$\mathsf{MA}(\mathcal{I}$) and a Failure of Separation on the third Level.

Separation is one of the three classical and fundamental properties of definable subsets of the reals with numerous applications in descriptive set theory, measure theory, functional analysis and ergodic theory (see e.g. the classical textbooks [14] or [12]).

Recall first the definition of separation.

Now we turn to uniformization. Recall that for an $A\subset 2^{\omega}\times 2^{\omega}$, we say that $f$ is a uniformization (or a uniformizing function) of $A$ if $f:pr_{1}(A)\rightarrow 2^{\omega}$, where $pr_{1}(A)$ is $A$’s projection on the first coordinate, and the graph of $f$ is a subset of $A$. In other words, $f$ chooses exactly one point of every non-empty section of $A$.

It is a classical result of Novikov that if $\Gamma$ has the uniformization property, then $\Gamma$ does not have the separation property.

Starting point for this work is the following, on the surface surprising theorem (see [6]):

It is natural to ask whether already weaker forcing axioms axioms are sufficient to settle separation on the third projective level in the same way. We show that they do not suffice. Let $\mathsf{MA}(\mathcal{I})$ be Martin’s axiom for the class of indestructible ccc forcings, i.e. for those forcing $\mathbb{P}$ such that $\mathbb{P}\times\mathbb{Q}$ is ccc for every ccc forcing $\mathbb{Q}$. The main theorem of this article is the following:

Forcing a failure of $\Pi$-separation in the presence of forcing axioms is considerably more delicate than it might appear on a first glance. Aside from theorem 1.4, this is additionally explained by the fact that there is a very general method to force the global $\Sigma$-uniformization property (see [8]) and $\Sigma$-uniformization implies $\Pi$-separation. Thus any attempt to force the failure of $\Pi$-separation together with some forcing axiom, must be such that it can not be combined with the method from [8], which is albeit technicalities a straightforward iteration with lots of flexibility. So failures of $\Pi$-separation in the presence of forcing axioms are necessarily in need of somewhat unconventional constructions. This article is an attempt to further explore the tension between forcing axioms and separation in the presence of the anti-large cardinal axiom $\omega_{1}=\omega_{1}^{L}$.

There exists literature already which deals with the failure of separation. L. Harrington, in handwritten notes from 1974 devised a forcing which forces the failure of ${\Pi^{1}_{n}}$ and ${\Sigma^{1}_{n}}$-separation for a fixed $n\geq 3$. This result was proved in detail in [11] using different methods. Both constructions are very specific universes of set theory and the proof of the failure of separation involves homogeneity considerations of the forcings involved, which will get lost once we start to throw in additional forcings in order to work for $\mathsf{MA}(\mathcal{I})$. Thus in order to approach a proof of our main theorem a new technique is in need. The method presented here approaches the problem from a very different angle and builds on some techniques and ideas from [6], [4], [7], [9], in that we construct an iteration which produces two pairs of set on the third projective level which diagonalize against all possible pairs of $\Sigma^{1}_{3}$ or $\Pi^{1}_{3}$ sets which are candidates for separating them.

The coding method of our choice utilizes Suslin trees, which can be generically destroyed in an independent way of each other.

Note that an independent sequence of Suslin trees $\vec{T}=(T_{\alpha}\,:\,\alpha<\kappa)$ has the property that whenever we decide to generically add branches to some of its members, then all the other members of $\vec{T}$ remain Suslin in the resulting generic extension. Indeed, if $A\subset\kappa$ and we form $\prod_{i\in A}T_{i}$ with finite support, then in the resulting generic extension $V[G]$, for every $\alpha\notin A$, $V[G]\models``T_{\alpha}$ is a Suslin tree$"$.

One can easily force the existence of independent sequences of Suslin trees with products of Jech’s or Tennenbaum’s forcing, or with just products of ordinary Cohen forcing. On the other hand independent sequences of length $\omega_{1}$ already exist in $L$.

Whenever we force with a Suslin tree $(T,<_{T})$, i.e. we force with its nodes to add an uncountable branch, we denote the forcing with $T$ again.

We briefly introduce the almost disjoint coding forcing due to R. Jensen and R. Solovay (see [10]). We will identify subsets of $\omega$ with their characteristic function and will use the word reals for elements of $2^{\omega}$ and subsets of $\omega$ respectively. Let $D=\{d_{\alpha}\,\>\,\alpha<\aleph_{1}\}$ be a family of almost disjoint subsets of $\omega$, i.e. a family such that if $r,s\in D$ then $r\cap s$ is finite. Let $X\subset\omega$ be a set of ordinals. Then there is a ccc forcing, the almost disjoint coding $\mathbb{A}_{D}(X)$ which adds a new real $x$ which codes $X$ relative to the family $D$ in the following way

We shall briefly discuss the $L$-definable, $\aleph_{1}^{L}$-sized almost disjoint family of reals $D$ we will use throughout this article. The family $D$ is the canonical almost disjoint family one obtains when recursively adding the $<_{L}$-least real $x_{\beta}$ not yet chosen and replace it with $d_{\beta}\subset\omega$ where that $d_{\beta}$ is the real which codes the initial segments of $x_{\beta}$ using some recursive bijections between $\omega$ and $\omega^{<\omega}$.

