geq k^{i}_{i}$, $k_{i}\geq i$,
$k\leq\mu\leq\lambda<{\rm Ded}^{*}k$.

Conjecture:
$\\{(2^{n_{i}},n_{i}):i<\omega\\}\rightarrow(2^{\mu},\mu)[n_{i}\geq i]$. (Some
information is suggested).

(B38) Graphs with prescribed asymmetry and minimal number of edges, Erdős
Symp. (infinite and finite sets), Vol.III(1975), 1241-1256.

We shall deal with non-directed graphs, without loops and double edges, and
having a finite number of vertices.

A graph is symmetric if it has a non-trivial automorphism = a permutation of
its vertices, such that a pair of vertices is connected iff their images are
connected. The asymmetry of a graph is the minimal number of changes (i.e.
adding and deleting of edges) which is necessary to make the graph symmetric.
Erdős and Renyi [ER63] defined and investigated this notion, and defined,
$F(n,k)\,[C(n,k)]$ for $k\geq 1,n>1$ as the minimal number of edges in a
[connected] graph with $n$ vertices which is asymmetric is $k$; if there is no
such graph the value of the function will be $\infty$ (if $n$ is too small,
this happens).

######  Theorem 2.12.

For $n$ sufficiently large

$F(n,2)=n+1,\ C(n,2)=n+2.$

######  Theorem 2.13.

For odd $k>2$ and $n$ sufficiently larger than $k$

$F(n,k)=C(n,k)=[(k+3)n/4]-0.5[2n/(k+3)+1/2].$

######  Theorem 2.14.

For even $k>2$ and $n$ sufficiently larger than $k$

$F(n,k)=C(n,k)=[(k+2)n/4+1/2].$

The proof is given only for $k\geq 41$, and uses a detailed analysis of
examples.

(B39) Differentially closed fields, IJM 16(1973), 314-328.

We prove that even the prime, differentially closed field of characteristic
zero, is not minimal; that over every differential radical field of
characteristic $p$, there is a closed prime one, and that the theory of closed
differential radical fields of characteristic $p(>0)$ is stable (hence the
prime model above is unique).

(B40) Notes on Partition Calculus, Erdős Symp. Vol.III, 1257-1276.

We deal here with some separate problems which appeared in the problem list of
Erdős and Hajnal [EH71].

In §1 we solve problem 3 of [EH71] affirmatively. Thus, if
$\aleph_{\omega}<2^{\aleph_{n(0)}}<\ldots<2^{\aleph_{n(k)}}<\ldots$, then
$\sum\limits_{n<\omega}2^{\aleph_{n}}\rightarrow(\aleph_{\omega},\aleph_{\omega})^{2}$.
We prove a canonization lemma for it.

In §2 we deal with problem 32 of [EH71] which asks whether there is a graph
$G$ with $\aleph_{1}$ vertices and with no $[[\aleph_{0},\aleph_{1}]]$
subgraph for which $\aleph_{1}\rightarrow(G,G)^{2}$. We provide a wide class
of such graphs, assuming CH. If $\mathbf{V}=\mathbf{L}$ is assumed we show
that $\aleph_{1}\rightarrow(G,G)^{2}$ iff $G$ has coloring number
$\leq\aleph_{0}$.

In §3 we deal with problems 48, 50 of [EH71] (asked by Erdős and Hajnal):
partition relations concerning coloring numbers.

In §4 we deal with problem 42 of [EH71]. We get compactness and incompactness
results on the existence of transversals and on property $B$.

We also find sufficient and necessary conditions for the existence of
transversals for a family of sets.

(B41) Some theorems on transversals Erdős Symp., Vol.III, 115-1126 (with E.C.
Milner).

We prove e.g. ${\rm NPT}(\lambda,\aleph_{0})\rightarrow{\rm
NPT}(\lambda^{+},\aleph_{0})$ where $P(\lambda)$: there is a family $F$ of
$\lambda$ countable sets such that $F$ has no transversals but every
$F^{\prime}\subseteq F$, $|F^{\prime}|<|F|$ has a transversal.

(B42) The monadic theory of order, Annals of Math 102(1975), 379-419.

First, a decidability method is developed and used to decide in a uniform and
simplified way all known decidable fragments of the monadic theory of order:
the monadic theory of $\omega$, the monadic theory of $\omega_{1}$, the
monadic theory of countable orders and so on. The method is used to reduce the
monadic theory of ordinals $\alpha<\lambda^{+}$ to that of $\lambda$ and to
reduce the monadic theory of $\omega_{2}$ to the first-order theory of the
Boolean algebra of subsets of $\omega_{2}$ module closed unbounded sets with
an operation $F(X)=\\{\alpha:{\rm cf}(\alpha)=\omega_{1}$ and $X\cap\alpha$ is
stationary in $\alpha\\}$ and similarly for any cardinal. The new proof of
decidability of the monadic theory of countable orders helps to prove that
countable orders cannot be characterized in the monadic logic. To develop the
method a useful Feferman-Vaught type theorem for the monadic theory of order
is proven and also two Ramsey type theorems are proven.

E.g.

######  Theorem 2.15.

1) If $c$ is an additive two-place function form a dense linear order (i.e. if
$x<y<z$, $f(x,z)$ is determined by $f(x,y),f(y,z)$) and $f$ has a finite
range, then there is a $c$-homogeneous set which is dense in some interval
(this is made to help prove decidability).

2) If $\lambda$ is regular, $c$ and additive two-place function from $\lambda$
with range of cardinality $<\lambda$, then there is a $c$-homogeneous set
which is unbounded in $\lambda$.

Second, the true first-order arithmetic is interpreted under the Continuum
Hypothesis in the monadic theory of the real line and in the monadic theory of
order.

There are more results and many conjectures. The introduction surveys
preceding results.

(B43) Generalized quantifiers and compact logic, Trans. AMS 204(1975),
342-364.

We show the existence of a logic stronger than first-order logic even for
countable models, but still satisfying the general compactness theorem,
assuming e.g. the existence of a weakly compact cardinal. We also discuss
several kinds of generalized quantifiers.

We prove the compactness of $\mathbb{L}(Q^{{\rm cf}}_{C_{1}},\dotsc,Q^{{\rm
cf}}_{C_{n}})$ when $C_{n}$ is a convex class of cardinals, $Q^{{\rm cf}}_{C}$
say the cofinality of an order belongs to $C$, and axiomatize
$L(Q^{\text{cf}}_{C_{1}})$. We also prove that $\aleph_{0}$-compactness of
$(Q^{\mathrm{stat}}_{\aleph_{1}}x)$ (being stationary for order of cofinality
$\aleph_{1}$) and transfer theorems.

We define generalized second-order quantifiers: allowing in forming formulas
the use of second-order variables, and second-order quantifiers $(QP)\psi(P)$,
such that its satisfaction depends only on the isomorphism type of $\\{P:\
\models\psi[P]\\}$. In particular, stationary logic (that is, the quantifier
$(\mathrm{aa}X)$) is introduced (developed subsequently by Barwise, Kaufman
and Makkai, [BKM78]) and we note we could have used diamonds instead of a
weakly compact cardinal in the main theorem.

(B44) Infinite abelian groups, Whitehead problem and some constructions, IJM,
18(1974), 243-256.

We deal with three problems: the existence of indecomposable and even rigid
(abelian) groups, the number of reduced separable $p$-groups and Whitehead
problem.

We prove that for any cardinal $\lambda>\aleph_{0}$ there are $2^{\lambda}$
(abelian) groups $G_{i}$ ($i<2^{\lambda}$) such that any homomorphism
$h:G_{i}\rightarrow G_{j}$ is zero or $i=j$, $h(x)=nx$. We do it by coding
deepness of well-founded trees and stationary sets.

For regular $\lambda>\aleph_{0}$ we build a family of $2^{\lambda}$ reduced
separable $p$-groups, each of power $\lambda$, no one embeddable into another,
by imitating the proof for non-superstable theories (see [1], Ch.VIII and
[Shear] for singular $\lambda$). The proof indicates that reduced separable
$p$-groups cannot be characterized by reasonable invariants.

An abelian group $G$ is $W$ (= Whitehead) if ${\rm Ext}(G,\mathbb{Z})=0$, or
equivalently, when: if $\mathbb{Z}\subseteq H$ ($\mathbb{Z}$ the integers),
$h:H\rightarrow G$ a homomorphism onto $G$ with kernel $\mathbb{Z}$, then for
some homomorphism $g:G\rightarrow H$ we have $hg=1_{G}$. It is proved that
“every Whitehead group of power $\aleph_{1}$ is free”, is independent of ${\rm
ZFC}$: it follows from $\mathbf{V}=\mathbf{L}$ (or even $\diamondsuit_{S}$ for
every stationary $S\subseteq\omega_{1}$) and is contradicted by ${\rm
MA}+2^{\aleph_{0}}>\aleph_{1}$. We partition the $\aleph_{1}$-free groups of
power $\aleph_{1}$ to three classes: I (“unstable in $\aleph_{0}$”), II
(intermediate) and III (the free ones). Class II is similar to “strongly
$\aleph_{1}$-free but not free” but somewhat wider.

For latter use let $\Gamma(G)=\\{i:G/G_{i}$ is not $|G|$-free$\\}$, where

$G=\bigcup\limits_{i<|G|}G_{i},$

$G_{i}$ increasing continuums, $|G_{i}|<G$, and $G$ is $|G|$-free ($\Gamma(G)$
is determined modulo the filter of closed unbounded subsets of $|G|$).

(B45) Existence of rigid-like families of abelian $p$-groups, model theory
and algebra, lecture notes in Math. 498, Springer-Verlag, 1975, 385-402.

We prove that for arbitrarily large $\lambda$ there is a family of
$2^{\lambda}$ (abelian) separable $p$-groups, each of power $\lambda$ with
only the necessary homomorphism (called simple) between them. A homomorphism
$h:G\rightarrow H$ is zero-like if there are no $m<\omega$ and $a_{n}\in G$ of
order $p^{m+m}$, $h(a)$ having order $\geq p^{n}$; we call $h$ simple if
$h=h_{1}+h_{2}$, $h_{1}$ zero-like, $h_{2}$ a multiplication by a $p$-adic
integer (if $H\neq G,h_{2}=0$).

We prove this for $\lambda$ strong limit of uncountable cofinality; the method
is that we can for each strong limit $\mu<\lambda$, ${\rm
cf}(\mu)=\aleph_{0}$, diagonalize over all approximation of a homomorphism; we
can do it as $2^{\mu}=\mu^{\aleph_{0}}$.

For $\lambda^{\aleph_{0}}=2^{\lambda}>2^{\aleph_{0}}$ we prove a weaker
theorem

  1. $(*)$

i.e. $\forall x\bigl{[}\bigvee\limits_{n<\omega}p^{n}x=0\bigr{]}$, and the
group has no divisible subgroup $\neq 0$.

(B46) Colouring without triangles and partition relation, IJM 20(1975), 1-12.

We say the colouring $h$ of $\lambda$ proves
$\lambda\rightarrow[\mu]^{2}_{\chi}$ if $h$ is a function from the edges of
the complete graph on $\lambda$ into $\chi$ such that whenever
$S\subseteq\lambda$, $|S|=\mu$, then the range of $h$ restricted to the edges
in $S$ is the whole of $\chi$.

Erdős and Hajnal (Problem 68) asked if it is true (assuming CH) that for every
colouring which witness $\aleph_{1}\rightarrow[\aleph_{1}]^{2}_{3}$ there is a
triangle whose edges have three colours. We answer this negatively and prove
the stronger result (Theorem 1.1) that, if
$2^{\aleph_{\alpha}}=\aleph_{\alpha+1}$, then there is a colouring which
witness
$\aleph_{\alpha+1}\rightarrow[\aleph_{\alpha+1}]^{2}_{\aleph_{\alpha}}$and for
which there is no triangle with three colours. We also show that, if
$\mathbf{V}=\mathbf{L}$, and $\aleph_{\alpha}$ is regular then there is even
such a colouring which witness
$\aleph_{\alpha+1}\rightarrow[\aleph_{\alpha+1}]^{2}_{\aleph_{\alpha+1}}$.

In §2, we deal with the corresponding finite problem and confirm a conjecture
of Erdős. We prove, for example, that if the edges of the complete graph on
$n$ points are colored with three colours so that no triangle has three
colours, then there is a set $A\subseteq n$, such that $|A|\geq n^{c}$ ($c$
some constant) and which contains only two colours. This should be compared
with the result $n\nrightarrow[c_{1}\log(n),n]^{2}_{3}$ proved by Erdős by a
probabilistic method.

