# Descriptive Choice Principles and How to Separate Them

Lucas Wansner This work forms part of the doctoral dissertations of the two
authors [Wan23, Won23]. The authors would like to thank their supervisor,
Benedikt Löwe, for his help throughout. Fachbereich Mathematik, University of
Hamburg, Hamburg, Germany<EMAIL_ADDRESS>Ned J. H.
Wontner11footnotemark: 1 Institute for Logic, Language and Computation,
University of Amsterdam, Amsterdam, the Netherlands<EMAIL_ADDRESS>
> Abstract. The axiom of countable choice for reals is one of the most basic
> fragments of the axiom of choice needed in many parts of mathematics.
> Descriptive choice principles are a further stratification of this fragment
> by the descriptive complexity of the sets. In this paper, we provide a
> separation technique for descriptive choice principles based on Jensen
> forcing. Our results generalise a theorem by Kanovei.

##  1 Introduction

The axiom of choice $\mathsf{AC}$ states that every family of non-empty sets
has a choice function. It can be stratified into fragments based on the index
set of the family and the domain the non-empty sets come from. If we write
$\mathsf{AC}_{X}(Y)$ for “every family indexed by elements of $X$ of non-empty
subsets of $Y$ has a choice function”, then $\mathsf{AC}$ is equivalent to
“$\mathsf{AC}_{X}(Y)$ holds for all sets $X$ and $Y$”.

Among these fragments of $\mathsf{AC}$, $\mathsf{AC}_{\omega}(\mathbb{R})$ or
_the countable axiom of choice for reals_ , has a special position. Most of
the uses of $\mathsf{AC}$ in ordinary mathematics are required because we want
theorems to hold in general; when specialising the results to concrete
mathematical objects, they can often be proved in $\mathsf{ZF}$. However, the
uses of $\mathsf{AC}$ in the basic foundations of analysis are needed even
when working with concrete mathematical objects such as the reals or the
complex numbers. The fragment of $\mathsf{AC}$ most fundamental in this
respect is the countable axiom of choice for reals
$\mathsf{AC}_{\omega}(\mathbb{R})$. Like $\mathsf{AC}$ itself,
$\mathsf{AC}_{\omega}(\mathbb{R})$ does not follow from $\mathsf{ZF}$ and
there are pathological models of
$\mathsf{ZF}+\neg\mathsf{AC}_{\omega}(\mathbb{R})$ where bizarre things can
happen: the reals can be a countable union of countable sets or all subsets of
the reals can be Borel ([FL63], or [Coh66, pg. 146]).111The first situation
implies the second, as singletons are closed, hence
${\mathbf{\Sigma}^{0}_{4}}=\mathcal{P}(\mathbb{R})$. These both occur in the
Feferman-Lévy model. This collapse is optimal in $\mathsf{ZF}$ by [Mil08,
Theorem 2.1].

In this paper, we shall look at fragments of this particular fragment called
_descriptive choice principles_. These restrict
$\mathsf{AC}_{\omega}(\mathbb{R})$ to those sequences of sets of reals that
have a particular description. Here, we are mainly interested in the
descriptions given by the projective pointclasses. Choice principles like this
were studied in [Kan79] who proved a first separation theorem for some of
these principles. In this paper, we improve on Kanovei’s result with a method
based on Jensen forcing.

In § 2, we give all necessary definitions, formally state our main theorem,
and derive its consequences. In § 3, we introduce the notion of an _$n$
-slicing forcing notion_ and prove our main theorem from the assumption that
there is a slicing forcing notion. In § 4, we generalise Jensen forcing to
construct $n$-slicing forcing notions, thus completing the proof of our main
theorem. Finally, in § 5, we list some open questions.

##  2 Definitions and the main results

Our basic theory is Zermelo-Fraenkel set theory $\mathsf{ZF}$ without any
fragment of $\mathsf{AC}$. In some proofs, we need fragments of set theory
without the power set axiom. We write $\mathsf{ZFC}^{-}$ for $\mathsf{ZFC}$
without the power set axiom and the collection scheme instead of the
replacement scheme.222Cf. [GHJ16] for a discussion of this theory. The
standard model of $\mathsf{ZFC}^{-}$ is the set of hereditarily countable sets
$\mathbf{HC}$.

###  2.1 Descriptive choice principles

As usual in descriptive set theory, we call the elements of Baire space
${\omega^{\omega}}$ _real numbers_ or _reals_. _Par abus de langage_ , we
write $\mathbb{R}$ to refer to Baire space even though it is not homeomorphic
to what is usually called the _real line_. The standard real line plays no
role in this paper and therefore, there is no possibility of confusion. We use
the standard notational convention that if $n\in\omega$ and $x\in\mathbb{R}$,
then

$nx(k):=\left\\{\begin{array}[]{cl}n&\mbox{if $k=0$ and}\\\ x(\ell)&\mbox{if
$k=\ell+1$;}\end{array}\right.$

similarly for finite sequences $s\in\omega^{<\omega}$.

The advantage of working with Baire space is that it is zero-dimensional and
therefore, we can easily identify subsets of $\mathbb{R}^{n}$ and even
$\mathbb{R}^{\omega}$ with elements of $\mathbb{R}$ via the Cantor pairing
function $\ulcorner.,.\urcorner:\omega\times\omega\to\omega$, e.g., by
encoding a sequence $a:=\langle a_{n}:n\in\omega\rangle$ by
$\widehat{a}(\ulcorner k,\ell\urcorner):=a_{k}(\ell)$. Similarly, if
$A=\\{A_{n}:n\in\omega\\}$ is a family of sets of reals indexed by natural
numbers, then $\widehat{A}:=\\{nx:x\in A_{n}\\}$ is a set of reals that
contains exactly the same information. If $A\subseteq\mathbb{R}^{2}$, we can
form its _projection_ $\mathsf{proj}(A):=\\{x\in\mathbb{R}:\exists y((x,y)\in
A)\\}$.

Since we are working in models without $\mathsf{AC}$, we have to be very
specific about our definitions. As mentioned, in pathological models of
$\mathsf{ZF}$, very basic features of descriptive set theory can fail to hold
(e.g., that there are sets which are not Borel). The theory of Borel sets
without $\mathsf{AC}$ is complicated: different standard definitions of the
Borel hierarchy that are equivalent in
$\mathsf{ZF}+\mathsf{AC}_{\omega}(\mathbb{R})$ fail to be equivalent in
general; also, the length of the Borel hierarchy is not determined by
$\mathsf{ZF}$ [Mil08, Theorem 1.2 & 1.3]. As a consequence, we focus on
projective descriptive classes in this paper.

A subset $T\subseteq\omega^{<\omega}$ is called a _tree_ if it is closed under
taking initial segments; if $r$ is a real, a tree is called _computable in
$r$_ if there is a Turing machine that computes its characteristic function
from the oracle $r$; it is called _computable_ if it is computable without an
oracle. If $T$ is a tree, we define the _set of branches of $T$_ to be
$[T]:=\\{x\in\mathbb{R}:\forall n$, $x{\upharpoonright}n\in T\\}$. A set of
reals is $\Pi^{0}_{1}(r)$ if it is the set of branches of a tree computable in
$r$. We define the _lightface projective hierarchy_ by recursion: we define
$\Pi^{1}_{0}(r):=\Pi^{0}_{1}(r)$ and say that a set is $\Sigma^{1}_{n+1}(r)$
if it is the projection of a $\Pi^{1}_{n}(r)$ set and it is $\Pi^{1}_{n}(r)$
if it is the complement of a $\Sigma^{1}_{n}(r)$ set. If $r$ is a computable
real, we write $\Sigma^{1}_{n}$ and $\Pi^{1}_{n}$ for $\Sigma^{1}_{n}(r)$ and
$\Pi^{1}_{n}(r)$, respectively. The _boldface projective hierarchy_ are
defined as usual by
$\boldsymbol{\Sigma}^{1}_{n}:=\bigcup_{r\in\mathbb{R}}\Sigma^{1}_{n}(r)$ and
$\boldsymbol{\Pi}^{1}_{n}:=\bigcup_{r\in\mathbb{R}}\Pi^{1}_{n}(r)$; we say
that sets in $\boldsymbol{\Sigma}^{1}_{1}$ are _analytic_. We furthermore let
$\Delta^{1}_{n}(r):=\Sigma^{1}_{n}(r)\cap\Pi^{1}_{n}(r)$ and
$\boldsymbol{\Delta}^{1}_{n}:=\boldsymbol{\Sigma}^{1}_{n}\cap\boldsymbol{\Pi}^{1}_{n}$.
A set is called _projective_ if it is in one of the boldface projective
classes; we denote the set of projective sets by $\mathbf{Proj}$.

A countable set of reals $C=\\{a_{n}:n\in\omega\\}$ is easily seen to be
$\Sigma^{1}_{1}(\widehat{a})$ by considering the tree $T:=\\{(s,nt):t\subseteq
a_{n}\mbox{ and $s$ and $t$ are compatible}\\}$ and observing that
$C=\mathrm{p}[T]$. Not all countable sets are lightface $\Sigma^{1}_{1}$. If
we denote the set of countable sets of reals by $\mathbf{Ctbl}$, we therefore
obtain the following chain of inclusions:

$\mathbf{Ctbl}\subseteq\boldsymbol{\Sigma}^{1}_{1}\subseteq...\boldsymbol{\Sigma}^{1}_{n}\subseteq\boldsymbol{\Sigma}^{1}_{n+1}\subseteq...\subseteq\mathbf{Proj}.$

If we define $\mathbf{F}_{\sigma}$ as the collection of countable unions of
closed sets, then $\mathbf{Ctbl}\subseteq\mathbf{F}_{\sigma}$. It is known
that in $\mathsf{ZF}$,
$\mathbf{F}_{\sigma}\subseteq\boldsymbol{\Sigma}^{1}_{1}$ [Fre08, 562D &
562F].

If $\Gamma$ is any of the descriptive classes defined above, we say that a
countable family $A=\\{A_{n}:n\in\omega\\}$ of sets of reals is in $\Gamma$ if
each of the elements of the family is in $\Gamma$. We say that it is
_uniformly in $\Gamma$_ if it is in $\Gamma$ and furthermore $\widehat{A}$ is
also in $\Gamma$. We write $\mathrm{unif}\Gamma$ for the collection of
countable families of sets of reals that are uniformly in $\Gamma$.