We have to first create a suitable ground model $W$ over which the actual iteration will take place. The construction is inspired by [4], where a very similar universe is used as well. $W$ will be a generic extension of $L$, satisfying $\mathsf{CH}$ and having the crucial property that in $W$ there is an $\omega_{2}$-sequence $\vec{S}$ of independent Suslin trees which is $\Sigma_{1}(\omega_{1})$-definable over $H(\omega_{2})^{W}$. The sequence $\vec{S}$ will enable a coding method which is independent of the surrounding universe, a feature we will exploit to a great extent in what follows.

To form $W$, we start with Gödels constructible universe $L$ as our ground model. Recall that $L$ comes equipped with a $\Sigma_{1}$-definable, global well-order $<_{L}$ of its elements. We first fix an appropriate sequence of stationary, co-stationary subsets of $\omega_{1}$ using Jensen’s $\diamondsuit$-sequence. The proof of the next lemma is well-known so we skip it.

The $\diamondsuit$-sequence can be used to produce an easily definable sequence of $L$-stationary, co-stationary subsets of $\omega_{1}$: we list the elements of $L_{\omega_{2}}$ in an $\omega_{2}$ sequence $(r_{\alpha}\,:\,\alpha<\omega_{2})$. We fix

and note that $R$ is stationary and co-stationary.

Then we define for every $\beta<\omega_{2}$, $\beta\neq 0$ a stationary, co-stationary set in the following way:

and

It is clear that $\forall\alpha\neq\beta(R_{\alpha}\cap R_{\beta}\in\hbox{NS}_{\omega_{1}})$ and that $R_{\beta}\cap R=\emptyset$. To avoid writing $\beta\neq 0$ all the time we re-index and confuse $(R_{\beta}\,:\,\beta<\omega_{2},\beta\neq 0)$ with $(R_{\beta}\,:\,\beta<\omega_{2})$. We derive the following standard result concerning the definability of the $R_{\beta}$’s. A detailed proof can be found in [6], Lemma 1.12.

We proceed with defining the universe $W$. First we generically add $\aleph_{2}$-many Suslin trees using of Jech’s Forcing $\mathbb{P}_{J}$. We let

using countable support. This is a $\sigma$-closed, hence proper notion of forcing. We denote the generic filter of $\mathbb{Q}^{0}$ with $\vec{S}=(S_{\alpha}\,:\,\alpha<\omega_{2})$ and note that $\vec{S}$ is independent. We fix a definable bijection between $[\omega_{1}]^{2}$ and $\omega_{1}$ and identify the trees in $(S_{\alpha}\,:\,\alpha<\omega_{2})$ with their images under this bijection, so the trees will always be subsets of $\omega_{1}$ from now on.

In a second step we code the trees from $\vec{S}$ into the sequence of $L$-stationary subsets $\vec{R}$ we produced earlier. It is important to note that the forcing we are about to define does preserve Suslin trees, a fact we will show later. The forcing used in the second step will be denoted by $\mathbb{Q}^{1}$ and will be a countable support iteration of length $\omega_{2}$ whose factors are itself countably supported iterations of length $\omega_{1}$. Fix first a definable bijection $h\in L_{\omega_{3}}$ between $\omega_{1}\times\omega_{2}$ and $\omega_{2}$ and write $\vec{R}$ from now on in ordertype $\omega_{1}\cdot\omega_{2}$ making implicit use of $h$, so we assume that $\vec{R}=(R_{\alpha}\,:\,\alpha<\omega_{1}\cdot\omega_{2})$.

$\mathbb{Q}^{1}$ is defined via induction in the ground model $L[\mathbb{Q}^{0}]$. Assume we are at stage $\alpha<\omega_{2}$ and we have already created the iteration $\mathbb{Q}^{1}_{\alpha}$ up to stage $\alpha$. We work with $L[\mathbb{Q}^{0}][\mathbb{Q}^{1}_{\alpha}]$ as our ground model and consider the Suslin tree $S_{\alpha}\subset\omega_{1}$. We define the forcing we want to use at stage $\alpha$, denoted by $\mathbb{Q}^{1}(\alpha)$, as the countable support iteration which codes the characteristic function of $S_{\alpha}$ into the $\alpha$-th $\omega_{1}$-block of the $R_{\beta}$’s. So $\mathbb{Q}^{1}(\alpha)=\Asterisk_{\gamma<\omega_{1}}\mathbb{R}_{\alpha}(\gamma)$ is again a countable support iteration in $L[\mathbb{Q}^{0}][\mathbb{Q}^{1}_{\alpha}]$, defined inductively via

and

Recall that we let $R$ be a stationary, co-stationary subset of $\omega_{1}$ which is disjoint from all the $R_{\alpha}$’s which are used. We let $\mathbb{Q}^{1}$ be the countably supported iteration,

which is an $R$-proper forcing. We shall see later that $\mathbb{Q}^{1}$ in fact is $\omega$-distributive, hence the iteration $\mathbb{Q}^{1}$ is in fact a countably supported product. This way we can turn the generically added sequence of Suslin trees $\vec{S}$ into a definable sequence of Suslin trees. Indeed, if we work in $L[\vec{S}\ast G]$, where $\vec{S}\ast G$ is $\mathbb{Q}^{0}\ast\mathbb{Q}^{1}$-generic over $L$, then