In §3, we show that examples for 1.1 should contain quite homogeneous sets;
and then introduce new partition relations, (of interest in the finite and
infinite case), give some results and suggest some problems.

It was proved by Erdős and Hajnal that any example of
$\aleph_{1}\rightarrow\bigl{[}[\aleph_{1},\aleph_{0}]\bigr{]}^{2}_{3}$ is
$\aleph_{0}$-universal. Erdős and Hajnal asked whether every example of
$\aleph_{2}\rightarrow\bigl{[}[\aleph_{2},\aleph_{1}]\bigr{]}^{2}_{3}$ is
$\aleph_{1}$-universal. We prove the consistency of the negative answer and
the method is applicable in many other cases.

(B47) $\Delta$-Logics and generalized quantifiers, AML 10(1976), 155-192 (with
J.A. Makowsky and Y. Stavi).

We study the $\delta$-closure of an abstract logic, which is its smallest
extension satisfying the $\Delta$-interpolation theorem or, equivalently, its
largest extension having the same $PC$-classes.

We study properties which are preserved by the $\Delta$-closure such as
compactness, and various Lowenheim and Hauf numbers.

We apply these to exhibit new logics with various properties, and we study
$\Delta$-closed fragments of $\mathbb{L}_{\omega_{1},\omega}$ in great detail.

(B48) Categoricity in $\aleph_{1}$ of sentences in
$\mathbb{L}_{\omega_{1},\omega}(Q)$, IJM 20(1975), 127-148.

We investigate the possible numbers of models of a sentence
$\psi\in\mathbb{L}_{\omega_{1},\omega}(Q)$ of cardinality $\aleph_{1}$.

Assume $2^{\aleph_{0}}<2^{\aleph_{1}}$ and $1\leq
I(\aleph_{1},\psi)<2^{\aleph_{1}}$, it is proved that there exists a class $K$
of models which are the atomic models of some first-order countable theory
$T_{\psi}$ such that $\forall\lambda[I(\lambda,K)\leq I(\lambda,\psi)]$ and
$I(\aleph_{1},K)\geq 1$.

Under the same assumptions it is proved that $K$ is $\aleph_{0}$-stable.

The Main Theorem is: assuming $\diamondsuit_{\aleph_{1}}$ if $1\leq
I(\aleph_{1},\psi)<2^{\aleph_{1}}$, then there exists a model of $\psi$ of
cardinality $\aleph_{2}$. This is proved by showing:

  * •

if the $\aleph_{0}$-amalgamation property fails for $K$, then using
$\diamondsuit_{\aleph_{1}}$ we can construct $2^{\aleph_{1}}$ non-isomorphic
models in $K$ of cardinality $\aleph_{1}$.

  * •

The $\aleph_{0}$-amalgamation property is equivalent to a suitable rank being
always $<\infty$ (and some other properties).

  * •

Assuming the rank is $<\infty$, we prove that the order property (= there are
$\aleph_{1}$ sequences in a model of $\psi$, ordered by some formula) is
equivalent to two variants of the symmetry property (one of them:
$R({\rm tp}(\bar{a},M\cup\bar{b}))=R({\rm tp}(\bar{a},M))$ iff $R({\rm
tp}(\bar{b},M,\bar{a}))=R({\rm tp}(\bar{b},M))$ essentially). The second
asymmetry property, enables us to prove $I(\aleph_{1},K)=2^{\aleph_{1}}$ by
coding stationary subsets of $\omega_{1}$. But the holding of the symmetry
property enable us to properly extend every model of power $\aleph_{1}$. Thus
by iteration, we get a model of power $\aleph_{2}$.

######  Remark 2.16.

1) We continue this work on $\mathbb{L}_{\omega_{1},\omega}$ sentences in
[Sh:87a], [Sh:87b] where we improve some of the theorems from this paper and
some of the claims here are preparation for them and also in [She87a].

2) The main difficulty is in 3).

(B49) A two-cardinal theorem and a combinatorial theorem, Proc. AMS 62(1977),
134-136.

We prove a new two-cardinal theorem, e.g.
$(\aleph_{\omega},\aleph_{0})\rightarrow(2^{\aleph_{0}},\aleph_{0})$.

For this we prove a new partition theorem, with “tree indiscernibility”.

The problem seemed to have been the right induction. Also compactness is
gotten.

######  Conjecture 2.17.

A)
$(\aleph_{\alpha+\omega+\omega},\aleph_{\alpha+\omega},\aleph_{\alpha})\rightarrow(\lambda,\mu,\chi)$
whenever $\chi<\mu<\lambda<\mathrm{Ded}^{*}\chi$.

B) If a countable theory $T$ has a $\lambda$-like model, $\lambda$ a limit
cardinal, and $|T|\leq\mu<\lambda_{1}{\rm Ded}^{*}(\mu)$, $\lambda_{1}$ a
singular cardinal, then $T$ has a $\lambda_{1}$-like model. If $\lambda$ is
$\omega$-Mahlo weakly inaccessible cardinal, we can remove the cardinal, we
can remove the singularity of $\lambda_{1}$.

C) If $\psi\in L_{\omega_{1},\omega}$ has a model of cardinality
$\aleph_{\omega_{1}}$, then $\psi$ has a model of cardinality $2^{\aleph_{0}}$
(but see [Sh:522]).

(B50) Decomposing uncountable squares to countably many chains, J. Comb. Th.
A. 21(1976), 110-114.

We construct an ordered set $I$ of cardinality $\aleph_{1}$, such that its
square is the union of $\aleph_{0}$ chains (in the natural partial order).
This gives a complete order, isomorphic to any open interval of itself but not
to its universe. This is proved by constructing a “very special” Aronszajn
tree.

######  Conjecture 2.18.

It is consistent that:

  * •

any Specker order contains a suborder as above

  * •

if $I,J$ are as above, $I,J$ or $I,J^{*}$ have uncountable isomorphic
suborders.

The consistency of 2) is proved in [AS85].

As for (1) it is equivalent to:

  1. $(*)$

if $T$ is an Aronszajn tree, $c:T\rightarrow 2$ then for some uncountable
$X\subseteq T$ and $\ell\neq\\{0,1\\}$ for every $x\neq y$ in $X$, the maximal
$z\leq_{T}x,y$ satisfies $c(z)=\ell$.

(B51) Why there are many non-isomorphic models for unsuperstable theories,
Proc. of the International Congress of Math. Vancouver (1974), 553-557.

We review the proof (see [She78]) that for unsuperstable $T$, $T\subseteq
T_{1}$, $I(\lambda,T_{1},T)=2^{\lambda}$ when e.g. $\lambda>|T_{1}|$ or
$\lambda=|T_{1}|+\aleph_{1}$, $T$ unsuperstable. To exemplify the generality
of the method, we use it to prove that in any uncountable cardinality
$\lambda$ there is a rigid Boolean algebra (for $\lambda$ regular satisfying
the countable chain condition), and a rigid order (with a rigid completion).

(B52) A compactness theorem in singular cardinals, free algebras, Whitehead
problem and transversals, IJM 21(1975), 319-349.

We prove in an axiomatic way a compactness theorem for singular cardinals,
concluding that if $\mathbf{V}=\mathbf{L}$ every Whitehead group is free. For
this we use index sets of submodels $\langle M_{u}:u\in\mathcal{P}(n)\rangle$
and prove various assertion by induction on the cardinality for all $n$ at
once.

There is an application for transversals, i.e. if $\kappa<\lambda$, $\lambda$
singular $\\{A_{i}:<\lambda\\}$ a family of $\lambda$ sets each of power
$\leq\kappa$, and every subfamily of smaller cardinality has a transversal
then the family has a transversal.

Similar conclusions hold for freeness of groups, freeness abelian groups and
colouring numbers.

The general theorem:

  1. $(*)$

“if $V$ is a variety with $<\lambda$ functions, $A\in V$ an algebra of power
$\lambda$, and every (or “many”) subalgebras of $A$ of cardinality $<\lambda$
are free, then $A$ is free”

and does not follow by the axiomatic treatment. However, we develop here some
filters on $S^{\delta}_{\kappa}(A)=\\{\langle A_{i}:i<\delta\rangle:A_{i}$ a
subset of $A$ of power $\leq\kappa$, $A_{i}$ increasing continuous$\\}$, using
those we prove $(*)$.

Further work on the situation in regular cardinals is in [She85a], and the
author eliminates $AxI^{*},V$ from the axiomatic system (see [She19], Hodges
[Hod81]).

######  Theorem 2.19.

For a $\lambda$ singular, $\chi^{*}<\lambda$ assume $\mathbf{F}$ is a set of
pairs $(A,B),(\mathbf{F}$ for free) $A,B\subseteq\lambda$ satisfying the
axioms (II-IV,VI,VII) below. Let $A^{*},B^{*}\subseteq\lambda$, then
$B^{*}/A^{*}\in\mathbf{F}$ if $B^{*}/A^{*}$ is $\lambda$-free in a weak sense
which means (see Definition below):

  1. $(*)_{0}$

for the $D_{\chi^{*}}(B^{*})$-majority of $B\in[B^{*}]^{<\lambda}$ we have
$B/A^{*}\in\mathbf{F}$ or just

  2. $(*)_{1}$

for a club of $\mu<\lambda$ we have

$\big{\\{}\mu<\lambda:\\{B\in[B^{*}]^{\mu}:B/A^{*}\in\mathbf{F}\\}\in
E^{\mu^{+}}_{\mu}(B^{*})\big{\\}}$

contains a club of $\lambda$ all $B\in[B^{*}]^{\mu}$ (see below),
$B/A^{*}\in\mathbf{F}$

or at least

  1. $(*)_{2}$

for some set $C$ of cardinals $<\lambda$, unbounded in $\lambda$ and closed
(meaningful only if cf$(\lambda)>\aleph_{0}$) for a
$E^{\mu^{+}}_{\mu}(B^{*})$-positive set of $B\in[B^{*}]^{\mu}$,
$B/A^{*}\in\mathbf{F}$.

The axioms are:

Ax II: $B/A\in\mathbf{F}\Leftrightarrow A\cup B/A\in\mathbf{F}$.

Ax III: if $A\subseteq B\subseteq C$, $B/A\in\mathbf{F}$, $C/B\in\mathbf{F}$
then $C/A\in\mathbf{F}$.

Ax IV: if $A_{i}$ is increasingly continuous for $i\leq\theta={\rm
cf}(\theta),A_{i+1}/A_{i}\in\mathbf{F}$ then $A_{\theta}/A_{0}\in\mathbf{F}$.

Ax VI: if $A/B\in\mathbf{F}$ then for the $D_{\chi^{*}}$-majority of
$A^{\prime}\subseteq A$, $A^{\prime}/B\in\mathbf{F}$.

Ax VII: if $A/B\in\mathbf{F}$ then for the $D_{\chi^{*}}$-majority of
$A^{\prime}\subseteq A$, $A/B\cup A^{\prime}\in\mathbf{F}$.

######  Definition 2.20.

1) Let for $D$ a function giving for any set $B^{*}$ a filter $D(B^{*})$ on
${\mathcal{P}}(B^{*})$ (or on $[B^{*}]^{\mu}$), then to say for the
$D$-majority of $B\subseteq B^{*}$ (or $B\in Y^{*}$) we have $\varphi(B)$
means $\\{B\subseteq B^{*}:\varphi(B)\\}\in D(B^{*})$ or $\\{B:B\in
Y^{*},\neg\varphi(B)\\}=\varnothing\mod D$.

2) Let $D_{\mu}(B^{*})$ be $\\{Y:Y\subseteq{\mathcal{P}}(B^{*})$ such that for
some algebra $M$ with universe $B^{*}$ and $\leq\mu$ functions,
$Y\supseteq\\{B\subseteq B^{*}:B\neq\varnothing$ closed under the functions of
$M\\}$.