We can now stratify the axiom $\mathsf{AC}_{\omega}(\mathbb{R})$ according to
the descriptive classes that we have considered: let $\Xi$ be any definable
class of countable collections of sets of reals, e.g., one of our descriptive
classes $\Gamma$ or one of the uniform versions $\mathrm{unif}\Gamma$. Then we
write

$\mathrm{AC}_{\omega}(\mathbb{R};\Xi)$

for the statement “every countable collection of non-empty sets of reals in
$\Xi$ has a choice function”. Note that
$\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{Ctbl})$ is the same principle that is
sometimes known as $\mathsf{CAC}_{\omega}(\mathbb{R})$ in the literature.
Clearly, if $\Xi^{\prime}\supseteq\Xi$, then
$\mathrm{AC}_{\omega}(\mathbb{R};\Xi^{\prime})$ implies
$\mathrm{AC}_{\omega}(\mathbb{R};\Xi)$, so our inclusions of descriptive
classes naturally give an implication diagram for the axiom fragments as
displayed in Figure 1. The diagram forms a rectangular solid where the back
side consists of the boldface principles, the front side of the lightface
principles, the left side of the non-uniform principles, and the right side of
the uniform principles. We shall state the main applications of our main
theorem in terms of this diagram below.

Before doing so, we introduce the final axiom family: for any of our
descriptive classes $\Gamma$, we write $\mathsf{DC}(\mathbb{R};\Gamma)$ for
the _Axiom of Dependent Choice for $\Gamma$ relations_, i.e. for any non-empty
set $X$ of reals in $\Gamma$ and any total relation $R\subseteq X\times X$
(i.e. for all $x$ there is a $y$ such that $x\mathrel{R}y$) that is in
$\Gamma$, there is a sequence $\langle x_{i}:i\in\omega\rangle\in X^{\omega}$
such that for all $i\in\omega$, we have $x_{i}\mathrel{R}x_{i+1}$.333Kanovei’s
definition of $\mathsf{DC}(\mathbb{R};\Gamma)$ is slightly different. However,
he showed in [Kan79] that Lemma 1 and Theorem 2 are also true for our
definition of $\mathsf{DC}(\mathbb{R};\Gamma)$.

######  Lemma 1 (Kanovei, [Kan79]).

If $\Gamma$ is one of the classes $\boldsymbol{\Pi}^{1}_{n}$ or $\Pi^{1}_{n}$,
then $\mathsf{DC}(\mathbb{R};\Gamma)$ implies
$\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\Gamma)$.

We can now formulate Kanovei’s separation result from the introduction:

######  Theorem 2 (Kanovei, [Kan79]).

For each $n\geq 1$, there is a model of
$\mathsf{ZF}+\mathsf{DC}(\mathbb{R};\boldsymbol{\Pi}^{1}_{n})+\neg\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\Pi^{1}_{n+1})$.

Our main theorem is the following separation of the principles, which
strengthens Kanovei’s Theorem 2.

######  Theorem 3.

For every $n\geq 1$, there is a model of
$\mathsf{ZF}+\mathsf{DC}(\mathbb{R};\boldsymbol{\Pi}^{1}_{n})+\neg\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\Pi^{1}_{n+1})+\neg\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{Ctbl})$.

We can represent this pictorially, with double arrows representing
implications, and dashed lines representing the separation between layers due
to Theorem 3.

---
$\textstyle{\mathsf{AC}_{\omega}(\mathbb{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{Proj})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\mathbf{Proj})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{AC}_{\omega}(\mathbb{R};\boldsymbol{\Pi}^{1}_{n+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\boldsymbol{\Pi}^{1}_{n+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{AC}_{\omega}(\mathbb{R};\Pi^{1}_{n+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\Pi^{1}_{n+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{AC}_{\omega}(\mathbb{R};\boldsymbol{\Pi}^{1}_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\boldsymbol{\Pi}^{1}_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{AC}_{\omega}(\mathbb{R};\Pi^{1}_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\Pi^{1}_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{AC}_{\omega}(\mathbb{R};\boldsymbol{\Pi}^{1}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\boldsymbol{\Pi}^{1}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{AC}_{\omega}(\mathbb{R};\Pi^{1}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\Pi^{1}_{1})}$$\textstyle{\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{Ctbl})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Figure 1: Implication diagram of the fragments of
$\mathsf{AC}_{\omega}(\mathbb{R})$

###  2.2 Consequences of our main theorem

We can now separate some of the sides of the solid quadrilateral in our Figure
1 by applying our Theorem 3.

######  Corollary 4.

There is a model of
$\mathsf{ZF}+\mathsf{DC}(\mathbb{R};\mathbf{Proj})+\neg\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{Ctbl})$.

###### Proof.

We argue by compactness. Let $T_{\infty}$ be the theory $\mathsf{ZF}$ along
with the sentences $\mathsf{DC}(\mathbb{R};\boldsymbol{\Pi}^{1}_{n})$ for each
$n\in\omega$ and the sentence
$\neg\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{Ctbl})$. Let $F\subsetneq
T_{\infty}$ be a finite subtheory of $T_{\infty}$. Then $F$ only contains
finitely many sentences of the form
$\mathsf{DC}(\mathbb{R};\boldsymbol{\Pi}^{1}_{n})$. Let $k$ be the maximum $n$
such that $\mathsf{DC}(\mathbb{R};\boldsymbol{\Pi}^{1}_{n})\in F$. By Theorem
3, there is a model $N_{k}$ which satisfies $\mathsf{ZF}$ and
$\mathsf{DC}(\mathbb{R};\boldsymbol{\Pi}^{1}_{k})$, and so $N_{k}\models F$.
Hence $T_{\infty}$ is finitely satisfiable. By compactness, there is a model
of
$\mathsf{ZF}+\mathsf{DC}(\mathbb{R};\mathbf{Proj})+\neg\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{Ctbl})$.
∎

This, and Lemma 1, yields two corollaries:

######  Corollary 5.

For every $n\geq 1$, there is a model of
$\mathsf{ZF}+\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\boldsymbol{\Pi}^{1}_{n})+\neg\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\Pi^{1}_{n+1})$.

######  Corollary 6.

There is a model of
$\mathsf{ZF}+\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\mathbf{Proj})+\neg\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{Ctbl})$.

Corollary 5 separates the horizontal slices of the diagram on the uniform side
from each other. In the diagram, this is indicated by the dashed line between
$\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\boldsymbol{\Pi}^{1}_{n})$ and
$\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\Pi^{1}_{n+1})$, indicating that
there is no implication from the lower horizontal slice to the higher slice.

Corollary 6 separates the boldface non-uniform part of the diagram from the
uniform part by showing that the strongest uniform principle does not imply
the weakest non-uniform boldface principle. This is indicated by the dashed
line between $\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{Ctbl})$ and
$\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\boldsymbol{\Pi}^{1}_{1})$ in
the diagram. So, in particular
$\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{\Sigma}^{1}_{n})$ implies but, is not
implied by,
$\mathsf{AC}_{\omega}(\mathbb{R};\mathsf{unif}\mathbf{\Sigma}^{1}_{n})$.

Furthermore, by the remark that
$\mathbf{Ctbl}\subseteq\mathbf{F}_{\sigma}\subseteq\boldsymbol{\Sigma}^{1}_{1}$,
our results separate all non-trivial non-uniform Borel descriptive choice
principles from the uniform descriptive choice principles.

Any weaker Borel descriptive choice principles are provable in $\mathsf{ZF}$:

######  Proposition 7 (Folklore).

The descriptive choice principle
$\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{F}_{\sigma}\cap\mathbf{G}_{\delta})$
is provable in $\mathsf{ZF}$.

###### Proof.

We can canonically pick an element from any
$\mathbf{F}_{\sigma}\cap\mathbf{G}_{\delta}$ set like so: by Hausdorff’s
Difference Lemma, every $\mathbf{F}_{\sigma}\cap\mathbf{G}_{\delta}$ set $A$
of reals can be written as an $\alpha$-difference of closed sets for some
$\alpha<\omega_{1}$, i.e. there is a sequence $\langle
C_{\beta}:\beta<\alpha\rangle$ of closed sets of reals such that

$\displaystyle
A=\mathrm{Diff}_{\beta<\alpha}C_{\beta}:=\\{x\in\bigcup_{\beta<\alpha}C_{\beta}:\text{
the least }\beta<\alpha\text{ such that }x\notin C_{\beta}\text{ is odd}\\}.$

With extra care, we can uniquely specify such a sequence (cf. [And20, 3.E.1]).
Let $A\not=\emptyset$ be a $\mathbf{F}_{\sigma}\cap\mathbf{G}_{\delta}$ set of
reals, and let $\langle C_{\beta}:\beta<\alpha\rangle$ be its sequence. So, it
suffices to canonically define a real in $A$ from $\langle
C_{\beta}:\beta<\alpha\rangle$. There is a minimal even $\beta<\alpha$ such
that $C_{\beta}\supsetneq C_{\beta+1}$. Let $\\{O_{n}:n\in\omega\\}$ enumerate
the basic open sets and let $n\in\omega$ be minimal such that $O_{n}\cap
C_{\beta+1}=\emptyset$ and $O_{n}\cap C_{\beta}\neq\emptyset$. Since
$O_{n}\cap C_{\beta}$ is closed, there is a unique pruned tree
$T\subseteq\omega^{<\omega}$ such that $[T]=O_{n}\cap C_{\beta}$. The left-
most branch of $T$ yields a real in $A$. ∎

##  3 Model construction for the main theorem

###  3.1 Slicing forcing notions

Here, we describe $n$-slicing forcing notions. These are key to our separation
results: in Section 3.2, we use $n$-slicing forcing notions to prove Theorem
3. Before we can introduce $n$-slicing forcing notions, we need two further
definitions, _$n$ -slice products_, and _$n$ -absoluteness for slices_. The
former is a special variant of an $\omega$-product of forcing notions. The
latter specifies a form of complexity-limited absoluteness for $n$-slice
products.

Let $\xi\leq\omega_{1}$ be an ordinal and let
$\langle\mathbb{P}_{\nu}:\nu<\xi\rangle$ be a sequence of forcing notions. We
define the _$\omega$ -slice product of
$\langle\mathbb{P}_{\nu}:\nu<\xi\rangle$ with finite support_ as the set
$\mathbb{P}$ of all partial functions $p$ such that
$\mathrm{dom}(p)\subseteq\xi\times\omega$ is finite and for every
$(\nu,k)\in\xi\times\omega$, $p(\nu,k)\in\mathbb{P}_{\nu}$ ordered by

$\displaystyle p\leq q:\iff\forall(\nu,k)\in\mathrm{dom}(q)(p(\nu,k)\leq
q(\nu,k)).$

We say $\mathbb{P}$ has _length_ $\xi$. If, for every
$\nu,\nu^{\prime}<\omega_{1}$, $\mathbb{P}_{\nu}=\mathbb{P}_{\nu^{\prime}}$,
then we say $\mathbb{P}$ is the _$\omega$ -slice product of
$\mathbb{P}_{\nu}$_ with finite support of length $\xi$. Let
$I\subseteq\xi\times\omega$. For $A\subseteq\mathbb{P}$, we define
$A{\upharpoonright}I:=\\{p{\upharpoonright}I:p\in A\\}$. Then for every
$\mathbb{P}$-generic filter $G$, $G{\upharpoonright}I$ is a
$\mathbb{P}{\upharpoonright}I$-generic filter.

Let $\xi=\omega_{1}$. We define $\mathrm{Aut}(\omega_{1}\times\omega,\omega)$
as the group of all bijections $\pi$ of $\omega_{1}\times\omega$ such that for
every $\nu\in\omega_{1}$, $\pi[\\{\nu\\}\times\omega]=\\{\nu\\}\times\omega$.
Let $p\in\mathbb{P}$ and let
$\pi\in\mathrm{Aut}(\omega_{1}\times\omega,\omega)$. We define
$\pi^{\ast}(p):=p^{\prime}$ with
$\mathrm{dom}(p^{\prime})=\pi[\mathrm{dom}(p)]$ and for
$i\in\mathrm{dom}(p^{\prime})$, $p^{\prime}(i)=p(\pi^{-1}(i))$. So, every
automorphism $\pi\in\mathrm{Aut}(\omega_{1}\times\omega,\omega)$ induces an
automorphism $\pi^{\ast}$ on $\mathbb{P}$. Let
$\mathcal{G}:=\\{\pi^{\ast}:\pi\in\mathrm{Aut}(\omega_{1}\times\omega,\omega)\\}$
be the group of all such automorphisms on $\mathbb{P}$.

We say $\mathcal{S}\subseteq\mathcal{P}(\omega_{1}\times\omega)$ is a _set of
slices_ if, for every $s\in\mathcal{S}$, $s$ is of the form $X\times\omega$,
where $X\subseteq\omega_{1}$. Let
$\mathcal{S}\subseteq\mathcal{P}(\omega_{1}\times\omega)$ be a set of slices
and for every $s\in\mathcal{S}$, let $H_{s}:=\\{\pi^{\ast}:\forall(\nu,k)\in
s(\pi(\nu,k)=(\nu,k))\\}\subseteq\mathcal{G}$ be the subgroup of all
$\pi^{\ast}$ such that $\pi$ point-wise fixes $s$. We define
$\mathcal{F}_{\mathcal{S}}$ as the filter on the subgroups of $\mathcal{G}$
generated by $\\{H_{s}:s\in\mathcal{S}\\}$. Then $\mathcal{F}_{\mathcal{S}}$
is normal. Let $G$ be a $\mathbb{P}$-generic filter. We denote the symmetric
extension we obtain from $G$ and $\mathcal{F}_{\mathcal{S}}$ by
$V(G,\mathcal{S})$.

######  Lemma 8.

Let $\langle\mathbb{P}_{\nu}:\nu<\omega_{1}\rangle$ be a sequence of forcing
notions, let $\mathbb{P}$ be the $\omega$-slice product of
$\langle\mathbb{P}_{\nu}:\nu<\omega_{1}\rangle$ with finite support, let
$\mathcal{S}\subseteq\mathcal{P}(\omega_{1}\times\omega)$ be a set of slices,
and let $G$ be a $\mathbb{P}$-generic filter. Then

  1. 1.

for every $s\in\mathcal{S}$, $V[G{\upharpoonright}s]\subseteq
V(G,\mathcal{S})$ and

  2. 2.

for every set of ordinals $X$, $X\in V(G,\mathcal{S})$ iff there is some
$s\in\mathcal{S}$ such that $X\in V[G{\upharpoonright}s]$.

###### Proof.

For the first part, let $X\in V[G{\upharpoonright}s]$. Then there is a
$\mathbb{P}{\upharpoonright}s$-name $\dot{X}$ such that
$\dot{X}_{G{\upharpoonright}s}=X$. Since
$\mathbb{P}{\upharpoonright}s\subseteq\mathbb{P}$, $\dot{X}$ is also a
$\mathbb{P}$-name and $\dot{X}_{G}=X$. Hence, it is enough to show that every
$\mathbb{P}{\upharpoonright}s$-name is hereditarily symmetric. Let $\sigma$ be
a $\mathbb{P}{\upharpoonright}s$-name, let $\pi^{\ast}\in H_{s}$, and let
$(\tau,p)\in\sigma$. By assumption $p\in\mathbb{P}{\upharpoonright}s$ and
$\pi^{\ast}$ fixes $s$, so $\pi^{\ast}(p)=p$. Hence,
$\pi^{\ast}(\sigma)=\\{(\pi^{\ast}(\tau),\pi^{\ast}(p)):(\tau,p)\in\sigma\\}=\\{(\pi^{\ast}(\tau),p):(\tau,p)\in\sigma\\}$.
By induction, $\sigma$ is hereditarily symmetric.

We prove the second part. The backwards direction follows directly from the
first part. For the forwards direction, let $\dot{X}$ be a symmetric
$\mathbb{P}$-name for $X$, let $s\in\mathcal{S}$ such that
$H_{s}\subseteq\mathrm{sym}(\dot{X})$, and let $p\in G$ such that $p\Vdash$
“$\dot{X}$ is a set of ordinals”. We define a
$\mathbb{P}{\upharpoonright}s$-name $\sigma$ such that
$\sigma_{G{\upharpoonright}s}=X$. Let
$\sigma:=\\{(\check{\xi},q{\upharpoonright}s):q\in\mathbb{P},$ and $q\leq p,$
and $q\Vdash\check{\xi}\in\dot{X}\\}$ and let
$X^{\prime}:=\sigma_{G{\upharpoonright}s}$.

First, we prove $X\subseteq X^{\prime}$. Let $\xi\in X$. Then there is a $q\in
G$ such that $(\check{\xi},q)\in\dot{X}$. As $G$ is a filter, there is an
$r\in G$ which witnesses that $p$ and $q$ are compatible. Hence, $r\leq p$ and
$r\Vdash\check{\xi}\in\dot{X}$. Then
$(\check{\xi},r{\upharpoonright}s)\in\sigma$ and so $\xi\in X^{\prime}$.

Next, we prove $X^{\prime}\subseteq X$. Let $\xi\in X^{\prime}$. Then there is
a $q\in G{\upharpoonright}s$ such that $(\check{\xi},q)\in\sigma$. Hence,
there is an $r\leq p$ such that $q=r{\upharpoonright}s$ and
$r\Vdash\check{\xi}\in\dot{X}$. We suppose for a contradiction, that
$q\not\Vdash\check{\xi}\in\dot{X}$. Then there is a $q^{\prime}\leq q$ such
that $q^{\prime}\Vdash\check{\xi}\notin\dot{X}$. Hence, $q^{\prime}$ and $r$
are incompatible, but agree on $s$ as functions. Let $\pi\in H_{s}$ such that
$\mathrm{dom}(r)\cap\pi[\mathrm{dom}(q^{\prime})]\subseteq s$. Then
$\pi^{\ast}(q^{\prime})$ and $r$ are compatible, as they agree on their common
domain. By construction $\pi^{\ast}\in H_{s}\subseteq\mathrm{sym}(\dot{X})$,
so $\pi^{\ast}(\dot{X})=\dot{X}$. Hence,
$\pi^{\ast}(q^{\prime})\Vdash\check{\xi}\notin\dot{X}$ which is a
contradiction since $r$ and $\pi^{\ast}(q^{\prime})$ are compatible.
Therefore, $q\Vdash\check{\xi}\in\dot{X}$ and so $\xi\in X$. ∎

######  Definition 9.

Let $\mathbb{P}$ be an $\omega$-slice product with finite support of a
sequence $\langle\mathbb{P}_{\nu}:\nu<\omega_{1}\rangle$ of forcing notions,
let $\mathcal{S}\subseteq\mathcal{P}(\omega_{1}\times\omega)$ be a set of
slices, and let $n\geq 1$. We say $\mathbb{P}$ is _$n$ -absolute for
$\mathcal{S}$-slices_ if, for every $s\in\mathcal{S}$, every $\Sigma^{1}_{n}$
formula with real parameters $V[G{\upharpoonright}s]$ is absolute between
$V[G{\upharpoonright}s]$ and $V[G]$. A set of slices
$\mathcal{S}\subseteq\mathcal{P}(\omega_{1}\times\omega)$ is _unbounded_ if,
for every $s\in\mathcal{S}$, $\\{\nu:\\{\nu\\}\times\omega\subseteq s\\}$ is
unbounded in $\omega_{1}$. We say $\mathbb{P}$ is _$n$ -absolute for slices_
if for every unbounded set of slices $\mathcal{S}$, $\mathbb{P}$ is
$n$-absolute for $\mathcal{S}$-slices.

Note that any $\omega$-slice product with finite support of length
$\omega_{1}$ is 2-absolute for slices, by Shoenfield absoluteness [Jec03,
Theorem 25.20].

######  Definition 10.

A forcing notion $\mathbb{P}$ is _$n$ -slicing_ if there is a sequence of
forcing notions $\langle\mathbb{P}_{\nu}:\nu<\omega_{1}\rangle$ such that

  1. 1.

$\mathbb{P}$ is the $\omega$-slice product of
$\langle\mathbb{P}_{\nu}:\nu<\omega_{1}\rangle$ with finite support,

  2. 2.

for every $\nu<\omega_{1}$, every $\mathbb{P}_{\nu}$-generic filter $G$ over
$V$ is uniquely determined by a real in $V[G]\setminus V$,

  3. 3.

$\mathbb{P}$ is $n$-absolute for slices, and

  4. 4.

for every $\mathbb{P}$-generic filter $G$, the set
$\\{(\ell,x_{G}^{(\ell,k)}):(\ell,k)\in\omega^{2}\\}$ is $\Pi^{1}_{n}$ in
$V[G]$, where $x_{G}^{(\ell,k)}$ is the generic real defined by
$G{\upharpoonright}\\{(\ell,k)\\}$.

######  Lemma 11 (Sandwiching Lemma).

Let $n\in\omega$, let $\mathbb{P}$ be $n$-slicing, let $G$ be a
$\mathbb{P}$-generic filter, and let
$\mathcal{S}\subseteq\mathcal{P}(\omega_{1}\times\omega)$ be an unbounded set
of slices. For every $s\in\mathcal{S}$, every $\Sigma^{1}_{n}$-formula with
real parameters in $V[G{\upharpoonright}s]$ is absolute between
$V[G{\upharpoonright}s]$, $V[G]$, and $V(G,\mathcal{S})$. So every
$\Sigma^{1}_{n}$-formula with real parameters in $V(G,\mathcal{S})$ is
absolute between $V[G]$ and $V(G,\mathcal{S})$.