$\begin{array}[]{clcr}E^{\mu^{+}}_{\mu}(B^{*})=\bigl{\\{}Y\subseteq[B^{*}]^{\mu}:&\text{for
some }\chi,\ x\text{ such that }\\{B^{*},x\\}\in{\mathscr{H}}(\chi)\\\
&\text{and if }\bar{M}=\langle M_{i}:i<\mu^{+}\rangle\text{ is an
increasingly}\\\ &\text{continuous sequence of elementary submodels of}\\\
&({\mathscr{H}}(\chi),\in)\text{ such that }x\in M_{0}\text{ and}\\\
&\bar{M}\restriction(i+1)\in M_{i+1}\text{ then for some club }C\text{ of
}\mu^{+},\\\ &i\in C\Rightarrow M_{i}\cap B^{*}\in Y\bigr{\\}}\end{array}$

(see [She75a], [BD78]). Now the proof in [She75a] of singular compactness used
two axioms which the author eliminated later; see [She19], Hodges [Hod81].

There are some cases of incompactness ([FS89], [She90]).

(B53) Models with few non-isomorphic expansions, IJM 28(1971), 331-338 (with
A. Litman).

If $M$ is a countable model (with countable language) the number of expansions
of $M$ by one one-place predicate is, up to isomorphism, $\aleph_{0}$ or
$2^{\aleph_{0}}$. Moreover, we characterize when the number is $\aleph_{0}$:
when $M$ is definable in $N\times\omega$ for some finite model $N$.

(B54) The lazy model theorist’s guide to stability, six days of model theory,
Proc. of a conference in Louvain-la-Neuve, March 1975, edited by Paul Henramd,
Pub. by Paul Castella, Switzerland, 1661 Albeuve (1978), 9-76.

This paper is an introduction to classification theory and an outline of the
main results updated to 1975. The main goal is to convince the reader that
stability is useful for algebraic theories, and though it was developed for
elementary classes, it can be adapted to others, mainly:

  * •

the existentially closed models of a universal theory with the joint embedding
property and the amalgamation property

  * •

the existentially closed models of a universal theory with the JEP.

We assume knowledge of only basic model theory (as in the book of Chang and
Keisler); stability is defined and some of the harder combinatorial theorems
are just quoted and appear with an outline of proof.

Morley’s categoricity theorem is proved, uniqueness of prime model for stable
countable theory, characterization of stability using rank, the basic
properties of non-forking (sometimes with simpler proofs than in the book
[She78]). We prove that for a ring $R$, if $K$ is the class of $R$-modules
then either every $R$-module $M$ is the direct sum of modules of
$\leq|R|+\aleph_{0}$, or in every $\lambda>\aleph_{0}$ there are $2^{\lambda}$
non-isomorphic $R$-modules; the first case occurs iff the theory is
superstable. Application to group theory (chain condition) and field are
discussed.

It is noted (with Macintyre) that by [She73], some theories of separably
closed field of characteristic $p$ are stable (ie. the reducts of the theory
of differentially closed fields of characteristic $p$), and by a similar
proof, all such theories are stable.

Since this paper is very condensed and is itself a summary of results, we
could not mention in this summary all the theorems. We gave just a
representative sample list.

(B55) Universal locally finite groups, J. of Alg. 43(1976), 168-175 (with
Macintyre).

Call $G$ a u.f.l. (universal locally finite) group if:

  1. (a)

every finitely generated subgroup is finite

  2. (b)

every finite group can be embedded into $G$

  3. (c)

every isomorphism by one finitely generated subgroup to another is induced by
an inner automorphism.

We prove that for $\lambda>\aleph_{0}$, there are $2^{\lambda}$ u.l.f. non-
isomorphic groups, no one embeddable into another (using [She72a]).

(B56) Refuting Ehrenfeucht Conjecture on Rigid models, IJM 25(1976), 273-288.

We prove that the class of cardinalities in which a first-order sentence has a
rigid model can be very complicated. In fact, we essentially characterized
those classes: if GCH holds any $\Sigma^{1}_{2}$ class (of pure second order
logic) occurs.

(B57) The complete finitely axiomatized theories of order are dense, IJM
23(1976), 200-208 (with A. Amit).

We prove that every sentence of the theory of linear order which has a model,
has a model with finitely axiomatized theory.

(B58) Decidability of a portion of the predicate calculus, IJM 28(1977).

During this century many fragments of first-order logic were proved to be
decidable or undecidable. The most natural fragments are classes of prenex
sentences defined by restrictions on prefix and numbers of monadic, dyadic,
etc., predicate and function symbols. Such classes are called standard below.
The decision problem for standard classes without equality is settled. See the
complete picture in the article, “The decision problem for standard classes”
by Y. Gurevich in JSL 41(1976). According to that article the only open case
of the decision problem for standard classes with equality and at least one
function symbol is the class (let’s call it $K$) of
$\exists\ldots\exists\forall\exists\ldots\exists$ sentences with arbitrary
predicate symbols and exactly one function symbol, which is unary. Here we
settle the decision problem for standard classes with equality and at least
one function symbol by proving decidability. Also the decision problem for
satisfiability in finite models is decidable. Let us say a few words about the
method. Given $\varphi\in K$ we study closure properties of the class of
models of $\varphi$ mainly under what operations it is preserved.

See the book by Berger Gurevich.

(B59) Singular Cohomology in $\mathbf{L}$, IJM 26(1977), 313-319 (with H.L.
Hiller).

######  Theorem 2.21.

$(\mathbf{V}=\mathbf{L})$. If $G$ is a $\kappa$-free non-free abelian group of
power $\kappa$ then ${\rm Ext}(G,\mathbb{Z})$ has power $\kappa^{+}$ hence for
no torsion free $G,{\rm Ext}(G,\mathbb{Z})$ has power $\aleph_{0}$.

A consequence is that there does not exist a topological space $X$ and integer
$n$ such that $H^{n}(X,\mathbb{Z})=\mathbb{Q}$ ($H^{n}$ – ordinary, singular
cohomology).

In [She81d] we construct a universe of set theory where there exists a
topological space $X$ and an integer such that
$H^{n}(X,\mathbb{Z})=\mathbb{Q}$.

(B60) Possible orderings of an indiscernible sequence, Bull. London, Math.
Soc. 9(1977), 212-215 (with W. Hodges and A. Lachlan).

Let $n$ be a positive integer and $X$ a set of cardinality $\geq\max(2n-3)$,
$n+2$. Suppose $(X,<)$ is indiscernible with respect to an $n$-relation $R$,
i.e. $<$ linearly orders $X$, and whether $Rx_{1},\dotsc,x_{n}$ holds depends
only on the relative order of $x_{1},\dotsc,x_{n}$ in $X$.

Then one of the following holds:

  * •

for all linear orderings $<_{1}$ of $X,(X,<_{1})$ is indiscernible w.r.t. $R$;

  * •

if $<_{1}$ linearly orders $R$ than $(X,<_{1})$ is indiscernible w.r.t. $R$
iff $<_{1}$ comes from $<$ by moving an initial segment to the end and maybe
reversing the order;

  * •

if $<_{1}$ linearly orderings $X$ then $(X,<_{1})$ is indiscernible w.r.t. $R$
iff $<_{1}$ is either $<$ or $>$, except that at each end of $(X,<)$ some
elements may be permuted (but at most $n$ in all). The theorem is equivalent
to an earlier result of Fransnay not mentioning indiscernibles, but our bounds
improve his.

(B61) Interpreting set theory in the endomorphism semi-group of a free algebra
or in a category, Ann. Sci. Univ. Clermont Sec. Math. fase 13(1976), 1-29.

In the category of any variety we interpret set theory, essentially. Moreover,
in the semi-group of endomorphism of $F_{\lambda}$, the free algebra with
$\lambda$ generators, we interpret $(\mathscr{H}(\lambda^{+}),\in)$,
essentially. The essentially indicate that we assume some terms call beautiful
does not exist, but even if they exist we get the best possible results.

The method is using stationary sets and deepness of trees.

A related problem is about the group of automorphism of $F$, ${\rm
Aut}(F_{\lambda})$, and we believe

######  Conjecture 2.22.

For varieties satisfying $(*)$ below, a similar answer holds, whereas if $(*)$
fails the group is like the group of permutations of $\lambda$ (see [24,25]).

  1. $(*)$

Let $F$ be freely generated by $\\{x_{i}:i<\omega+\omega\\}$. Then there is an
automorphism $h$ of $F$, $h(x_{i})=x_{i}$ for $i<\omega$ and
$h(x_{\omega})\notin\langle x_{i}:\omega\leq i<\omega+\omega\rangle_{F}$.

(B62) The theorems of Beth and Craig in Abstract Model theory I, The abstract
setting, Trans AMS 256(1979) (with J.A. Makowsky).

In the context of abstract model theory various definability properties, their
interrelations and their relation to compactness are investigated.

Beth’s theorem together with a Feferman-Vaught theorem for tree-like sums
implies a weak form of Robinson’s consistency lemma, hence e.g. assuming
compactness, Craig theorem follows. So in many cases Craig and Beth’s theorems
are equivalent. Also, essentially, Robinson consistency lemma together with
the Feferman-Vaught theorem for pairs implies full compactness.

(B63) The Hanf number of the first-order theory of Banach spaces, Trans AMS
244(1978) (with J. Stern).

In this paper, we discuss the possibility of developing a nice, i.e. first
orer theory for Banach spaces: the restrictions on the set of sentences for
recent compactness arguments applied to Banach spaces as well as for other
model-theoretic results are both natural and necessary; without them we rove
here that we essentially get a second order logic with quantification over
countable sets. Especially, the Hanf number for sets of sentences of the
first-order theory of Banach spaces is exactly the Hanf number for the second
order logic of binary relations (with the second order quantifiers ranging
over countable sets).

(B64) Whitehead group may not be free even assuming Ch.I, IJM 29(1978),
239-247.

In spite of its name this is a paper in set theory. Its purpose is to prove
that many consequences of MA are consistent with GCH. More specifically, it is
possible that GCH holds, $S\subseteq\omega_{1}$ is a stationary, co-stationary
set of limit ordinals, $\omega_{1}\setminus S$ is not small in fact
$\diamondsuit^{*}_{\omega_{1}\setminus S}$ holds, but e.g. every ladder system
$\bar{\eta}=\langle\eta_{\delta}:\delta\in S\rangle$ can be
$\aleph_{0}$-uniformized where “$\bar{\eta}=\langle\eta_{\delta}:\delta\in
S\rangle$ is a ladder system” means each $\eta_{\delta}$ is an increasing
$\omega$-sequence converging to $\delta$ and $\kappa$-uniformized means that
for every $\bar{c}=\langle c_{\delta}:\delta\in S\rangle$,
$c_{\delta}\in{}^{\omega}\kappa$ there is $h:\omega_{i}\rightarrow\kappa$ such
that for every $\delta\in S$, for large enough $n$,
$c(\eta_{\delta}(n))=c_{\delta}(n)$. By [DS78], $\omega_{1}\setminus S$ has to
be stationary. So we prove that
$\diamondsuit_{\omega_{1}}\neq\diamondsuit_{S}$ ($S\subseteq\omega_{1}$
stationary).

We use straightforwardly countable support iteration of length $\omega_{2}$;
the main problem is why $S$ remains stationary. For this we take
$N\prec({\mathscr{H}}(\aleph_{3}),\in)$ and build a sequence of
$\bar{q}^{n}=\langle q_{\eta}:\eta\in T_{n}\rangle$, $T_{n}$ a finite tree,
and in the end get by inverse limit, a tree of candidates for being a
condition generic over $N$. Using $\omega_{1}\setminus S$ we then prove at
least one is contained in a condition.

(B65) A weak form of the diamond which follows from
$2^{\aleph_{0}}<2^{\aleph_{1}}$ IJM 29(1978), 239-247 (with K. Devlin).

We prove that weak CH (i.e. $2^{\aleph_{0}}<2^{\aleph_{1}}$) implies a weak
variant of the diamond, explicitly:

  1. $(\Phi)$

for each function $F:{}^{\omega_{1}>}2\rightarrow 2$ there is
$g\in{}^{\omega_{1}}2$, such that for any $f\in{}^{\omega_{1}}2$

$\\{\alpha<\omega_{1}:F(f\restriction\alpha)=g(\alpha)\\}\text{ is
stationary}.$

Note that the “guessing power” is being able to say “yes or no” (i.e. 0 or 1),
i.e. the range of $g$ is 2 (not even 3). Also the guesses are on $\omega_{1}$,
not any stationary subset $S$, so we can phrase $\Phi(S)$, and let us call $S$
small if it fails. The small sets form a normal idea. There are generalization
for higher cardinals.