###### Proof.

The second part follows from the first and Lemma 8. The first part follows by
induction, the base case is analytic absoluteness [Jec03, 25.4]. ∎

The following theorem will be proved in Section 4.2.

######  Theorem 12.

Let $n\geq 2$. In $L$, there is an $n$-slicing forcing notion.

###  3.2 Theorem 12 implies the main theorem

In this section, we prove Theorem 3 under the assumption that Theorem 12 has
been established. In particular, we construct our model from $L$ which then,
by Theorem 12 has an $n$-slicing forcing notion for every $n\geq 2$.

We fix a natural number $n\geq 1$ and aim to construct a model of
$\mathsf{ZF}+\mathsf{DC}(\mathbb{R};\boldsymbol{\Pi}^{1}_{n})+\neg\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\Pi^{1}_{n+1})+\neg\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{Ctbl})$.
As $n+1\geq 2$, let $\mathbb{P}$ be an $(n+1)$-slicing forcing notion, let $G$
be a $\mathbb{P}$-generic filter over $L$, let
$Z:=\\{(F\cup(\omega_{1}{\setminus}\omega))\times\omega:F\subseteq\omega$ is
finite$\\}$, let $N:=L(G,Z)$, for every $\ell\in\omega$, let
$A_{\ell}:=\\{x_{G}^{(\ell,k)}:k\in\omega\\}$ and let
$A:=\\{A_{\ell}:\ell\in\omega\\}$. Then, in $L[G]$, each $A_{\ell}$ is
countable and $A$ is in $\mathrm{unif}\Pi^{1}_{n+1}$. We shall show that, in
$N$, $A$ has no choice function, each $A_{\ell}$ is still countable, and $A$
is still in $\mathrm{unif}\Pi^{1}_{n+1}$.

######  Lemma 13.

In $N$, $A$ is $\mathrm{unif}\Pi^{1}_{n+1}$ and for every $\ell\in\omega$,
$A_{\ell}$ is countable.

###### Proof.

For every $\ell\in\omega$, let
$z:=(\\{\ell\\}\cup(\omega_{1}\setminus\omega))\times\omega$. Then $A_{\ell}$
is countable in $L[G{\upharpoonright}z]$ by the enumeration defined by
$G{\upharpoonright}z$, hence $A_{\ell}$ is countable in $N$ by Lemma 8. So we
show the first part. Since $\mathbb{P}$ is $(n+1)$-slicing, $\widehat{A}$ is
$\Pi^{1}_{n+1}$ in $L[G]$. Let $\varphi$ be a $\Pi^{1}_{n+1}$ formula defining
$\widehat{A}$. By Lemmas 8 and 11, $N$ contains each $x_{G}^{(\ell,k)}$ and
$\varphi$ is absolute between $L[G]$ and $N$. Therefore, $\varphi$ defines
$\widehat{A}$ in $N$ and so $A$ is in $\mathrm{unif}\Pi^{1}_{n+1}$. ∎

######  Proposition 14.

In $N$, $\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\Pi^{1}_{n+1})$ and
$\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{Ctbl})$ fail.

###### Proof.

By Lemma 13, it suffices to show that $A$ has no choice function in $N$.
Suppose there is a choice function $f:\omega\to\mathbb{R}$ for $A$ in $N$.
Since $f$ is a countable sequence of reals, we can code $f$ as a real and so
as a subset of $\omega$. By Lemma 8, there is some $z\in Z$ such that $f\in
L[G{\upharpoonright}z]$. Hence, there is a $\mathbb{P}{\upharpoonright}z$-name
$\dot{f}$ for $f$. As $z\in Z$, there is an $\ell\in\omega$ such that $z$ does
not meet any $\\{\ell^{\prime}\\}\times\omega$ with $\ell^{\prime}\geq\ell$.
Let $\ell$ be minimal with that property and let $k\in\omega$ such that
$f(\ell)=x^{(\ell,k)}_{G}$. Then there is a $p\in G$ such that $p\Vdash``f$ is
a choice function for $A"\land\dot{f}(\ell)=\dot{x}^{(\ell,k)}_{G}$, where
$\dot{x}^{(\ell,k)}_{G}$ is the canonical $\mathbb{P}$-name for
$x^{(\ell,k)}_{G}$. Let $k^{\prime}\neq k\in\omega$ such that
$(\ell,k^{\prime})\in(\\{\ell\\}\times\omega){\setminus}\mathrm{dom}(p)$ and
let $\pi\in\mathrm{Aut}(\omega_{1}\times\omega,\omega)$ which only swaps
$(\ell,k)$ and $(\ell,k^{\prime})$. Then $\pi^{\ast}\in
H_{z}\subseteq\mathrm{sym}(\dot{f})$ and $p$ and $\pi^{\ast}(p)$ are
compatible. But
$\pi^{\ast}(p)\Vdash\dot{f}(\ell)=\pi^{\ast}(\dot{x}^{(\ell,k)}_{G})=\dot{x}^{(\ell,k^{\prime})}_{G}$,
which is impossible as these are distinct. ∎

######  Proposition 15.

In $N$, $\mathsf{DC}(\mathbb{R};\boldsymbol{\Pi}^{1}_{n})$ holds.

###### Proof.

Let $X\subseteq\mathbb{R}$ be a $\mathbf{\Pi}^{1}_{n}$ set of reals in $N$ and
let $R\subseteq(\mathbb{R})^{2}$ be a total relation such that $R$ is
$\mathbf{\Pi}^{1}_{n}$ in $N$. Then there are $\Pi^{1}_{n}$ formulas $\varphi$
and $\psi$ with real-parameter $a$ and $b$ in $N$ which define $X$ and $R$ in
$N$, respectively. We can assume $a=b$. By Lemma 8, there is a $z\in Z$ such
that $a\in L[G{\upharpoonright}z]$. Let $X_{\varphi}$ and $R_{\psi}$ be the
sets defined by $\varphi(a)$ and $\psi(a)$ in $L[G{\upharpoonright}z]$, and
let $\chi(x,a)$ be the formulate $\varphi(x,a)\to\exists
x^{\prime}\psi((x,x^{\prime}),a)$. Then $\chi(x,a)$ is $\Sigma^{1}_{n+1}$ and
so absolute by Lemma 11. By downwards-absoluteness, for every $x$,$\chi(x,a)$
is true in $L[G{\upharpoonright}z]$. As usual
$L[G{\upharpoonright}z]\models\mathsf{ZFC}$, so there is a sequence $\langle
x_{i}:i\in\omega\rangle$ in $L[G{\upharpoonright}z]$ such that
$x_{i}\mathrel{R_{\psi}}x_{i+1}$ for every $i\in\omega$. By Lemma 8, $\langle
x_{i}:i\in\omega\rangle\in N$, and $\varphi(a)$ and $\psi(a)$ are absolute
between $L[G{\upharpoonright}z]$ and $N$, so $x_{i}\mathrel{R}x_{i+1}$ in $N$
for every $n\in\omega$. ∎

This finishes the proof of Theorem 3. In the classical case, when aiming to
construct a model of
$\mathsf{ZF}+\neg\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{Ctbl})$, we construct
a symmetric extension from an $\omega\times\omega$-product, where we fix
finitely many rows from the $\omega\times\omega$-matrix of the product. Here,
we modify this construction by instead using an
$\omega_{1}\times\omega$-product. The extra $\omega_{1}$-many rows are used to
prove Theorem 12 (via the Kanovei-Lyubetsky Lemma, Lemma 32). We then ‘ignore’
the later reals (i.e. those added by the $\alpha\times k$-component for
$\alpha\in\omega_{1}\backslash\omega$). This motivates our choice of a filter,
$\mathcal{F_{S}}$, which contains subgroups of permutations which can move any
finite initial segment of $\omega_{1}$, but fix everything beyond $\omega$.
Hence $A$ is then the countable $\omega\times\omega$-sized bottom corner of
the $\omega_{1}\times\omega$ grid, formed by ‘chopping off’ the end
$\omega_{1}\backslash\omega$ tail of each $\omega_{1}$-length sequence of
reals.

##  4 Jensen forcing and $n$-Jensen forcing

###  4.1 Jensen forcing

The aim of Section 4 is to prove Theorem 12, i.e. to prove that for every
$n\geq 2$, there is an $n$-slicing forcing notion in $L$. In preparation for
this, here we briefly introduce Jensen forcing, which we will then generalise
in Section 4.2 to construct $n$-slicing forcing notions. Our presentation and
notation are based on that of [FGK19].

Jensen forcing was originally introduced by Jensen in [Jen70] (see also
[Jec03, §28]) to construct a model with especially simple non-constructible
reals. After Cohen’s totemic proof of the consistency of $\mathsf{ZFC}$ with
the existence of non-constructible reals, it was naturally asked how simple
these non-constructible reals could be. Shoenfield’s absoluteness theorem
showed that the minimum possible complexity was $\mathbf{\Delta}^{0}_{3}$.
Jensen used Jensen forcing to construct such a model, where there was a
$\mathbf{\Delta}^{0}_{3}$ real, with the minimal non-trivial degree of
constructibility (if $x\in L[a]$ and $a\not\in L[x]$, then $x\in L$). We are
interested in Jensen forcing because its generic reals are unique (i.e. there
is only one Jensen real in the extension) and the set of its generic reals
over $L$ is $\Pi^{1}_{2}$. Therefore, products of Jensen forcing can be used
to add a set of complexity $\Pi^{1}_{2}$ in a controlled way. To be able to
add sets of higher complexity, we generalise Jensen’s construction of Jensen
forcing in Section 4.2.

Before we can begin with the construction of Jensen forcing, we need some
additional notation for trees. Let $\mathbb{S}$ be Sacks forcing, let
$T\in\mathbb{S}$, and let $t\in T$. We define $T_{t}:=\\{s\in T:s\subseteq
t\lor t\subseteq s\\}$. Then $T_{t}\in\mathbb{S}$ and every node in $T_{t}$ is
compatible with $t$. Let $\mathbb{C}:=\\{(2^{<\omega})_{t}:t\in
2^{<\omega}\\}$, ordered by inclusion, be the arboreal version of Cohen
forcing. The _meet_ of two perfect trees, $S$ and $T$, denoted $S\land T$, is
the largest perfect tree contained in $S\cap T$, if it exists (see [FGK19, pg.
5]). We say a subset $P$ of $\mathbb{S}$ is a _perfect poset_ if,
$\mathbb{C}\subseteq P$, $P$ is closed under finite unions, and for every
$S,T\in P$, if $S$ and $T$ are compatible in $\mathbb{S}$, then $S\land T\in
P$. If $P$ is a perfect poset, we order $P\times\omega$ by

$\displaystyle(S,n)\leq(T,m):\iff S\leq T\land m\leq n\land 2^{m}\cap
S=2^{m}\cap T.$

We denote the poset $(P\times\omega,\leq)$ by $\mathbb{Q}(P)$.