A corollary is that if $\Gamma(G)\subseteq\omega_{1}$ (see [She74]) is not
small then $G$ is not Whitehead. Also if $S\subseteq\omega_{1}$ is not small,
$\bar{\eta}=\langle\eta_{\delta}:\delta\in S\rangle$ a ladder system then it
is not even 2-uniformizable (see [She77]).

(B66) End extensions and numbers of countable models, JSL 43(1978), 550-562.

We prove that:

  * •

every model of $T=\mathrm{Th}(\omega,<,\ldots)$ for $T$ countable has a
(proper) end extension (so addition and multiplication are not used, hence
definition by induction is not available);

  * •

every countable theory with an infinite order and Skolem functions has
$2^{\aleph_{0}}$ non-isomorphic countable models; (instead of Skolem
functions, existence of arbitrarily large finite intervals can be assumed);

  * •

if every model of $T$ has an end extension, then every $|T|$-universal model
of $T$ has an end extension definable with parameters.

(B67) On the number of minimal models, JSL 43(1978), 475-480.

We prove that for every $\kappa$, $1\leq\kappa\leq\aleph_{0}$, there is a
(countable) complete theory $T$, with no prime model, and exactly $\kappa$
minimal models (up to isomorphism). The examples are specifically built for
this purpose.

It is clear that if the number is $>\aleph_{1}$ it is $2^{\aleph_{0}}$.

(B68) Jónsson algebras in successor cardinals, IJM 30(1978), 57-64.

We shall show here that in many successor cardinals $\lambda$, there is a
Jónsson algebra (in other words $\mathrm{Jn}(\lambda)$ or $\lambda$ is not a
Jónsson cardinal). In connection with this we show that, e.g., for every
ultrafilter $D$ over $\omega$, in $(\omega_{\omega},<)^{\omega}/D$ there is no
increasing sequence of length $\aleph_{(2^{\aleph_{0}})^{+}}$ and a lemma on
the existence of l.u.b. We prove e.g., $\mathrm{Jn}(\aleph_{\omega+1})$ when
$2^{\aleph_{0}}\leq\aleph_{\omega+1}$ and $\mathrm{Jn}(2^{\aleph_{0}})$ when
$2^{\aleph_{0}}=\aleph_{\alpha+1}$, $\alpha<\omega_{1}$ and similar results
for higher cardinals.

(B69) On a problem of Kurosh, Jónsson groups and applications work problem,
II, edited by Adjan, Boone and Higman North Holland Publ. Col., 1979

We prove some results in group theory in model theoretic spirit.

1) Construct Jónsson groups of cardinality $\aleph_{1}$, and others as well
($G$ is Jónsson group of cardinality $\lambda$ ($>\aleph_{0}$) if it has no
proper subgroup of cardinality $\lambda$); this solves a problem of Kurosh.

2) Our group is simple with no maximal subgroup; so it follows that taking
Frathini subgroups does not commute with direct products.

3) Assuming $2^{\aleph_{0}}=\aleph_{1}$, our group has no non-trivial
Hausdorff topology; this answers a question of Markov.

We use small cancellation theory, but quote exactly what we use; we assume
only knowledge of naive set theory and little group theory (except the theorem
we quote).

(B70) Modest theory of short chains, II, JSL 44(1979), 491-502 (with Yuri
Gurevich).

A chain (i.e. a linearly ordered set) is _short_ if it embeds neither
$\omega_{1}$ nor $\omega^{*}_{1}$. Shortness is easily expressible in the
monadic language of order. Given a chain $C$ and a positive integer $p$ we
define (in the monadic language of order) $p$-_modest_ subsets in $C$. A
subset of $C$ is _modest_ in $C$ if it is $p$-modest in $C$ for every $p$. A
chain is _modest_ if it is modest in itself, it is _absolutely modest_ if it
is modest in its Dedekind completion.

######  Theorem 2.23.

The monadic theory of modest short chains coincides with that of countable
chains.

Decidability of the latter theory is due to Rabin; it is reproved here.

######  Theorem 2.24.

The monadic theory of any non-modest short chain is undecidable under the
Continuum Hypothesis. (This use of the Continuum Hypothesis is eliminated in a
later paper “Monadic theory of order and topology in ZFC” by the same
authors.)

The _modest theory_ of a chain $C$ is the theory of $C$ in the monadic
language of orders when the set variables range over absolutely modest subsets
of $C$.

######  Theorem 2.25.

The modest theory of the real line $R$ is decidable; it coincides with the
theory of $R$ with quantification over countable sets, with the theory of $R$
with quantification over sets of cardinality $<2^{\aleph_{0}}$, and with the
modest theory of any complete short chain without jumps and ends having an
everywhere dense absolutely modest subset.

It is proved also that Theorem 3 is in some sense best possible.

(B71) A note on cardinal exponentiation, JSL 45(1980), 56-66.

For an $\aleph_{1}$-complete filter $D$ on $I$, for $f:I\rightarrow{\rm Ord}$,
let: $\|f\|_{D}$ (the rank of $f$) be $\sup\\{\|g\|_{D}:g<_{D}f\\}$ and

$T_{D}(f)=\sup\big{\\{}|G|:g\subseteq I({\rm Ord}),\ (\forall g\in G)g<_{D}f,\
(\forall g_{1}\neq g_{2}\in G)[g_{1}\neq_{D}g_{2}]\big{\\}}.$

Our results are:

  * •

improvements of the bounds on $\|f\|_{D}$, e.g. if $|\alpha|=\beth_{\omega}$
and $|I|<\beth_{\omega}$ then $\|\alpha\|_{D}<\beth^{+}_{\omega}$, hence

  * •

corresponding bound on $2^{\aleph_{\alpha}},\aleph_{\alpha}$ strong limit,
${\rm cf}(\alpha)\geq\aleph_{1}$

  * •

bounded on $2^{\aleph_{\alpha}}$ in new cases: if
$\aleph_{\lambda}(\aleph_{0})$ is the first $\aleph_{\alpha}=\alpha$ of
cofinality $\lambda$ and $\aleph_{\omega_{1}}(\aleph_{0})$ is strong limit
then $2^{\aleph_{\omega_{1}}(\aleph_{0})}<\aleph_{\omega_{2}}(\aleph_{0})$
provided that the Chang conjecture holds.

The method is that in the proof rather than fixing a filter (say the club
filter) we look at all possible filters, say normal filters on $\omega_{1}$
(doing induction together) and that we get bounds on the rank by cases of
$T_{D}(-)$ (a kind of converse to previous results).

(B72) Models with second order properties I. Boolean Algebras with no
definable automorphisms, AML 14(1978), 57-72.

This is (the first) part of a series of papers, in which we try to build
models with various second order properties.

Examples of such properties are:

  1. (a)

let $T$ be a (first order) theory, and we want a model $M$, such that
$M\models T$ and every Boolean algebra definable in $M$, all its automorphisms
are definable in $M$

  2. (b)

we extend first order logic by allowing quantification over automorphism of
Boolean algebras, and we want to show this logic is compact (see [She75d, Th.
2.6, pg. 353], [She72b, Th. 25]) for a definition of a second order
quantification, and this amounts to assuming $T$ satisfies various schemes,
e.g., $T$ has a model expanding $({\mathscr{H}}(\kappa^{+}),\in)$

  3. (c)

we have a model $M$ of $T$ in which a (definable) Boolean algebra is rigid,
and we want to build other such models.

We want more concretely to build those models in specific cardinals, and
replace “automorphisms of Boolean algebras” by automorphisms of other
structures (e.g. ordered fields) branches of trees, etc.

Note that (a) is stronger than (b) which is stronger than (c); and in (b), (c)
we can expand the language, so without loss of generality the theory has
Skolem functions, and has enough build in set theory. But in (a) this is
forbidden. Note that if we e.g. want to get a $(\lambda^{+},\lambda)$-model in
which a tree of height $\lambda^{+}$ has only definable branches, Chang’s
original proof of his two-cardinal theorem is not appropriate, as he extends
the language (so to encode finite sets). Note also that (b) is good for giving
examples of compact logic. For example, allowing quantification over
automorphisms of Boolean algebras, gives us a logic stronger than first order
even on finite models (we can say a Boolean algebra is atomic, and has an
automorphism of order two, which moves every atom. This distinguishes among
the finite Boolean algebras between those with an even number of atoms. We
shall prove this logic is compact (assuming GCH).

Let us return to this paper.

In [RS80] we prove (assuming $\diamondsuit_{\aleph_{1}}$) that for a Boolean
algebra $B$, the (two-sorted) model $(B,{\rm Aut}(B))$ (with the Boolean
operations, and the operations of applying an automorphism (where Aut$(B)$ is
the group of automorphisms of $(B)$) is elementarily equivalent to
$(B^{\prime},\text{Aut}(B^{\prime}))$ for some $B^{\prime}$ of cardinality
$\aleph_{1}$. We can add more relations and make other strengthening. Our
first aim was to generalize this to uncountable cardinalities. This is done in
§1, for an appropriate $\lambda$ our results are stronger; because we prove a
theory $T$ has a model in $\lambda^{+}$ in which every automorphism of
$(P(M)\cup Q(M),R^{M})$ is inner (i.e. definable is the model) and we do not
expand the language to have, essentially, some replacement axioms. Here the
requirements on $\lambda$ are severe; but this paper was written in order to
exemplify the technique developed for solving the problem.

In §2 we use our technique to prove (a slightly stronger version of) the Chang
two-cardinal theorem without expanding the language.

In some sense our problem is to omit types of power $\lambda$ in models of
power $\lambda^{+}$.

(B73) Models with second order properties. II. Trees with no undefined
branches, AML 14(1978), 73-87.

We prove several theorems of the form: a first order theory $T$ has a model
$M$ (sometimes with additional conditions) such that (some) trees defined in
$M$, have no branches except those defined in it. For this we omit types of
cardinality $\lambda$ for constructing models in $\lambda^{+}$ using “strong
splitting”. We have some applications, e.g. an example for compact logic
$\mathbb{L}(Q)$, where in $\mathbb{L}_{\omega_{1},\omega}(Q)$ well-ordering is
definable, its compactness is proved in ZFC, and it is stronger than first
order logic even for countable models; and eliminate extra set-theoretic
axioms (like $\diamondsuit_{\aleph_{1}}$) from some theorems. The paper also
deals with constructing models of cardinality $\aleph_{1}$.

Also we prove the absoluteness of the answer to “does
$\psi\in\mathbb{L}_{\omega_{1},\omega}(\exists^{\geq\aleph_{1}})$ have a model
$M$ such that every tree with sets of levels of cofinality $\aleph_{1}$ has no
(full) branch which is not (first order) not definable (with parameters)?”. So
if we can find such a model, assuming $\diamondsuit_{\aleph_{1}}$, then we can
really construct one222if we use a countable fragment ${\mathscr{L}}$ of
$\mathbb{L}_{\omega_{1},\omega}(\exists^{\geq\aleph_{1}})$ to which $\psi$
belongs and consider ${\mathscr{L}}$-definability with parameters,
straightforwardly there is such a model if $\diamondsuit_{\aleph_{1}}$ hence
provably in ZFC (e.g. a model of ZF or PA: the “classes” of it are branches of
such a tree).

(B74) Appendix, Vaught two cardinal theorem revisited.

We gave an alternative proof to Vaught’s theorem
$(\lambda^{+},\lambda)\rightarrow(\aleph_{1},\aleph_{0})$ which reveal the
corresponding partition theorems from [She71c], i.e. we characterize them.

(B75) A Banach space with few operators, IJM 30(1978), 181-191.

Assuming the axiom (of set theory) $\mathbf{V}=\mathbf{L}$ (explained below),
we construct a Banach space with density character $\aleph_{1}$ such that
every (linear bounded) operator $T$ from $B$ to $B$ has the form $aI+T_{1}$,
where $I$ is the identity and $T_{1}$ has a separable range. The axiom
$\mathbf{V}=\mathbf{L}$ means that all the sets in the universe are in the
class $\mathbf{L}$ of sets constructible from ordinals; in a sense this is the
minimal universe. In fact, we make use of just one consequence of this axiom,
$\diamondsuit_{\aleph_{1}}$ proved by Jensen, which is widely used by
mathematical logicians.

(B76) Independence of strong partition relations for small cardinals and the
free subset problem, J.45(1980), 505-509.