######  Definition 16 (Jensen’s Operation, $P^{H}$).

Let $M$ be a countable transitive model (ctm) of
$\mathsf{ZFC}^{-}+``\mathcal{P}(\omega)$ exists$"$, let $P\in M$ be a perfect
poset, let $\mathbb{Q}(P)^{<\omega}$ be the $\omega$-product of
$\mathbb{Q}(P)$ with finite support, let $H$ be a
$\mathbb{Q}(P)^{<\omega}$-generic filter over $M$, and let
$\langle\mathcal{T}_{k}:k\in\omega\rangle$ be the sequence of generic trees
added by $H$. We define $P^{H}$ as the closure of

$\displaystyle P\cup\\{\mathcal{T}_{k}\land S:S\in P\text{, }k\in\omega\text{,
and }\mathcal{T}_{k}\land S\neq\emptyset\\}$

under finite unions, and order $P^{H}$ by inclusion.

Jensen’s operation expands $P$ to a new poset $P^{H}$, which has many nice
properties. Crucially, predense subsets of $P$ remain predense in $P^{H}$:

######  Lemma 17.

Let $M$, $P$, $H$, and $\langle\mathcal{T}_{k}:k\in\omega\rangle$ be as in
Definition 16. Then:

  1. 1.

$P^{H}$ is a perfect poset.

  2. 2.

$\\{\mathcal{T}_{k}:k\in\omega\\}$ is a maximal antichain in $P^{H}$.

  3. 3.

Every predense set $D\subseteq P$ in $M$ remains predense in $P^{H}$.

Let $P^{<\omega}$ and $(P^{H})^{<\omega}$ be the finite support
$\omega$-product of $P$ and $P^{H}$, respectively. Then:

  1. 4.

Every predense set $D\subseteq P^{<\omega}$ in $M$ remains predense in
$(P^{H})^{<\omega}$.

###### Proof.

See [FGK19, Propositions 2.4 & 2.5]. ∎

Nowadays, Jensen forcing is usually constructed using a
$\diamondsuit$-sequence (cf. e.g. [Jec03, Theorem 28.1]). However, when Jensen
forcing was introduced, the $\diamondsuit$-principle had not been defined yet
and so Jensen had to do the construction by hand. We follow Jensen’s approach
and do not use a $\diamondsuit$-sequence because then the construction is
easier to generalise. Jensen forcing is defined as the union of a sequence
$\langle P_{\xi}:\xi<\omega_{1}\rangle$ of perfect posets. We define the
sequence recursively in $L$: let $P_{0}$ be the closure of $\mathbb{C}$ under
finite unions and let $\gamma_{0}$ be the least countable ordinal such that
$L_{\gamma_{0}}$ is a model of $\mathsf{ZFC}^{-}+``\mathcal{P}(\omega)$ exits”
containing $P_{0}$. We assume that for $\xi^{\prime}<\xi$, we have already
defined a pair $(L_{\gamma_{\xi^{\prime}}},P_{\xi^{\prime}})$ such that
$L_{\gamma_{\xi^{\prime}}}$ is a model of
$\mathsf{ZFC}^{-}+``\mathcal{P}(\omega)$ exits” and $P_{\xi^{\prime}}$ is a
perfect poset in $L_{\gamma_{\xi^{\prime}}}$. If $\xi$ is a limit ordinal,
then we set $P_{\xi}:=\bigcup_{\xi^{\prime}<\xi}P_{\xi^{\prime}}$. If
otherwise $\xi=\xi^{\prime}+1$, we set $P_{\xi}:=(P_{\xi^{\prime}})^{H}$,
where $H$ is the $<_{L}$-least
$\mathbb{Q}(P_{\xi^{\prime}})^{<\omega}$-generic filter over
$L_{\gamma_{\xi^{\prime}}}$. In both cases, let $\gamma_{\xi}$ be the least
countable ordinal such that $P_{\xi}\in L_{\gamma_{\xi}}$ and $\mathbb{R}\cap
L_{\gamma_{\xi}+1}\nsubseteq L_{\gamma_{\xi}}$. Finally, we define Jensen
forcing as $\mathbb{J}:=\bigcup_{\xi<\omega_{1}}P_{\xi}$. Note that for every
$\mathbb{J}$-generic filter $G$, $\bigcap\\{[T]:T\in G\\}$ is a real. We call
such reals _Jensen reals_.

######  Theorem 18 (Jensen, [Jen70]).

Jensen forcing is ccc, the set of Jensen reals over $L$ is $\Pi^{1}_{2}$ in
every inner model of $\mathsf{ZF}$ containing $L$, and for every
$\mathbb{J}$-generic filter $G$ over $L$, the set of Jensen reals in $L[G]$ is
a singleton.

###  4.2 $n$-Jensen forcing notions

In this section, we define a certain iteration of Jensen’s operation, a
Jensen-sequence, and then prove that each forcing notion associated to a
Jensen-sequences, which we call a _Jensen-like forcing notion_ , exhibits
generalisations of the properties Jensen forcing. Then we construct a forcing
notion to prove Theorem 12, using an analogue of [KL20, Theorem 13], which we
call the _Kanovei-Lyubetsky Lemma_ (Lemma 32) by using a witnessing property
for the underlying Jensen-sequence, _$n$ -completeness_, to ensure that the
products of resulting Jensen-like forcing notions are $n$-absolute for slices,
and hence can be used to separate the required fragments of choice.

First, we generalise Jensen’s construction from Section 4.1.

######  Definition 19.

Let $\zeta\leq\omega_{1}$. We say
$\langle(L_{\gamma_{\xi}},P_{\xi}):\xi<\zeta\rangle$ is a _Jensen-sequence_
if, for every $\xi<\zeta$,

  1. 1.

$P_{0}$ is the closure of $\mathbb{C}$ under finite unions,

  2. 2.

$\gamma_{\xi}$ is a countable ordinal such that
$L_{\gamma_{\xi}}\models\mathsf{ZFC}^{-}+``\mathcal{P}(\omega)$ exists” and
$\mathbb{R}\cap\mathrm{L}_{\gamma_{\xi}+1}\nsubseteq\mathrm{L}_{\gamma_{\xi}}$,

  3. 3.

$P_{\xi}$ is a perfect poset in $L_{\gamma_{\xi}}$,

  4. 4.

if $\xi=\zeta+1$, then there is a $\mathbb{Q}(P_{\zeta})^{<\omega}$-generic
filter $H_{\zeta}\in L_{\gamma_{\xi}}$ over $L_{\gamma_{\zeta}}$ such that
$P_{\xi}=(P_{\zeta})^{H_{\zeta}}$, and

  5. 5.

$\langle P_{\xi}:\xi\in\zeta\rangle$ is continuous at limits, i.e. if $\xi$ is
a limit, then $P_{\xi}=\bigcup_{\xi^{\prime}<\xi}P_{\xi^{\prime}}$.

As in the construction of Jensen forcing in Section 4.1, we can construct
Jensen-sequences of length $\omega_{1}$ by recursion. Whilst Jensen uses the
$<_{L}$-least generic filter, we consider all possible forcing notions which
are defined by Jensen-sequences, using any generic filter. We call a forcing
notion $\mathbb{P}$ _Jensen-like_ if there is a Jensen-sequence
$\langle(L_{\gamma_{\xi}},P_{\xi}):\xi<\omega_{1}\rangle$ such that
$\mathbb{P}:=\bigcup_{\xi<\omega_{1}}P_{\xi}$. Note that, as with Jensen
forcing, if $G$ is a generic filter for $\mathbb{P}$, then $\bigcap\\{[T]:T\in
G\\}$ is a real. We call such reals _$\mathbb{P}$ -generic reals_.

We show that the properties of Jensen-like forcing notions generalise those of
Jensen forcing in Theorem 18. Obviously, predense sets remain predense at
successor steps along a Jensen-sequence as those steps are instances of
Jensen’s operation; moreover predense sets remain predense at limit steps and
for the union of the sequence, and suitable products along a Jensen-sequence
are ccc:

######  Proposition 20.

If $\mathbb{P}$ is a Jensen-like forcing notion, then for every $\xi<\omega$,
every predense set $D\subseteq P_{\xi}$ (or $P_{\xi}^{<\omega}$) in
$L_{\gamma_{\xi}}$ is predense in $\mathbb{P}$ (or $\mathbb{P}^{<\omega}$).

###### Proof.

Let $\langle(L_{\gamma_{\xi}},P_{\xi}):\xi<\omega_{1}\rangle$ be a Jensen-
sequence such that $\mathbb{P}:=\bigcup_{\xi<\omega_{1}}P_{\xi}$. Suppose that
there is some $p\in\mathbb{P}$ where for all $d\in D$, $d\bot p$. By
definition, there is some $\xi<\omega_{1}$ such that $p\in P_{\xi}$. Then $D$
is not predense in $P_{\xi}$. Let $\zeta$ be minimal such that $\xi<\zeta$ and
$D$ is not predense in $P_{\zeta}$. Then $\zeta$ is a successor. Since
$\xi<\zeta$, $D\in L_{\gamma_{\xi}}\subseteq L_{\gamma_{\zeta-1}}$. But this
contradicts Lemma 17. The case $D\subseteq P_{\xi}^{<\omega}$ is similar. ∎

######  Proposition 21.

If $\mathbb{P}$ is a Jensen-like forcing notion, then the $\omega$-product
$\mathbb{P}^{<\omega}$ with finite support is ccc in $L$.

###### Proof.

By a condensation argument we can reduce this to Proposition 20 (cf. [Jen70,
Lemma 6]). ∎

######  Corollary 22.

Let $\mathbb{P}$ be a Jensen-like forcing notion, and $\mathbb{Q}$ be the
$\omega$-slice product of $\mathbb{P}$ with finite support of length
$\omega_{1}$. Then for every set $I\subseteq\omega_{1}\times\omega$,
$\mathbb{Q}{\upharpoonright}I$ is ccc in $L$.

###### Proof.

This follows from Proposition 21 and a $\Delta$-system argument. ∎

Next, we consider the uniqueness of generic reals for Jensen-like forcing
notions. With an argument similar to that of Jensen forcing, one can show that
for every generic filter $G$ for a Jensen-like forcing notion $\mathbb{P}$,
the set of $\mathbb{P}$-generic reals in $L[G]$ is a singleton. However, we
are more interested in what happens after forcing with a product of
$\mathbb{P}$. Kanovei and Lyubetsky [KL17] extended the uniqueness of Jensen
reals to countable products of Jensen forcing. We generalise their result to
Jensen-like forcing notions.

######  Lemma 23 (Kanovei-Lyubetsky).

Let $M$ be a ctm of $\mathsf{ZFC}^{-}+``\mathcal{P}(\omega)$ exists$"$, $P\in
M$ be a perfect poset, and let $\dot{y}\in M$ be a $P^{<\omega}$-name for a
real such that for every $k\in\omega$,
$1_{P^{<\omega}}\Vdash\dot{y}\neq\dot{x}^{k}_{G}$. Then for every
$\mathbb{Q}(P)^{<\omega}$-filter $H$ over $M$ and every generic tree
$\mathcal{T}_{k}$ added by $H$, in $M[H]$ the set of conditions forcing that
$\dot{y}\notin[\mathcal{T}_{k}]$ is dense in $(P^{H})^{<\omega}$.

###### Proof.

See [FGK19, Theorem 3.1]. ∎

######  Proposition 24.

Let $\mathbb{P}$ be a Jensen-like forcing notion and let $G$ be a
$\mathbb{P}^{<\omega}$-generic filter over $L$. Then for every
$\mathbb{P}$-generic real $y\in L[G]$ over $L$, there is some $k\in\omega$
such that $y=x_{G}^{k}$.

###### Proof.

By a condensation argument we can reduce this to Lemma 23. ∎

######  Corollary 25.

Let $\mathbb{P}$ be a Jensen-like forcing notion and let $\mathbb{Q}$ be the
$\omega$-slice product of $\mathbb{P}$ with finite support of length
$\omega_{1}$, let $I\subseteq\omega_{1}\times\omega$, and let $G$ be a
$\mathbb{Q}{\upharpoonright}I$-generic filter over $L$. Then for every
$\mathbb{P}$-generic real $y\in L[G]$ over $L$, there is some $i\in I$ such
that $y=x^{i}_{G}$.

###### Proof.

Let $y\in L[G]$ be $\mathbb{P}$-generic over $L$ and let $\dot{y}$ be a nice
$\mathbb{Q}{\upharpoonright}I$-name for $y$. By Corollary 22,
$\mathbb{Q}{\upharpoonright}I$ is ccc is and so $\dot{y}$ is countable. Hence,
there is a countable infinite $I^{\prime}\subseteq I$ such that $\dot{y}$ is a
$\mathbb{Q}{\upharpoonright}I^{\prime}$-name. Since $I^{\prime}$ is countable,
$\mathbb{Q}{\upharpoonright}I^{\prime}$ is order isomorphic to
$\mathbb{P}^{<\omega}$. Hence, we can apply Proposition 24 and so there is an
$i\in I^{\prime}$ such that $y=x^{i}_{G}$. ∎

Hence Jensen-like forcing notions satisfy generalisations of two of the three
properties in Theorem 18. The following proposition helps us with the
remaining one, to determine the complexity of the set of generic reals.

######  Proposition 26.

Let $\mathbb{P}$ be a Jensen-like forcing notion, let
$\langle(L_{\gamma_{\xi}},P_{\xi}):\xi<\omega_{1}\rangle$ be a Jensen-sequence
with $\mathbb{P}:=\bigcup_{\xi<\omega_{1}}P_{\xi}$, and let
$\langle\mathcal{T}_{k}^{\xi}:k\in\omega\rangle$ be the
$\mathbb{Q}(P_{\xi})^{<\omega}$-generic sequence used to construct
$P_{\xi+1}$. The following are equivalent:

  1. 1.

A real $x$ is $\mathbb{P}$-generic over $L$,

  2. 2.

for every $\xi<\omega_{1}$, $x$ is $P_{\xi}$-generic over $L_{\gamma_{\xi}}$,
and

  3. 3.

for every $\xi<\omega_{1}$, there is some $k\in\omega$ such that
$x\in[\mathcal{T}^{\xi}_{k}]$.

###### Proof.

We start with $(1.\Rightarrow 3.)$. Let $\xi<\omega_{1}$. By Lemma 17,
$\\{\mathcal{T}_{k}^{\xi}:k\in\omega\\}$ is a maximal anitchain in $P_{\xi+1}$
and by Proposition 20, it remains predense in $\mathbb{P}$. Hence, there is
some $k\in\omega$ such that $x\in[\mathcal{T}^{\xi}_{k}]$.

For $(3.\Rightarrow 2.)$, let $\xi<\omega_{1}$ and let $D\subseteq P_{\xi}$ be
dense in $L_{\gamma_{\xi}}$. By assumption, there is some $k\in\omega$ such
that $x\in[\mathcal{T}^{\xi}_{k}]$. We define

$\displaystyle E:=\\{q\in\mathbb{Q}(P_{\xi})^{<\omega}:q(k)=(T,n)\land\forall
s\in 2^{n}\cap T(T_{s}\in D)\\}.$

First, we prove that $E$ is dense in $\mathbb{Q}(P_{\xi})^{<\omega}$: let
$q\in\mathbb{Q}(P_{\xi})^{<\omega}$ and let $q(k)=(T,n)$. For every $s\in
2^{n}\cap T$, $T_{s}\in P_{\xi}$, so there is an $S_{s}\in D$ with $S_{s}\leq
T_{s}$. Let $S:=\bigcup_{s\in 2^{n}\cap T}S_{s}$ and
$q^{\prime}\in\mathbb{Q}(P_{\xi})^{<\omega}$ by $q^{\prime}(k):=(S,n)$ and for
$l\neq k$, $q^{\prime}(l):=q(l)$. Then $(S,n)\leq(T,n)$ and so $q^{\prime}\leq
q$ and $q^{\prime}\in E$. Hence, $E$ is dense in
$\mathbb{Q}(P_{\xi})^{<\omega}$. Let $H_{\xi}$ be the
$\mathbb{Q}(P_{\xi})^{<\omega}$-generic filter corresponding to
$\langle\mathcal{T}_{k}^{\xi}:k\in\omega\rangle$. Then $E$ meets $H_{\xi}$, so
there is a pair $(T,n)\in\mathbb{Q}(P_{\xi})$ such that for every $s\in
2^{n}\cap T$, $T_{s}\in D$ and $\mathcal{T}_{k}^{\xi}\subseteq T$. Let $s\in
2^{n}\cap T$ such that $s\subseteq x$. Then $x\in[T_{s}]$ and $T_{s}\in D$.
Thus, $x$ is $P_{\xi}$-generic over $L_{\gamma_{\xi}}$.

Finally, for $(2.\Rightarrow 1.)$, let $\mathcal{A}\subseteq\mathbb{P}$ be a
maximal antichain. Since $\mathbb{P}$ is ccc, $\mathcal{A}$ is countable, so
let $\xi<\omega_{1}$ be such that $\mathcal{A}\subseteq P_{\xi}$ and
$\mathcal{A}\in L_{\gamma_{\xi}}$. By assumption $x$ is $P_{\xi}$-generic over
$L_{\gamma_{\xi}}$, so there is a $T\in P_{\xi}\subseteq P$ with $x\in[T]$. ∎

######  Corollary 27.

Let $\mathbb{P}$ be a Jensen-like forcing notion, let
$J:=\langle(L_{\gamma_{\xi}},P_{\xi}):\xi<\omega_{1}\rangle$ be a Jensen-
sequence with $\mathbb{P}:=\bigcup_{\xi<\omega_{1}}P_{\xi}$, and let $n>2$. If
there is an unbounded subsequence
$J^{\prime}:=\langle(M_{\xi},Q_{\xi}):\xi<\omega_{1}\rangle$ of $J$ which is
$\Delta^{\mathbf{HC}}_{n-1}$, then the set of all $\mathbb{P}$-generics over
$L$ is $\Pi^{1}_{n}$ in every inner model of $\mathsf{ZFC}$ containing $L$.

###### Proof.

By Proposition 26, a real $x$ is $\mathbb{P}$-generic over $L$ iff for every
$\xi<\omega_{1}$, $x$ is $P_{\xi}$-generic over $L_{\gamma_{\xi}}$. If $x$ is
$P_{\xi}$-generic over $L_{\gamma_{\xi}}$, then $x$ is also
$P_{\xi^{\prime}}$-generic over $L_{\gamma_{\xi^{\prime}}}$ for every
$\xi^{\prime}<\xi$. Hence, $x$ is $\mathbb{P}$-generic over $L$ iff for every
$\xi<\omega_{1}$, $x$ is $Q_{\xi}$-generic over $M_{\xi}$. Let $M$ be an inner
model of $\mathsf{ZFC}$ containing $L$. Note that $\omega_{1}^{L}$ and
$L_{\omega_{1}^{L}}$ are $\Sigma^{\mathbf{HC}}_{1}$ in $M$. Hence,
$J^{\prime}$ is $\Delta^{\mathbf{HC}}_{n-1}$ in $M$ and so the set of
$\mathbb{P}$-generic reals over $L$ is $\Pi^{\mathbf{HC}}_{n-1}$ in $M$. By
[Jec03, Lemma 25.25], the set of $\mathbb{P}$-generic reals over $L$ is
$\Pi^{1}_{n}$. ∎

Next, we aim to show that $\omega$-slice products of Jensen-like forcing
notions are $n$-absolute for slices. Kanovei and Lyubetsky proved an analogous
result for almost disjoint forcing ([KL20, $\S 4$]), building their product
from an increasing sequence of forcing notions. Our method is somewhat
analogous; instead of almost disjoint forcing, we use Jensen-like forcing
notions. We first store together all of the Jensen-sequences in a single set.

######  Definition 28 ($\mathcal{M}_{\mathcal{J}}$ and $\preccurlyeq$).

Let $\mathcal{M}_{\mathcal{J}}$ be the set of all $(M,P)$ such that there is a
Jensen-sequence $J\in M$, such that $J^{\frown}(M,P)$ is a Jensen-sequence. We
define an ordering
$\preccurlyeq\,\,\subseteq\mathcal{M}_{\mathcal{J}}\times\mathcal{M}_{\mathcal{J}}$
by $(N,Q)\preccurlyeq(M,P)$ iff $N\subseteq M$, and either $Q=P$ or else there
is a Jensen-sequence $J\in M$ such that

  1. 1.

there is some $\xi\in\mathrm{dom}(J)$ such that $J(\xi)=(N,Q)$ and

  2. 2.

$J^{\frown}(M,P)$ is a Jensen-sequence.

Let $\vartheta\leq\omega_{1}$. A $\preccurlyeq$-increasing sequence
$\langle(M_{\xi},P_{\xi}):\xi<\vartheta\rangle$ in $\mathcal{M}_{\mathcal{J}}$
is _strict_ if $M_{\xi}\subsetneq M_{\xi+1}$ and $P_{\xi}\subsetneq P_{\xi+1}$
for every $\xi<\vartheta$.