We prove the consistency of ZFC + GCH with the following (assuming
$\mathrm{con}(\mathrm{ZFC})$ \+ “there are $\omega$ measurable cardinals”):

  1. $(*)$

for any function $f$ from finite subsets of $\aleph_{\omega}$ to $\omega$
there are pairwise disjoint $S_{n}\subseteq\omega_{2n}$ $(1\leq n<\omega)$
satisfying $|S_{n}|=\aleph_{2n}$ such that: if $u,v$ are finite subsets of
$\aleph_{\omega}$ and $(\forall n)[|S_{n}\cap u|=|S_{n}\cap v|\leq 1]$ then
$f(u)=f(v)$.

On the free subset problem we prove the following: $(*)$ implies that any
algebra with countably many functions and power $\aleph_{\omega}$ has a free
subset $A$ of power $\aleph_{0}$, i.e. $(\forall a\in A)$ ($a$ is not in the
subalgebra generated by $A\setminus\\{a\\}$).

We get similar independence results for $\aleph_{\alpha}>|\alpha|$, i.e. a
free subset of power $|\alpha|$, and prove we cannot get a free set of power
$|\alpha|^{+}$ in $\aleph_{\alpha}$. Continued in [She81a].

(B77) Existentially-closed groups in $\aleph_{1}$ with special properties,
Bull. Greek Math. Soc. 18(1977), 17-27.

We prove: for any countable e.c. group (= existentially closed group) $G^{*}$,
there is an e.c. group $M\equiv{}_{\infty,\omega}G^{*}$ of cardinality
$\aleph_{1}$ such that:

  1. (i)

$M$ is complete; i.e. every automorphism is inner

  2. (ii)

(CH) $M$ has no uncountable abelian subgroup.

(B78) Hanf number of omitting types for simple first-order theories, JSL
44(1979), 319-324.

Let $T$ be a complete countable first-order theory such that every ultrapower
of a model of $T$ is saturated. If $T$ has a model omitting a type $p$ in
every cardinality $<\beth_{\omega}$, then $T$ has a model omitting $p$ in
every cardinality. There is also a related theorem, and an example showing the
$\beth_{\omega}$ cannot be improved. Continued in [Sh:334].

(B79) On uniqueness of prime models, JSL 44(1979), 215-220.

We prove there are theories (stable or countable) for which over every $A$
there is a prime model but it is not necessarily unique. We also give a
simplified proof of the uniqueness theorem for countable stable theories.

(B80) A weak generalization of MA to higher cardinals, IJM 30(1978), 297-306.

We generalize MA to $\aleph_{1}$-complete forcing notions satisfying a
strengthening of the $\aleph_{2}$-c.c. described in $(*)$ below, and prove its
consistency with ZFC + CH. (The number of dense sets is $<2^{\aleph_{1}}$, of
course). Where

  1. $(*)$

for any $p_{i}\in P$ (for $i<\aleph_{2}$) there are regressive functions
$f_{n}:\omega_{2}\rightarrow\omega_{2}$, such that if
$\alpha,\beta<\aleph_{1}$,
$\bigwedge\limits_{n<\omega}f_{n}(\alpha)=f_{n}(\beta)$ then
$p_{\alpha},p_{\beta}$ have a least upper bound.

We get similar consistency results (with
$ZFC+CH+``2^{\aleph_{1}}=2^{\aleph_{2}}$”) like a uniformization property for
suitable $\langle\eta_{\delta}:\delta\in S^{2}_{1}\rangle$.

As a result we get the consistency of the decidability of the monadic theory
of $\omega_{2}$.

(B81) The consistency with CH of some consequences of Martin Axiom plus
$2^{\aleph_{0}}<2^{\aleph_{1}}$, 31(1978), 19-33 (with U. Abraham and K.
Devlin).

The main conclusion is the consistency with GCH of:

  1. $(*)(1)$

for any ladder system $\langle\eta_{\delta}:\delta<\omega_{1}\rangle$
($\delta$ limit, $\eta_{\delta}$ an $\omega$-sequence converging to $\delta$)
and $c_{\delta}\in{}^{\omega}\omega$, for some $h:\omega_{1}\rightarrow\omega$
we have
$(\forall\delta)(\exists^{\infty}k)[h(\eta_{\delta}(k))=c_{\delta}(k)]$

  2. $(*)(2)$

if $G$ is a graph, $\omega_{1}$ its set of models,

$(\forall\alpha,\beta)\big{[}\alpha+\omega\leq\beta\rightarrow(\exists^{<\aleph_{0}}\gamma<\alpha)\langle\gamma,\beta\rangle\in
G\big{]}$

then $G$ has chromatic number $\leq\aleph_{0}$.

The method the known iteration of forcing by Suslin trees following a work of
Jensen; and a general principle SAD implying $(*)$ is phrased, and its
consistency proved.

(B82) Models with second order properties, III: omitting types for
$\mathbb{L}(Q)$, Archive f. Math Logic, 21(1981), 1-11.

A known technique for building model a power $\aleph_{1}$ is by building an
increasing continuous chain $M_{\alpha}$ $(\alpha<\omega_{1})$ of countable
models, and in stage $\alpha$ we are allowing to promise to omit countably
many types with parameters from $M_{\alpha}$ “strongly omitted” by
$M_{\alpha}$. We prove the parallel theorem with $\aleph_{0},\aleph_{1}$
replaced by $\lambda,\lambda^{+}$ provided that $\diamondsuit_{\lambda}$ holds
(or $\lambda$ is strongly inaccessible). However, we prove
$\diamondsuit_{\lambda}$ holds for any successor $\lambda\neq\aleph_{1}$ if
GCH holds.

The obstacle for straightforward generalization is preserving the induction
hypothesis in limit stages of cofinality $<\lambda$.

The main idea is to use $\diamondsuit_{\lambda}$, while constructing
$M_{\alpha+1}$ by approximation of power $<\lambda$, to “guess” obstacles in
the future.

(B83) Existentially closed structures in the power of the continuum, Ann. Pure
Appl. Logic 26(1984), 123-148 (with Giorgetta)

We deal with three cases:

  * •

$G$ is a countable universal locally finite group

  * •

$G$ is a countable existentially closed group,

  * •

$G$ is a countable existentially closed division ring.

In each case $K=\\{H:H$ is $\mathbb{L}_{\infty,\omega}$-equivalent to $G\\}$
which here means: every embedding of a finitely generated subalgebra of $G$
into $H$ can be extended to an embedding of $G$ into $H$, and every countable
subalgebra of $H$ can be embedded into $G$.

We prove that $K$ has members of power $2^{\aleph_{0}}$; in fact, many quite
complicated members which e.g. do not have $\aleph_{1}$ pairwise commuting
elements. The method is general: we build the model of power $2^{\aleph_{0}}$
as, essentially, an inverse limit of
$\langle M^{n}_{u}:u\in\mathcal{P}(n)\rangle$, $M^{n}_{u}$ finitely generated,
so the problems become finitary: for $N\subseteq N_{0}$, $N_{1}$ finitely
generated, amalgamate $N_{0},N_{1}$ over $N$ with some extra requirements. For
those we have some criteria, but the main work is specific to each case.

(B84) On elementary equivalence of automorphism groups of Boolean Algebras,
downward Skolem-Löwenheim theorems and compactness of related quantifiers,
JSL 45(1980), 265-283 (with M. Rubin).

(B85) A note on the normal Moore space conjecture Can. J. Math., Vol.XXXI,
2(1979), 241-251 (with Devlin).

In this paper we show that if CH be assumed, then Jones’ spaces are not normal
and that the GCH does not lead to a positive solution to the Jones conjecture.
A brief survey of the progress on the problem to date is also included. Jones’
spaces are closely related to Aronszayn trees.

(B86) Souslin properties and Tree Topologies Bull. London Math Soc., (3),
39(1979), 537-552.

(B87) Classification theory for non-elementary classes. I. The number of
uncount- able models of $\psi\in\mathbb{L}_{\omega_{1},\omega}$, Israel J.
Math., 46(1983), 212-273.

######  The Main Theorem 2.26.

Assuming $\forall n<\omega$, $2^{\aleph_{n}}<2^{\aleph_{n+1}}$; if
$\psi\in\mathbb{L}_{\omega_{1},\omega}$ ($\tau_{\psi}$ countable) has an
uncountable model then:

  * •

there exists $n$ positive such that $I(\aleph_{n},\psi)=2^{\aleph_{n}}$ or

  * •

$\psi$ has models in every uncountable cardinality, and iff
$(\exists\lambda>\aleph_{0})I(\lambda,\psi)=1$ then
$(\forall\lambda>\aleph_{0})I(\lambda,\psi)=1$.

As in [She75c] we consider a class of atomic models $K$ of some first order
complete theory.

It is proved that it is enough to prove the Main Theorem for $K$. Assuming
$2^{\aleph_{0}}<2^{\aleph_{1}}$ and $I(\aleph_{1},\psi)<2^{\aleph_{1}}$ basic
stability machinery is introduced, i.e. a substitute to non-forking; and a
substitute to saturated models in elementary classes.

The key to classify the complexity of the class $K$ instead by stability, to
do it by generalized amalgamation properties (remember the role of
$\aleph_{0}$ \- A.P. in [She75c]). The problem to find A.P.-s such that the
failure of those A.P.-s will imply existence of many non-isomorphic models and
if those generalized A.P.-s holds then to conclude existence of models in
higher cardinalities.

The paper logically is divided to three parts: first developing basic
stability theory. Second, introducing various amalgamation properties proving
some interrelations among them and proving that if all the A.P.-s holds them
$K$ has models in every cardinality and the Morley categoricity theorem for
$K$. The third part is by combinatorial work to prove that the failure of A.P.
implies the existence of many non-isomorphic models.

We try to minimalize the use of set theoretical assumptions: the main result
of [She75c] is proved from $2^{\aleph_{0}}<2^{\aleph_{1}}$ only. Most parts of
the last part of the paper can be proved from the assumption $(\forall
n<\omega)$ that $2^{\aleph_{n}}<2^{\aleph n+1}$.

It is an open problem to prove the Main Theorem in ZFC only, see also
[She87a]. Continued in [She09].

(B88) Classification theory for non-elementary classes. II. Abstract
elementary
classes, Lecture Notes in Math. 1292(1987) 419-497.

We deal with a class of models $K$ all of the same similarity type, and a
partial order $<$ on $K$ (which stands e.g. for elementary submodels) the only
demand except the obvious is existence of $\lambda(K)$ which is a downward
Skolem-Lowenheim number, and $(K,<)$ satisfies Tarski-Vaught elementary chain
theorems. It is proved from ZFC only that $I(\aleph_{1},K)=1\Rightarrow
I(\aleph_{2},K)\geq 1$ (this improves [She75c], [She83a], [She83b]) and
assuming $2^{\aleph_{0}}<2^{\aleph_{1}}$ we show that $1\leq
I(\aleph_{1},K)<2^{\aleph_{1}}$ implies $I(\aleph_{2},K)\geq 1$, the methods
are classifying by the amalgamation properties (as in [She75c] and [She83a],
[She83b]). Most of the work is preparation to generalize the Main Theorem from
[She83a], [She83b] to this context. Continued in [She01], [She09].

(B89) Boolean algebras with few endomorphisms, Proc. AMS 14(1979), 135-142.

Any Boolean algebra has some trivial endomorphism - the identity, an
endomorphism into $\\{0,1\\}$ corresponding to an ultrafilter, and some
combinations of them. We build (assuming $\diamondsuit_{\aleph_{1}}$ and then
CH only) Boolean algebra of power $\aleph_{1}$, such that every endomorphic is
definable from finitely many elements and ultrafilters. We also generalize a
construction of Rubin of some special Boolean Algebras in $\aleph_{1}$ to
higher cardinals. Continued in [Shear, Ch.I].

(B91) The structure of Ext$(G,\mathbb{Z})$ and $\mathbf{V}=\mathbf{L}$, Math
Zeitschrift 162(1978), 39-50 (with H.L. Hiller and M.K. Huber).

Let $G$ be a torsion free abelian group, then it is well know that ${\rm
Ext}(G,\mathbb{Z})$ is divisible, hence determine by its $p$-ranks ($p$ zero
or prime) $\nu_{p}(G)$.

We prove that for non-free $G$ (assuming $\mathbf{V}=\mathbf{L}$):
$\nu_{0}(G)$ is $\min\\{2^{|H|}:H$ a subgroup of $G$, $G/H$ free$\\}$, hence
$\nu_{0}(G)$ is a successor cardinal and $\nu_{p}(G)\leq\nu_{0}(G)$.