######  Lemma 29.

Let $\langle(M_{\xi},P_{\xi}):\xi<\omega_{1}\rangle$ be a strictly
$\preccurlyeq$-increasing sequence where $\langle
P_{\xi}:\xi<\omega_{1}\rangle$ is continuous at limits. Then
$\bigcup_{\xi<\omega_{1}}P_{\xi}$ is a Jensen-like forcing notion.

###### Proof.

It is enough to show that there is a Jensen-sequence which contains
$\langle(M_{\xi},P_{\xi}):\xi<\omega_{1}\rangle$ as an unbounded subsequence.
We build a sequence $\langle J_{\zeta}:\zeta\leq\omega_{1}\rangle$ of Jensen-
sequences such that for every $0<\zeta\leq\omega_{1}$,

  1. 1.

$\langle(M_{\xi},P_{\xi}):\xi<\zeta\rangle$ is an unbounded subsequence of
$J_{\zeta}$,

  2. 2.

if $\zeta^{\prime}<\zeta<\omega_{1}$, then $J_{\zeta}$ extends
$J_{\zeta^{\prime}}$, and

  3. 3.

if $\zeta\leq\theta$ is a limit, then
$J_{\zeta}=\bigcup_{\zeta^{\prime}<\zeta}J_{\zeta^{\prime}}$.

Then $J_{\omega_{1}}$ is the desired Jensen-sequence. We define the
$J_{\zeta}$ recursively. By definition, there is a Jensen-sequence $J_{0}\in
M_{0}$, such that ${J_{0}}^{\frown}(M_{0},P_{0})$ is a Jensen-sequence. We set
$J_{1}:={J_{0}}^{\frown}(M_{0},P_{0})$. Suppose $J_{\zeta^{\prime}}$ is
already defined for $\zeta^{\prime}<\zeta$. We distinguish cases:

Case 1: $\zeta=\zeta^{\prime}+1$ and $\zeta^{\prime}=\zeta^{\prime\prime}+1$.
Then $\mathrm{dom}(J_{\zeta^{\prime}})$ is a successor ordinal
$\eta=\eta^{\prime}+1$ and
$J_{\zeta^{\prime}}(\eta^{\prime})=(M_{\zeta^{\prime\prime}},P_{\zeta^{\prime\prime}})$.
By construction
$(M_{\zeta^{\prime\prime}},P_{\zeta^{\prime\prime}})\preccurlyeq(M_{\zeta^{\prime}},P_{\zeta^{\prime}})$
and the sequence is strict, so there is a Jensen-sequence
$J^{\prime}_{\zeta}\in M_{\zeta^{\prime}}$ such that there is some
$\xi\in\mathrm{dom}(J^{\prime}_{\zeta})$ with
$J^{\prime}_{\zeta}(\xi)=(M_{\zeta^{\prime\prime}},P_{\zeta^{\prime\prime}})$
where ${J^{\prime}_{\zeta}}^{\frown}(M_{\zeta^{\prime}},P_{\zeta^{\prime}})$
is a Jensen-sequence. Let $J^{\prime\prime}_{\zeta}$ be the restriction of
$J^{\prime}_{\zeta}$ to $\mathrm{dom}(J^{\prime}_{\zeta}){\setminus}(\xi+1)$.
We set
$J_{\zeta}:={J_{\zeta^{\prime}}}^{\frown}{J^{\prime\prime}_{\zeta}}^{\frown}(M_{\zeta^{\prime}},P_{\zeta^{\prime}})$.

Case 2: $\zeta=\zeta^{\prime}+1$ and $\zeta^{\prime}$ is a limit. We set
$J_{\zeta}:={J_{\zeta^{\prime}}}^{\frown}(M_{\zeta^{\prime}},P_{\zeta^{\prime}})$.
Since $(M_{\xi},P_{\xi}):\xi<\zeta^{\prime}\rangle$ is an unbounded
subsequence of $J_{\zeta^{\prime}}$ and
$P_{\zeta^{\prime}}=\bigcup_{\xi<\zeta^{\prime}}P_{\xi}$, $J_{\zeta}$ is a
Jensen-sequence.

Case 3: $\zeta$ is a limit. We set
$J_{\zeta}:=\bigcup_{\zeta^{\prime}<\zeta}J_{\zeta^{\prime}}$. ∎

Only certain Jensen-sequences suffice for generating Jensen-like forcing
notions whose $\omega$-slice products are $n$-absolute for slices. To this
end, we define a kind of Jensen-sequence which will suffice, in analogy to
[KL20, Definition 15].

######  Definition 30.

Let $n>2$. A strictly $\preccurlyeq$-increasing sequence
$\langle(M_{\xi},P_{\xi}):\xi<\omega_{1}\rangle$ is called _$n$ -complete_ if,
for every $\Sigma^{\mathbf{HC}}_{n-2}$ set
$D\subseteq\mathcal{M}_{\mathcal{J}}$, there is a $\xi<\omega_{1}$ such that
either $(M_{\xi},P_{\xi})\in D$ or there is no $(N,Q)\in D$ extending
$(M_{\xi},P_{\xi})$.

######  Lemma 31.

For every $n>2$, there is a $\Delta^{\mathbf{HC}}_{n-1}$, $n$-complete
strictly $\preccurlyeq$-increasing sequence
$\langle(M_{\xi},P_{\xi}):\xi<\omega_{1}\rangle$ in $L$ .

###### Proof.

Let $n>2$, and let $\Gamma\subset\omega_{1}\times\mathbf{HC}$ be a universal
$\Sigma^{\mathbf{HC}}_{n-2}$ set. We define the required sequence recursively.
Let $(M_{0},P_{0})$ be the $<_{L}$-least pair such that
$(M_{0},P_{0})\in\mathcal{M}_{\mathcal{J}}$. Suppose that
$\langle(M_{\xi^{\prime}},P_{\xi^{\prime}}):\xi^{\prime}<\xi\rangle$ is
already defined. If $\xi$ is a limit, then we set
$P_{\xi}:=\bigcup_{\xi^{\prime}<\xi}P_{\xi^{\prime}}$ and let $M_{\xi}$ be the
$<_{L}$-least ctm of $\mathsf{T}$ such that
$(M_{\xi},P_{\xi})\in\mathcal{M}_{\mathcal{J}}$ and $M_{\xi}$ contains
$\langle(M_{\xi^{\prime}},P_{\xi^{\prime}}):\xi^{\prime}<\xi\rangle$. If
$\xi=\xi^{\prime}+1$ is a successor, let $(M_{\xi},P_{\xi})$ be the
$<_{L}$-least pair such that

  1. 1.

$(M_{\xi^{\prime}},P_{\xi^{\prime}})$ is strictly
$\preccurlyeq(M_{\xi},P_{\xi})$,

  2. 2.

either $(M_{\xi},P_{\xi})\in
D_{\xi^{\prime}}:=\\{m\in\mathcal{M}:(\xi^{\prime},m)\in\Gamma\\}$ or there is
no $(N,Q)\in D_{\xi^{\prime}}$ extending $(M_{\xi},P_{\xi})$.

Note that ${<_{L}}\cap\mathbf{HC}^{2}$ is $\Delta^{\mathbf{HC}}_{1}$. By
definition $\mathcal{M}_{\mathcal{J}}$ and $\preccurlyeq$ are
$\Delta^{\mathbf{HC}}_{1}$, and $\Gamma$ is $\Sigma^{\mathbf{HC}}_{n-2}$, so
$\langle(M_{\xi},P_{\xi}):\xi<\omega_{1}\rangle$ is
$\Delta^{\mathbf{HC}}_{n-1}$. By 2., the sequence is $n$-complete. ∎

The following lemma is the analogue of [KL20, Corollary 11 and Theorem 13] for
$n$-Jensen forcing notions. To prove their Theorem 13, Kanovei and Lyubetsky
defined a forcing-like relation which allowed them to approximate the forcing
relation for almost disjoint forcing when restricted to $\Sigma^{0}_{n}$
formulas ([KL20, $\S 5$]). One can replicate this construction for generalised
Jensen forcing; and with suitable checking of several properties which are
proved in the almost disjoint case in [KL20], the proof of the following lemma
goes through for forcing notions defined by $\Delta^{\mathbf{HC}}_{n-1}$,
$n$-complete sequences in $\mathcal{M}_{\mathcal{J}}$.444For details, see
[Wan23, Lemma 3.3.18] and its proof.

######  Lemma 32 (Kanovei-Lyubetsky Lemma).

For $n>2$, let $\langle(M_{\xi},P_{\xi}):\xi<\omega_{1}\rangle$ be a
$\Delta^{\mathbf{HC}}_{n-1}$, $n$-complete, and strictly
$\preccurlyeq$-increasing $\mathcal{M}_{\mathcal{J}}$ sequence. Let
$\mathbb{P}^{\ast}$ be the $\omega_{1}$-product of
$\bigcup_{\xi<\omega_{1}}P_{\xi}$, $G$ be $\mathbb{P}^{\ast}$-generic over
$L$, and $e\subseteq\omega_{1}$ be unbounded and in $L$. Then every
$\Sigma^{1}_{n}$ formula with parameters in $L[G{\upharpoonright}e]$ is
absolute between $L[G]$ and $L[G{\upharpoonright}e]$.

###### Proof.

The proof is similar to [KL20, Corollary 11 and Theorem 13]. ∎

Next, we provide a forcing-equivalence argument to show that Lemma 32 can be
used to prove $n$-absoluteness for slices of an $\omega$-slice product.

######  Proposition 33.

For $n>2$, let $\langle(M_{\xi},P_{\xi}):\xi<\omega_{1}\rangle$ be a
$\Delta^{\mathbf{HC}}_{n-1}$, $n$-complete, and strictly
$\preccurlyeq$-increasing $\mathcal{M}_{\mathcal{J}}$ sequence. Let
$\mathbb{P}$ be the $\omega_{1}$-product of $\bigcup_{\xi<\omega_{1}}P_{\xi}$,
$G$ be $\mathbb{P}$-generic over $L$. Let $\mathbb{P}$ be the $\omega$-slice
product $\bigcup_{\xi<\omega_{1}}P_{\xi}$ of length $\omega_{1}$. Then
$\mathbb{P}$ is $n$-absolute for slices.