We determine the possible $\nu_{p}(G)$ when ${\rm Hom}(G,\mathbb{Z})=0$, and
we characterize the abelian group to which there exists co-Moore spaces; and
assert that instead $\mathbf{V}=\mathbf{L}$ it is enough to assume no
stationary subset of a regular cardinal is small. Continued in [MRS99].

(B92) Remarks on Boolean Algebras, Algebra Universalis, 11(198), 77-89.

In §4, we show that if $B$ satisfies the $\kappa$-chain condition,
$|B|>\lambda=\lambda^{<\kappa}$ then $B$ has large independent subsets, e.g.
of power $\geq\lambda^{+}$. We can replace $\lambda^{+}$ by $\chi={\rm
cf}(\chi)$ if $(\forall\alpha<\chi)(|\alpha|^{<\kappa}<\lambda)$, and find an
independent subset $A^{\prime}$ of $A\in[B]^{\chi}$ of cardinality $\chi$. If
$\chi$ is singular, $(\forall\alpha<\chi)(|\alpha|^{<\kappa}<\lambda)$ and
$A\in[B]^{\chi^{+}}$ we can find an independent $A^{\prime}\subseteq A$ of
cardinality $\chi$.

In fact, our results are almost complete.

We prove some theorems on Boolean Algebras. In §1, we prove that if $S$ is a
subset of a (Boolean Algebra) $B$, $|S|$ singular strong limit, then
$S^{-}=\\{a-b:a,b\in S\\}$ has a subset of power $|S|$ which is a pie subset
(a subset no two elements of which are comparable).

In §2 we prove the consistency of $ZFC+2^{\aleph_{0}}>\aleph_{1}$ with “there
is a Boolean Algebra $B$ of cardinality $2^{\aleph_{0}}$, with no uncountable
pie subset, nor an uncountable chain.”

In §3, we prove that if $|B|=\lambda^{+}$, $\lambda=\lambda^{<\kappa}$, $B$
satisfies the $\kappa$-chain condition, then $B\setminus\\{0\\}$ is the union
of $\lambda$ ultrafilters (proper, of course). We shall show in [She81b] that
this is not necessarily true when $|B|=\lambda^{++}$, and other connected
results.

(B93) Simple unstable theories, Ann. Math. Logic, 19(1980), 177-203.

Being simple is a property of first order complete theories weaker than
stability. We start their investigation, using a a test question asking if
$|T|<\kappa$, $\mu=\mu^{\kappa}<\lambda=\lambda^{|T|}\leq 2^{\mu}$, $M\models
T$, $\|M\|=\lambda$ implies $M$ has a $\kappa$-saturated elementary extension
of cardinality $\lambda$. For non-simple the answer is no, for simple theories
the answer is consistently yes. We start to develop a parallel of [She78,
Ch.II,III].

(B94) Weakly compact cardinals: a combinatorial proof, JSL 44(1979), 559.

We give here a direct purely combinatorial proof that weak compactness is
equivalent to a combinatorial property (2). This property (2) is apparently
stronger, and from it, all other usual equivalent definitions and usual
properties of weakly compact cardinals can be deduced. So this proof may be
useful for books which want to present weakly compact cardinals, but not
logic.

######  Remark 2.27.

For $\mu$ the first inaccessible we can define a $\mu$-tree with no
$\mu$-branch by:

$\begin{array}[]{clcr}T=\bigl{\\{}h:&\text{Dom }h\text{ an ordinal
}\alpha<\mu,\ h(i)<1+i,\\\ &\text{and for strong limit }i,j\in\text{ Dom }h,\
h(i)\neq h(j)\bigr{\\}}.\end{array}$

######  Theorem 2.28.

For $\mu$ strongly inaccessible the following are equivalent:

  * •

$\mu$ is weakly compact, i.e., every $\mu$-tree has a $\mu$-branch

  * •

for every family of functions
$f_{\alpha}:\alpha\rightarrow\alpha\,(\alpha<\mu)$ there is a function
$f:\mu\rightarrow\mu$ such that:
$(\forall\alpha<\mu)(\exists\beta)\alpha\leq\beta<\mu$ and
$[f_{\beta}\restriction\alpha=f\restriction\alpha]$.

(B95) Canonization theorems and applications, JSL, 45(1981), 345.

We prove canonization theorems generalizing Erdős-Rado partition theorems. We
introduce a very general partition relation which generalizes
$\lambda\rightarrow(\kappa)^{\chi}_{\mu}$ in many aspects. We prove a
corresponding partition theorem (which works simultaneously for sequences of
cardinals). E.g. assume $\langle\lambda_{i}:i<\alpha\rangle$ a sequence of
cardinals, $|\alpha|+\kappa<\lambda_{i}$,
$\langle\mu_{i}=\beth_{n}(\lambda_{i})^{+}:i<\alpha\rangle$ is strictly
increasing,
$F:\big{[}\bigcup\limits_{i}\mu_{i}\big{]}^{<\omega}\rightarrow\theta$, then
we can find $\langle A_{i}:i<\alpha\rangle$,
$A_{i}\in[\mu_{i}\setminus\bigcup\limits_{j<i}\mu_{j}]^{\lambda^{+}_{i}}$ and
$F$ is restricted to increasing sequences from
$\bigcup\limits_{i<\alpha}A_{i}$ does not depend on the last $n$ arguments.
Moreover, in the inductive proof if we restrict how the sequences are
distributed between the $A_{i}$’s, e.g. to those of $\bar{n}$-kind, where
$\bar{a}$ is of the $\bar{n}$-kind if

$n_{\ell}=\Big{|}\Big{\\{}k<\ell g(\bar{a}):\ell
g(\bar{n})-k+1=\big{\\{}i<\alpha:\varnothing\neq\bar{a}\cap\mu_{i}\setminus\bigcup\limits_{j<i}\mu_{j}\big{\\}}\Big{\\}}\Big{|}.$

If in the last one there is one element (e.g. $n_{\ell g(\bar{n})-1}=1$, we
can gain 2 places provided we increase the function
$\beth_{n}(\lambda_{i})^{+}$ which mainly consists of $n$ exponentiation, by
some plusses. If $\alpha<\omega$, we can gain more.

As an application we complete the answer to the following question:

Given a Hausdorff space $X$ of cardinality $\lambda$, characterize the pairs
of cardinals $(\lambda,\chi)$ such that $X$ has a subspace $Y$, of cardinality
$\chi$, which is discrete. Also, generalize Hajnal’s free subset theorem for
tuples instead of one-place functions.

(B96) Algebraically closed groups of large cardinality, JSL 44(1979), 232-552
(with Martin Ziegler).

Let $M$ be a countable algebraically closed group, $\kappa$ an uncountable
cardinal. We will prove in this paper the following theorems.

######  Theorem 2.29.

There is an algebraically closed group $N$ of cardinality $\kappa$ which is
$\mathbb{L}_{\infty,\omega}$-equivalent to $M$.

######  Theorem 2.30.

There is an algebraically closed group $N$ of cardinality $\kappa$ which is
$\mathbb{L}_{\infty,\omega}$-equivalent to $M$, and contains a free abelian
group of cardinality $\kappa$.

######  Theorem 2.31.

There are $2^{\kappa}$ non-isomorphic algebraically closed groups of
cardinality $\kappa$ which are $\mathbb{L}_{\infty,\omega}$-equivalent to $M$.

######  Theorem 2.32.

There is an algebraically closed group $N$ of cardinality $\kappa$ which is
$\mathbb{L}_{\infty,\omega}$-equivalent to $M$ and satisfies: every subgroup
of $N$ of uncountable regular cardinality contains a free subgroup of the same
cardinality.

The method is to construct things parallel to Ehrenfeucht-Mostowski models,
using things like centralizers of elements.

(B97) Unordered types of ultrafilters, Topology Proc. 3(1979), 199-204 (with
M.E. Rudin).

It is proved in ZFC that there are the maximal number of incomparable
ultrafilters on $\omega$, in the Rudin-Kaisler ordering (and similarly for
other cardinals).

(B98) Whitehead groups may not be free even assuming CH,II

(A) We prove the consistency of $ZFC+2^{\aleph_{0}}=2^{\aleph_{1}}+$ club
(i.e. $\clubsuit$), where club is the following weakening of diamond:

  1. $(*)$

there is $\langle A_{\delta}:\delta<\omega_{1}\rangle$, $A_{\delta}$ an
unbounded subset of $\delta$ such that for any unbounded
$A\subseteq\omega_{1}$, $\\{\delta:A_{\delta}\subseteq A\\}$ is stationary.
(Under CH, club and diamond were known to be equivalent).

The method is to start with a model of ZFC + GCH, blow up $2^{\aleph_{1}}$ and
then collapse $\aleph_{1}$.

(B) For simplicity let $\Phi$ be a family of $\omega$-sequences from a set
called ${\rm Dom}(\Phi)$; $(\Phi,h)$ has the uniformization property if for
any functions $f_{\eta}:{\rm Rang}(\eta)\rightarrow$ ordinals satisfying
$f_{\eta}(i)<h(i)$, there is $f:{\rm Dom}(\Phi)\rightarrow$ ordinals, such
that for every $\eta\in\Phi$ for all but finitely many $n$,
$f(\eta(n))=f_{\eta}(n)$.

We investigate such properties.

(B1) We prove that for a stationary $S\subseteq\omega_{1}$ and $\eta_{\delta}$
converging to $\delta$, then property “$(\langle\eta_{\delta}:\delta\in
S\rangle,2)$ has the uniformization property” does not depend on $S$ only: in
fact, seemingly trivial changes in $\eta_{\delta}$ can change its truth value.

(B2) We prove that if in the weak diamond (see [DS78]), restricting ourselves
to a stationary co-stationary $S$, replacing 2 by 3 give a non-equivalent
statement.

(B3) We prove that ${\rm Dom}(\Phi)$ may be $\omega,\Phi$ uncountable but
still $(\Phi,2)$ has the uniformization property, i.e. this is consistent with
$ZFC+2^{\aleph_{0}}=2^{\aleph_{1}}$. Though this looks like a consequence of
$MA+2^{\aleph_{0}}>\aleph_{1}$, it in fact contradicts MA. So the result says
that there are infinite $A_{\alpha}\subseteq\omega$ for $\alpha<\omega_{1}$
almost disjoint (i.e. $\alpha<\beta\Rightarrow|A_{\alpha}\cap
A_{\beta}|<\aleph_{0}$) and for any $h_{\alpha}\in{}^{(A_{\alpha})}2$ for
$\alpha<\omega_{1}$ there is $h\in{}^{\omega}2$ such that
$\alpha<\omega_{1}\Rightarrow h_{\alpha}\subseteq^{*}h$. In fact we have a
condition on $\langle A_{\alpha}:\alpha<\alpha^{*}\rangle$ preserved by finite
support iteration such that it guarantees that the natural forcing of adding
$h$ for any given $\bar{h}=\langle h_{\alpha}:\alpha<\alpha^{*}\rangle$
satisfies it.

Group Theory:

(C) We prove that occurance of non-free Whitehead groups, and $(\Phi,2)$ with
the uniformization property $\Phi$ “non-trivial” are closely related (for
power $\aleph_{1}$-equivalent). Hence by (B1), the property of being Whitehead
is quite delicate, maybe depending on seemingly trivial changes.

We prove a set theoretic result which shows that for many non-free $G$, ${\rm
Ext}(G,\mathbb{Z})$ is large (compare with [HHS78]).

(B99) Equi-consistency results, Notre Dame J. Formal Logic 26(1985), 178-188
(with L. Harrington).

There are two topics:

MA and measurability and the Baire property: The following are equi-
consistent:

  * •

ZFC + there is a weakly compact cardinal

  * •

ZFC + MA for every real $r$, $\aleph^{L[r]}_{1}<\aleph_{1}$

  * •

ZFC + MA + every $\Delta^{1}_{3}$-set of reals is measurable

  * •

ZFC + MA + every $\Delta^{1}_{3}$-set of reals has the Baire property.

We can replace $\Delta^{1}_{3}$ by first order definable with a real and
ordinal parameter.

Reflection of stationary sets: The following are equi-consistent:

  * •

ZFC + there is a Mahlo cardinal

  * •

ZFC + GCH + every stationary $S\subseteq S^{2}_{0}$ has an initial segment
stationary.

(B100) Independence results, JSL 45(1980), 563-573.

THE METHOD: We introduce the oracle chain condition which (for $\aleph_{1}$)
helps us to use iterated forcing of length $\aleph_{2}$, each step increasing
the set of reals, by enabling us to omit types (i.e. to promise no real
satisfying $\aleph_{1}$ conditions will be added later on).

We then introduce proper forcing, which is a very general condition
guaranteeing $\aleph_{1}$ is not collapsed and is preserved by countable
support iteration.

In fact we also introduce a blending of the two.

THE RESULTS: We prove the consistency of: $2^{\aleph_{0}}=\aleph_{2}$ and:

  * •

there is a universal linear order of power $\aleph_{1}$

  * •

there are countable complete $T\subseteq T_{1}$, $T$ is not
$\aleph_{0}$-stable, not superstable, but still there is a unique
$L(T)$-reduct of a model of $T_{1}$ of power $\aleph_{1}$.

The results are complimentary to [She78, Ch.VII]. More see [She89].

(B101) The theorems of Beth and Craig in abstract model theory. II. Compact
Logics, Archive Math Logik 21(1980), 13-36 (with J.A. Makowski).

Various compact logics such as stationary logic, positive logic, logics with
various cardinality quantifiers and cofinality quantifiers are studied.
Counterexmples to the theorems of Beth and Craig are given. Back and forth
arguments are studied for the first two logics, transfer theorems presented
for positive logic and a new compactness proof for the cofinality quantifier
is given.

(B102) Forcing with stable posets, JSL 47(1982), 37-42 (with U. Abraham).

(B103) On partitions of the real line, IJM 32(1979), 299-304 (with D.H.
Fremlin).

We proved that the real line is not necessarily the disjoint union of
$\aleph_{1}$ non-empty $G_{\delta}$-sets. This follows from known results and:
if the real line can be partitioned into $\kappa$ sets which are
$G_{\delta}$-sets ($\kappa$ uncountable) then the real line can be covered by
$\kappa$ nowhere dense closed sets.

(B104) The $\aleph_{2}$-Souslin hypothesis, Trans. AMS 264(1981), 411-419
(with R. Laver).

We prove, assuming con(ZFC + there is a weakly compact cardinal) the
consistency of
“$\mathrm{ZFC}+\mathrm{CH}+2^{\aleph_{1}}=2^{\aleph_{2}}=\aleph_{3}$ \+ there
is no $\aleph_{2}$-Souslin tree.”

(B105) On uncountable abelian groups, IJM 32(1979), 311.

In §1 we characterize the Whitehead groups of power $<2^{\aleph_{0}}$,
assuming Martin Axiom: they are the $\aleph_{1}$-free groups satisfying
possibility II or III from [She74]; and, equivalently, they are
$\aleph_{1}$-coseparable or equivalently ${\rm Ext}(-,\mathbb{Z}_{\omega})=0$.

In §2 we construct an $\aleph_{1}$-free group satisfying possibility II which
is not strongly $\aleph_{1}$-free. Hence $MA+2^{\aleph_{0}}>\aleph_{1}$
implies there is a Whitehead group which is not strongly $\aleph_{1}$-free.

We also prove (assuming $\mathbf{V}=\mathbf{L}$ or even
$2^{\aleph_{0}}<2^{\aleph_{1}}$) that there is a strongly $\aleph_{1}$-free,
separable, not $\aleph_{1}$-separable group of cardinality $\aleph_{1}$. At
last we construct an $\aleph_{2}$-free (hence strongly $\aleph_{1}$-free) non-
separable, non-Whitehead group of cardinality $2^{\aleph_{1}}$. The method is
using a tree which imitates $\diamondsuit_{\aleph_{1}}$ in a sense.

In §3 we deal with hereditarily separable groups. If $\mathbf{V}=\mathbf{L}$
(or every stationary subset of a regular cardinal is not small they are just
the free groups). (This strengthens the theorem: if $\mathbf{V}=\mathbf{L}$
every Whitehead group is free.) But $MA+2^{\aleph_{0}}>\aleph_{1}$ implies
there are non-Whitehead, hereditarily separable groups of cardinality. We also
prove, assuming $2^{\aleph_{0}}<2^{\aleph_{1}}$, that any hereditarily
separable group is strongly $\aleph_{1}$-free (a little more, in fact).

CORRECTIONS: Omit 3.3,(iii) and 3.6.

(B106) Martin axiom does not imply that every two $\aleph_{1}$-dense sets of
reals are
isomorphic, IJM 38(1981), 161-176 (with U. Abraham).

RESULTS: We prove the consistency of the following with ZFC +
“$2^{\aleph_{0}}=\aleph_{2}$” (assuming con(ZFC), of course).

1) There are two $\aleph_{1}$-dense sets of reals which cannot be made
isomorphic by any c.c.c. forcing notion (so without loss of generality MA
holds).

2) There is an $\aleph_{1}$-dense set of reals such that any $f:A\rightarrow
A$ is order preserving on some uncountable $A^{\prime}\subseteq A$.

3) For every $A\subseteq R$ of power $\aleph_{1}$, any one-to-one function
$f:A\rightarrow R$ is the union of countably many monotonic functions.

4) MA + there are entangled sets of reals (defined there).

METHOD: A basic fact is that in order to make two $\aleph_{1}$-dense subsets
of $R$ isomorphic, there is a reasonably canonical forcing notion doing it, if
we are given a small enough closed unbounded subset as a parameter.

(B108) On successor of singular cardinals, study in logic and the foundation
of
Math, Vol. 97(ed. boffa, Van dalen and McAllon) No. Holland, 1979, 357-386.

Let us for simplicity assume $\lambda=\aleph_{\omega+1}$, $\aleph_{\omega}$
strong limit. We find a set $S^{*}\subseteq\lambda$, determine modulo the
closed unbounded filter, prove the equivalence of some definitions and:

  * •

if $P$ is $\aleph_{n}$-complete forcing notion,
$S_{1}\subseteq\\{\delta<\lambda,{\rm cf}(\delta)<\aleph_{n},\delta\in
S^{*}\\}$ is stationary, then $S_{1}$ remains stationary after forcing with
$P$. However, there is an $\aleph_{n}$-complete forcing notion which makes
$S^{*}$ non-stationary.

  * •

$S^{*}$ has no initial segment $\delta$ stationary if GCH holds

  * •

$S^{*}=\varnothing$ (i.e. not stationary) iff a weak version of squares holds

  * •

it is consistent (with ZFC + GCH) that $S^{*}$ is stationary (if supercompact
cardinals are)(the large cardinals are necessary).

Using (2) and (3) we get a proof, using different methods according to whether
$S^{*}$ is stationary or not, of

COROLLARY: ($\aleph_{\omega}$ strong limit) there is an
$\aleph_{\omega+1}$-free non-free group of power $\aleph_{\omega+1}$
(similarly for transversal). For higher $\lambda$ we get less information.

Donder noted that as if there is a special Aronszajn on $\aleph_{\omega+1}$
then the weak squares (from 3) holds we can (from (4), (3)) deduce: ZFC + GCH
is consistent with “there is no special Aronszajn tree on $\aleph_{\omega+1}$”
(if a supercompact is consistent).

(B109) Infinite games and reduced powers, AML 20(1981), 77-108 (with W.
Hodges).

By [She71a], if $A,B$ are elementarily equivalent structures then there are
ultrafilters $D,E$ such that $A^{I}/D\cong B^{J}/E$. The paper proves
analogues of this theorem where the language can be infinitary and $D,E$ are
$\kappa$-complete for $\kappa$ a regular cardinal.

1) Let $\mathrm{PL}_{\kappa}$ be the language of prenex formulas of
$L_{\kappa,\kappa}$ with arbitrary well-ordered game quantifiers of length
$<\kappa$. If $A,B$ have mutually consistent $PL_{\kappa}$-theories and
$\kappa$ is strongly compact then $A,B$ have isomorphic limit ultrapowers
where the filter involved are $\kappa$-complete.

2) Let $\mathrm{PH}_{\kappa}$ be the language of Horn formulas in
$PL_{\kappa}$. If for every regular $\kappa$, $A$ and $B$ have mutually
consistent $\mathrm{PH}_{\kappa}$-theories then $A,B$ have isomorphic limit
reduced powers with $\kappa$-complete filters.

3) If there is a proper class of measurable cardinals, then it is consistent
that (1), (2) hold with “limit” deleted; this uses game-theoretic arguments of
Galvin and Laver.

4) The natural analogue of (2) for classes of structures and limit reduced
products also holds and gives interpolation and preservation theorems for
$\mathrm{PH}_{\kappa}$ which had been derived proof-theoretically by Hodges.

5) Necessary and sufficient conditions are given for a structure $A$ to have a
limit reduced power with $\kappa$-complete filters in which a given sentence
is true; as a corollary, when $\kappa>\omega$. The amalgamation property fails
for $\mathrm{PH}_{\kappa}$-elementary extensions.

6) Under various conditions, examples are given of sentences which are
preserved in reduced products over $\kappa$-complete filters but are not
logically equivalent to sentences in $\mathrm{PH}_{\kappa}$, also answering a
related question of Kueker.

7) A non-standard infinitary language is described which has good
interpolation and Feferman-Vaught properties.

(B110) Better quasi-order for uncountable cardinals, IJM 42(1982), 177-226.

We generalize Nash-Williams theory on better quasi-order to uncountable
cardinals.

Let the well-ordering number of a quasi-order $Q$ be the minimal $\lambda$
such that for any $q_{i}\in Q$ ($i<\lambda$) for some $i<j$ we have $q_{i}\leq
q_{j}$, and then we say $Q$ is $\lambda$-well ordered. We define some variants
of $\lambda$-better quasi-order, and prove for them preservation theorems
under the known infinitary operations, (like Seq, ${\mathcal{P}}$). We then
compute the well ordering number of some classes and of the result of the
infinitary operations on arbitrary quasi-order.

The main results are:

  * •

the well-ordering number of the class of graphs ordered by embeddings (taking
edges to edges rather than to paths (or disjoint paths)) is the first
beautiful cardinal $\kappa_{0}$ (it is a mild large cardinal, somewhat bigger
than weakly compact, and still compatible with $L$)

  * •

also the well-ordering number of the class of models with one-place
predicates, under elementary embeddings, is the same

  * •

the well-ordering number of $\\{M:M\text{ a linear order and is the union of
}\leq\lambda\text{ scattered ordered}\\}\,\,(\lambda>\aleph_{0})$, ordered by
embeddability, is the first beautiful cardinal $>\lambda$.

(B112) $S^{-}$ Forcing, a black box theorem for morasses with applications:
super-
Souslin trees and generalized MA, Israel J. Math 43(1982), 185-224 (with L.
Stanley).

(A) Quite a few consistency results can be gotten in the following way:
starting with $\mathbf{V}\models GCH$, we force an example of power
$\aleph_{2}$ by countable approximations. So a countable approximation depends
on a countable set $\alpha$ of ordinals, so it is $\tau(\alpha)$ for some term
$\tau$, and the number of $\tau$-s is $2^{\aleph_{0}}=\aleph_{1}$. We state a
black box principle, which says that many such proofs can be carried in the
universe $\mathbf{V}$ (if it satisfies the principle), moreover the proof that
the black box principle applies is the same as that the forcing works.

Of course, $\aleph_{2}$ can be replaced by higher cardinals.

(B) We prove that the principle follows from $\mathbf{V}=\mathbf{L}$, really
much less is needed: the morass

(C) We got some consequences of the principle, among them the morass, so they
are equivalent.

(D) Another application is the existence of super-Souslin trees. An
$\aleph_{2}$-super-Souslin tree $T$ is an $\aleph_{2}$-tree $T$, such that for
any $\alpha<\omega_{2}$,
$\bar{\alpha}=\langle\alpha_{n}:\alpha<\omega\rangle$, and $\omega$ distinct
members of $T_{\alpha}$, we have a function

$F_{\bar{\alpha}}:\Big{\\{}\langle\beta_{b}:n<\omega\rangle:\alpha_{n}\leq\beta_{n}\wedge(\exists\beta)\bigwedge\limits_{n}\beta_{n}\in
T_{\beta}\Big{\\}}\rightarrow\omega_{1}$

with $F_{\bar{\alpha}}(\beta)=F_{\bar{\alpha}}(\gamma)$ implying
$\beta_{n},\gamma_{n}$ are comparable for infinitely many $n$’s. (So it is
essentially Souslin as a special Aron tree is an Aron tree).