###### Proof.

In $L$, there is a canonical bijection
$b:\omega_{1}\to\omega_{1}\times\omega$. This $b$ induces an isomorphism
between $\mathbb{P}$ and $\mathbb{P}^{\ast}$, the $\omega_{1}$-product from
Lemma 32. Now let $G^{\ast}$ be the $\mathbb{P}^{\ast}$-generic filter over
$L$ corresponding to $G$ induced by $b$, let
$\mathcal{S}\subseteq\mathcal{P}(\omega_{1}\times\omega)$ be an unbounded set
of slices, and let $s\in\mathcal{S}$ be a slice. Note that
$L[G{\upharpoonright}s]=L[G^{\ast}{\upharpoonright}{b^{-1}(s)}]$ and
$b^{-1}(s)$ is unbounded in $\omega_{1}$. So, by Lemma 32, $\mathbb{P}$ is
$n$-absolute for $\mathcal{S}$-slices. ∎

######  Definition 34.

Let $n\geq 2$. We call a Jensen-like forcing notion $\mathbb{P}$ $n$-Jensen if

  1. 1.

the set of $\mathbb{P}$-generic reals over $L$ is $\Pi^{1}_{n}$ in every inner
model of $\mathsf{ZFC}$ containing $L$, and

  2. 2.

the $\omega$-slice product of $\mathbb{P}$ with finite support of length
$\omega_{1}$ i $n$-absolute for slices.

Clearly, Jensen forcing is $2$-Jensen. Moreover, if $n>2$ and
$\langle(M_{\xi},P_{\xi}):\xi<\omega_{1}\rangle$ is a
$\Delta^{\mathbf{HC}}_{n-1}$, $n$-complete, and strictly
$\preccurlyeq$-increasing $\mathcal{M}_{\mathcal{J}}$ sequence, then
$\mathbb{P}:=\bigcup_{\xi<\omega_{1}}P_{\xi}$ is $n$-Jensen. Hence, by Lemma
31, for every $n\geq 2$, there is an $n$-Jensen forcing notion in $L$. It is
not known whether, up to forcing equivalence, there is exactly one $n$-Jensen
forcing. We expect not.

Our final flourish is to prove Theorem 12, i.e. to show that $n$-slicing
forcing notions exist in $L$. Good candidates are $\omega$-slice products of
$n$-Jensen forcing notions. However, we do not know whether the set
$\\{(\ell,x_{G}^{(\ell,k)}):(\ell,k)\in\omega^{2}\\}$ is $\Pi^{1}_{n}$. Recall
that this was crucial in showing that the required amount of choice fails in
Theorem 3. The problem is that this does not follow from the fact that the set
of all $\mathbb{P}$-generic reals is $\Pi^{1}_{n}$ and Corollary 25, because
we cannot tell from which slice $\mathbb{P}$-generic reals are. But we solve
this issue by modifying $\omega$-slice products of $n$-Jensen forcing notions
to use the first bit of generic reals to ‘track’ which slice they are from.

###### Proof of Theorem 12.

Let $\mathbb{P}^{\ast}$ be $n$-Jensen, let $\mathbb{P}$ be the $\omega$-slice
product of $\mathbb{P}^{\ast}$ with finite support of length $\omega_{1}$, and
let $\mathbb{Q}^{k}:=\\{T\in\mathbb{P}^{\ast}:\langle
k\rangle\subseteq\mathrm{stem}(T)\\}$. Then $\mathbb{Q}^{k}$ is a perfect
poset and if $D$ is dense in $\mathbb{P}^{\ast}$, then $D\cap\mathbb{Q}^{k}$
is dense in $\mathbb{Q}^{k}$ for every $k\in\omega$. Hence, for every
$k\in\omega$, every $\mathbb{Q}^{k}$-generic real is also
$\mathbb{P}^{\ast}$-generic.

Let $\mathbb{Q}$ be the $\omega$-slice product of
$\langle\mathbb{Q}_{\nu}:\nu<\omega_{1}\rangle$ with finite support, where

$\displaystyle\mathbb{Q}_{\nu}=\begin{cases}\mathbb{Q}^{\nu+1}&\nu\in\omega\\\
\mathbb{Q}^{0}&\text{otherwise,}\end{cases}$

let $H$ be a $\mathbb{Q}$-generic filter over $L$, and let
$H^{\prime}:=\\{p\in\mathbb{P}:\exists q\in H(q\leq p)\\}$. We show that
$H^{\prime}$ is $\mathbb{P}$-generic over $L$. Let $D$ be dense in
$\mathbb{P}$. Then $D\cap\mathbb{Q}$ is dense in $\mathbb{Q}$ and so $H$ meets
$D\cap\mathbb{Q}$. Therefore, $H^{\prime}$ meets $D$. Thus, $H^{\prime}$ is
$\mathbb{P}$-generic over $L$. As $H^{\prime}$ can be constructed in $L[H]$
and vice versa, $L[H]=L[H^{\prime}]$. By Shoenfield absoluteness or
Proposition 33 respectively, $\mathbb{Q}$ is $n$-absolute for slices.

It remains to show that $\\{(\ell,x_{H}^{(\ell,k)}):(\ell,k)\in\omega^{2}\\}$
is $\Pi^{1}_{n}$ in $L[H]$. Let $B$ be the set of all
$\mathbb{P}^{\ast}$-generics in $L[H]$. By Corollary 25 , a real $y\in L[H]$
is $\mathbb{P}^{\ast}$-generic iff there is some
$(\nu,k)\in\omega_{1}\times\omega$ such that $y=x^{(\nu,k)}_{H}$. Hence
$B=\\{x^{(\nu,k)}_{H^{\prime}}:(\nu,k)\in\omega_{1}\times\omega\\}$. By
definition of $H^{\prime}$, $x_{H}^{(\nu,k)}=x_{H^{\prime}}^{(\nu,k)}$ for
every $(\nu,k)\in\omega_{1}\times\omega$, so
$B=\\{x^{(\nu,k)}_{H}:(\nu,k)\in\omega_{1}\times\omega\\}$. By Corollary 27,
$B$ is $\Pi^{1}_{n}$ in $L[H]$. Therefore,
$\\{(\ell,x_{H}^{(\ell,k)}):(\ell,k)\in\omega^{2}\\}=\\{(\ell,x):x(0)=\ell+1\land
x\in B\\}$ is $\Pi^{1}_{n}$ in $L[H]$. Hence $\mathbb{Q}$ is $n$-slicing. ∎

We could use Kanovei and Lyubetsky’s variant of almost disjoint forcing from
[KL20] to prove Corollary 4, by adapting the proof of Theorem 3 to almost
disjoint rather than Jensen forcing. This yields a model of
$\mathsf{ZF}+\neg\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{Ctbl})$. As the
product of this variant is $n$-absolute for slices (see [KL20, Theorem 9]),
$\mathsf{DC}(\mathbb{R};\Pi^{1}_{n})$ also holds in this model. A compactness
argument then yields Corollary 4. However, our proof of Theorem 3 does not
work for almost disjoint forcing, as the generics are not unique: we can swap
finitely many bits in a generic to generate another generic (as in [KL20,
Lemma 9vi)]). In which case, we do not know whether
$\mathsf{AC}(\mathbb{R};\Pi^{1}_{n+1})$ holds.

##  5 Open questions

So far, we have only considered the hierarchy of uniform choice principles for
‘$\mathbf{\Pi}$’ projective point classes. This suffices, as the
‘$\mathbf{\Pi}$’- and ‘$\mathbf{\Sigma}$’-type uniform choice principle
hierarchies are equivalent in the following sense:

######  Proposition 35 (Kanovei, [Kan79, pg. 6]).

Under $\mathsf{ZF}$,
$\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\mathbf{\Pi}^{1}_{n})$ is
equivalent to
$\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\mathbf{\Sigma}^{1}_{n+1})$.

By contrast, for the non-uniform choice principles, we know only that
$\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{\Pi}^{1}_{n+1})$ implies
$\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{\Pi}^{1}_{n})$ and
$\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{\Sigma}^{1}_{n})$.

######  Question 36.

Is $\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{\Pi}^{1}_{n})$
$\mathsf{ZF}$-equivalent to
$\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{\Sigma}^{1}_{n+1})$?

Such an equivalence does not follow from the uniform equivalence (i.e.
Proposition 35), as Corollary 6 shows that
$\mathsf{AC}_{\omega}(\mathbb{R};\Gamma)$ is not $\mathsf{ZF}$-equivalent to
$\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\Gamma)$.

As well as the equivalences of non-uniform descriptive choice principles, we
can ask about separating non-uniform descriptive choice principles. By
Corollary 6, we can separate uniform choice for all projective pointclasses
from the minimal non-uniform descriptive choice principle beyond $\mathsf{ZF}$
(namely $\mathsf{AC}_{\omega}(\mathbf{F}_{\sigma})$). It is not known whether
the non-uniform projective choice principles can be separated from one
another:

######  Question 37.

Is there a model $M$ which satisfies
$\mathsf{ZF}+\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{\Pi}^{1}_{n})$ which
violates $\mathsf{AC}_{\omega}(\mathbb{R};\Pi^{1}_{n+1})$?

Lastly, we defined our principles for any pointclass, so we can ask about
Borel choice principles. In $\mathsf{ZFC}$, several definitions are equivalent
to the ordinary definition of the Borel hierarchy (i.e. where
$\mathbf{\Sigma}^{0}_{\alpha}:=\\{\bigcup_{n\in\omega}A_{n}:A_{n}\in\bigcup_{\beta<\alpha}\mathbf{\Pi}^{0}_{\beta}\\}$).
These need not coincide under $\mathsf{ZF}$, hence their corresponding Borel
choice principles may be distinct. For example, the ‘closure’ definition
(where $\widehat{\mathbf{\Sigma}}^{0}_{\alpha}$ is the closure of
$\bigcup_{\beta<\alpha}\widehat{\mathbf{\Pi}}^{0}_{\beta}$ under countable
unions) is possibly distinct in $\mathsf{ZF}$, e.g. possibly
$\widehat{\mathbf{\Sigma}}^{0}_{2}\supsetneq\mathbf{\Sigma}^{0}_{2}$ (see
[Mil08, Mil11] for examples). The codeable-Borel hierarchy is a further such
definition (see [Fre08, §562]). We can ask whether the various Borel choice
principles can be separated, e.g.:

######  Question 38.

Is there a model $M$ which satisfies $\mathsf{ZF}$ and
$\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{\Sigma}^{0}_{\alpha})$ which violates
$\mathsf{AC}_{\omega}(\mathbb{R};\mathbf{\Sigma}^{0}_{\alpha+1})$? Is there a
model which satisfies $\mathsf{ZF}$ and
$\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\mathbf{\Sigma}^{0}_{\alpha})$
which violates
$\mathsf{AC}_{\omega}(\mathbb{R};\mathrm{unif}\mathbf{\Sigma}^{0}_{\alpha+1})$?

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