Now an $\aleph_{2}$-super-Souslin tree is $\aleph_{2}$-Souslin (ignoring
possibly finitely many branches), and it remains so in any extension not
adding reals.

Hence: if ${\rm CH}+{\rm SH}_{\aleph_{2}}$ then $\aleph_{2}$ is inaccessible
in $\mathbf{L}$.

(B114) Isomorphism types of Aronszajn trees, IJM 50(1985), 75-113 (with U.
Abraham).

We investigate Aronszajn trees, under embeddability and isomorphism when a
restriction to a club (closed unbounded set of levels) $C$ is allowed, get
consequences of $2^{\aleph_{0}}<2^{\aleph_{1}}$, and consistency results with
CH and with $2^{\aleph_{0}}=2^{\aleph_{1}}$. We also formulate a combinatorial
principle that follows from the weak diamond (which is equivalent to
$2^{\aleph_{0}}<2^{\aleph_{1}}$) and hopefully will serve to obtain results
like those in (A) below.

The results have variants speaking on Specker orders.

(A) Consequences of $2^{\aleph_{0}}<2^{\aleph_{1}}$ (§1).

If CH or even if $2^{\aleph_{0}}<2^{\aleph_{1}}$ holds, then:

  1. (a)

there are $2^{\aleph_{1}}$ pairwise really different Aronszajn trees, i.e. not
one of these trees is embeddable on a club into the other

  2. (b)

there is an Aronszajn tree $T$ such that for every closed unbounded
$C\subseteq\omega_{1}$, $T\restriction C$ is rigid, (i.e. the only embedding
of $T\restriction C$ into $T\restriction C$ is the identity, so $T$ is a
_really rigid_ tree).

We can combine (b) with (a):

  1. (c)

for every Aronszajn tree $T^{1}$ there is an Aronszajn tree $T^{2}$ such that
$T^{1}$ is not embeddable into $T_{2}$ on a closed unbounded set, i.e., there
is no prime Aronszajn tree.

The method is the combinatorial principle mentioned above.

(B) Consistency results with CH (§2).

  1. (a)

GCH + the following is consistent:

    1. $(a_{1})$

there is a universal Aronszajn tree $T$ i.e., a tree $T$ such that for every
Aronszajn tree $T^{*},T^{*}\restriction C$ is order embeddable into
$T\restriction C$ for some closed unbounded $C\subseteq\omega_{1}$. (The
universal tree $T$ is a special Aronszajn tree), so in this model there are no
Souslin trees. Thus $(a_{1})$ is a strengthening of Jensen’s “CH is consistent
with Souslin’s hypothesis,” whose method we use.

    2. $(a_{2})$

Every two Aronszajn trees contain subtrees which are isomorphic on a closed
unbounded set.

  2. (b)

We have some consistency result with CH concerning Souslin trees. For example,
CH + “there is a Souslin tree” do not imply that there are $2^{\aleph_{1}}$
really different Souslin trees. Using this construction, Shelah proved that
the Malitz-Magidor logic is not necessarily compact.

Problem: Does the existence of two really different Souslin trees follow from
the existence of a Souslin tree?

(C) Consistency results with $2^{\aleph_{0}}=2^{\aleph_{1}}$.

  1. (a)

Martin’s axiom + $2^{\aleph_{0}}=\aleph_{2}$ \+ every two Aronszajn trees are
isomorphic on a closed unbounded set is consistent.

  2. (b)

Martin’s axiom + $2^{\aleph_{0}}>\aleph_{1}$ does not imply that every two
Aronszajn trees are isomorphic on a closed unbounded set.

  3. (c)

Martin’s axiom + $2^{\aleph_{0}}=\kappa$ \+ every two Aronszajn trees are
isomorphic on a closed unbounded set is consistent. $\kappa$ here is “any”
regular cardinal such that $\kappa^{<\kappa}=\kappa$.

(The difference between this item and (a) is that in (a) we get
$2^{\aleph_{0}}=\aleph_{2}$ and here $2^{\aleph_{0}}$ is as big as we want;
moreover, the proofs are different: in (a) we use proper forcing and in (c), a
technique of using generic reals).

(B115) Superstable fields and groups, AML 18(1980), 227-270 (with G. Cherlin).

The main object is to prove: for any infinite field $F$, ${\rm Th}(F)$ is
superstable iff $F$ is algebraically closed. This is done by investigating
connected groups ($G$ connected if it has no proper definable subgroup of
finite index) and indecomposable subgroups (same rank and multiplicity one).
Also extend some known results for $\omega$-stable groups with Morley rank
$\leq 3$ to superstable groups (i.e. the theory of the group is superstable).

An important ingredient is looking at definable groups in models of a stable
theory and deal with local ranks $\mathrm{rk}(-,\Delta,\kappa)$,
$\kappa\in\\{2,\aleph_{0}\\}$ for $\Delta$-s which are translation closed.

(B116) Positive results in abstract model theory: a theory of compact logics,
Ann. Pure Appl. Logic 25(1983), 263-299 (with J.A. Makowski).

We prove that compactness is equivalent to the amalgamation property provided
the occurency number of the logic is smaller than the first measurable
cardinal. We also relate compactness to the existence of certain regular
ultrafilters associated with the logic and develop a general theory of
compactness and its consequences. Continued in [She85b].

(B117) Combinatorial problems on trees: Partitions, $\Delta$-systems and large
free
subsets, Ann. Pure Appl. Logic 33(1987), 43-81 (with M. Rubin).

We prove partition theorems on trees and generalize to a setting of trees the
theorems of Erdős and Rado on $\Delta$-systems and the theorems of Fodor and
Hajnal on free sets. For simplicity we ignore some of the results, e.g. on
trees with a non-fix splitting, or with $>\omega$ levels. Let $\mu$ be an
infinite cardinal and $T_{\mu}$ be the tree of finite sequences of ordinals
$<\mu$, with the partial ordering of being an initial segment: $\eta<\nu$
denotes that $\eta$ is an initial segment of $\nu$. A subtree of $T_{\mu}$ is
a non-empty subset of $T_{\mu}$ closed under initial segments. $T\leq T_{\mu}$
means that $T$ is a subtree of $T_{\mu}$ and $\langle T,\leq\rangle\cong
T_{\mu}$.

The following are extracts:

######  Theorem 2.33.

(1) A partition theorem. Suppose ${\rm cf}(\lambda)\neq{\rm cf}(\mu)$,
$F:T_{\mu}\rightarrow\lambda$ and for every branch $b$ of $T_{\mu}$ we have
$\sup(\\{F(\alpha):\alpha\in b\\})<\lambda$. Then there is $T\leq T_{\mu}$
such that $\sup(\\{F(\alpha):\alpha\in T\\})<\lambda$.

######  Theorem 2.34.

(2) A theorem on large free subtrees. Let $\lambda^{+}\leq\mu$,
$F:T_{\mu}\rightarrow P(T_{\mu})$, for every branch $b$ of
$T_{\mu}:|\bigcup\\{F(\alpha):\alpha\in b\\}|<\lambda$, and for every
$\alpha\in T_{\mu}$ and $\beta\in f(\alpha)$, $\beta\nless\alpha$. Then there
is $T\leq T_{\mu}$ such that for every $\alpha,\beta\in T$, $\beta\notin
F(\alpha)$.

Let $P_{\lambda}(C)$ denote the ideal in $P(C)$ of all subsets of $C$ whose
power is less than $\lambda$. Let ${\rm cov}(\mu,\lambda)$ mean that $\mu$ is
regular, $\lambda<\mu$, and for every $\kappa<\mu$ there is $D\subseteq
P_{\lambda}(\kappa)$ such that $|D|<\mu$ and $D$ generates the ideal
$P_{\lambda}(\kappa)$ of $P(\kappa)$. Note that if for every
$\kappa<\kappa^{<\lambda}<{\rm cf}(\mu)=\mu$, then ${\rm cov}(\mu,\lambda)$
holds. Let $\alpha\wedge\beta$ denote the maximal common initial segment of
$\alpha$ and $\beta$.

######  Theorem 2.35.

(3) A theorem on $\Delta$-systems. Suppose ${\rm cov}(\mu,\lambda)$ holds,
$F:T_{\mu}\rightarrow P(C)$ and for every branch $b$ of $T_{\mu}$ we have
$|\bigcup\\{F(\alpha):\alpha\in b\\}|<\lambda$, then there is $T\leq T_{\mu}$
and a function $K:T\rightarrow P_{\lambda}(C)$ such that for every
incomparable $\alpha,\beta\in T$, $F(\alpha)\cap F(\beta)\subseteq
K(\alpha\wedge\beta)$.

(B118) On the expressibility hierarchy of Magidor-Malitz quantifiers, JSL
48(1983), 542-557 (with M. Rubin).

We prove that the logics of Magidor Malitz and their generalization by Rubin
are distinct even for PC classes; during this we prove the existence of
forcing notions satisfying one variant of the countable chain condition but
not another and use the preservation of the satisfaction of sentences in
$\mathbb{L}^{n}$ (see below) under appropriate chain condition.

Let $M\models Q^{n}x_{1}\ldots x_{n}\varphi(x_{1}\ldots x_{n})$ mean that
there is an uncountable subset $A$ of $|M|$ such that for every
$a_{1},\dotsc,a_{n}\in A,M\models\varphi[a_{1},\dotsc,a_{n}]$.

######  Theorem 2.36.

(1) 1) $(\diamondsuit_{\aleph_{1}})$. For every $n\in\omega$ the class
$K_{n+1}=\\{\langle A,R\rangle:\langle A,R\rangle\models Q^{n+1}x_{1}\ldots
x_{n+1}R(x_{1},\dotsc,x_{n+1})\\}$ is not an $\aleph_{0}-PC$-class in the
logic $L^{n}$, obtained by closing first order logic under
$Q^{1},\dotsc,Q^{n}$, i.e., for no countable $L^{n}$-theory $T,K_{n+1}$ is the
class of reducts of the models of $T$.

2) (CH) For every $n\in\omega$ there is a forcing notion $(P,\leq)$ satisfying
$(Q^{i}x_{1}\ldots x_{i})(x_{1},\dotsc,x_{i}$ have no upper bound) iff $i>n$.

######  Theorem 2.37.

(2) $(\diamondsuit_{\aleph_{1}})$. Let $M\models(Q^{E}x,y)\varphi(x,y)$ mean
that there is $A\subseteq|M|$ such that $E_{A,\varphi}=:\\{\langle
a,b\rangle:a,b\in A$ and $M\models\varphi[a,b]\\}$ is an equivalence relation
on $A$ with uncountably many equivalence classes, and such that each
equivalence class is uncountable. Let $K^{E}=\\{\langle A,R\rangle:\langle
A,R\rangle\models(Q^{E}x,y)R(x,y)\\}$, then $K^{E}$ is not an
$\aleph_{0}-PC$-class in the logic gotten by closing first order logic under
the set of quantifiers $\\{Q^{n}:n\in\omega\\}$ which were defined in Theorem
1.

A more general version of the second theorem is proved in the paper.

(B119) Iterated forcing and changing cofinalities, IJM 40(1981), 1-32.

(B120) Free limits and Aronszajn trees, IJM 38(1981) 33-38.

Our main result is the consistency of the “ZFC + Souslin hypothesis + not
every Aronszajn tree is special.”

For this end we introduce some variant of being special, investigate the
connections between them, and for one “being $S$-special”
($S\subseteq\omega_{1}$ stationary co-stationary) we prove that if we
$S$-specialize all Aronszajn trees (by iterated forcing), some “bad” tree we
specify does not become special; as some related property of the forcing is
preserved.

We use (as clearer though CS is O.K.) an iteration in which we take a kind of
limit called free (it is even bigger than inverse limit).

(B121) On the standard part of non-standard models of set theory, JSL
48(1983), 33-38 (with M. Magidor and J. Stavi).

We characterize the ordinals $\alpha$ of uncountable cofinality such that
$\alpha$ is the standard part of a non-standard model of ZFC iff $\alpha$ is
weakly compact in some sense; more exactly:

  1. $(*)$

there is $\gamma>\alpha$ such that:

    1. (a)

for arbitrarily large $\beta<\gamma$, $\mathbf{L}_{\beta}$ is a model of ZFC

    2. (b)