Note here that the above formula can be used to make every $S_{\alpha}$ $\Sigma_{1}(\omega_{1},\alpha)$ definable over $L[\vec{S}\ast G]$ as is shown with the next lemma.

Let us set

which will serve as our ground model for a second iteration of length $\omega_{2}$. We shall need the following well-known result (see [1], Theorem 2.10 pp. 20.)

Note that by the last lemma, the second forcing $\mathbb{Q}^{1}$ which is a countably supported iteration of the appropriate club shooting forcings is in fact just a countably supported product of its factors, i.e. at every stage of the iteration we can force with the associated $\mathbb{P}_{R}$, as computed in $L$.

Our goal is to use $\vec{S}$ for coding again. For this it is essential that the sequence remains independent in $W$. To see this we shall argue that forcing with $\mathbb{Q}^{1}$ over $L[\mathbb{Q}^{0}]$ preserves Suslin trees. Recall that for a forcing $\mathbb{P}$, $\theta$ sufficiently large and regular and $M\prec H(\theta)$, a condition $q\in\mathbb{P}$ is $(M,\mathbb{P})$-generic iff for every maximal antichain $A\subset\mathbb{P}$, $A\in M$, it is true that $A\cap M$ is predense below $q$. In the following we will write $T_{\eta}$ to denote the $\eta$-th level of the tree $T$ and $T\upharpoonright\eta$ to denote the set of nodes of $T$ of height $<\eta$. The key fact is the following (see [13] for the case where $\mathbb{P}$ is proper, see also [6] for a detailed proof).

With a routine adjustment one can show that Theorem 1.3 of [13] holds true if one replaces proper by $R$-proper for $R\subset\omega_{1}$ a stationary subset, i.e. that a countable support iteration of $R$-proper forcings which preserve Suslin trees results in a forcing which preserves Suslin trees.

Invoking the generalized version of Miyamoto’s theorem we obtain:

We end with a straightforward lemma which is used later in coding arguments.

The following is taken entirely from [9], which itself is a simplification of the coding machinery first constructed in [4] and [5]. We continue with the construction of the appropriate notions of forcing which we want to use in our proof.

The objective is to first define a coding forcing $\operatorname{Code}(x,\eta,i)$ for real numbers $x$ and ordinals $\eta<\omega_{2}$ and $i\in 3$, which will force a certain $\Sigma^{1}_{3}$-formula $\Phi(x,i)$ to hold in the resulting generic extension. The forcing will be a two step iteration $\mathbb{Q}_{0}\ast\mathbb{Q}_{1}$ an will be defined uniformly for all four $\omega_{2}$-sequences of Suslin trees $\vec{S}^{0},\vec{S}^{1},\vec{S}^{2}$ and $\vec{S}^{3}$. In order to reduce notation we will define it just for $i=1$.

In the first step, we pick our given real $x\subset\omega$ and eliminate all elements of $\vec{S}^{1}$ starting at the $\eta$-th $\omega$-block of $\vec{S}^{1}$, by generically adding an $\omega_{1}$-branch according $x\subset\omega$. Specifically, we first form $E_{x,\eta}:=\{\omega\cdot\eta+2n\mid n\in\omega,n\in x\}$ and $F_{x,\eta}:=\{\omega\cdot\eta+2n+2\mid n\in\omega,n\notin x\}$. Then we force with

with finite support with ground model $W$. This is an $\aleph_{1}$-sized, ccc forcing over $W$, so in the generic extension, $\aleph_{1}$ is preserved. We let $g$ be a $\mathbb{Q}_{0}$-generic filter and work in $W[g]$.

Let $X\subset\omega_{1}$ be the least set (in some fixed well-order of $H(\omega_{2})^{W}[g]$) that encodes the following objects:

• •

  The $<$-least set of $\omega_{1}\cdot\omega\cdot\omega_{1}$-many club subsets through $\vec{R}$, our $\Sigma_{1}(\{\omega_{1}\})$-definable sequence of $L$-stationary subsets of $\omega_{1}$, which are necessary to compute every tree $S_{\beta}\in\vec{S}^{1}$ in the interval $[\omega\cdot\eta,\omega\cdot\eta+\omega)$, using the $\Sigma_{1}(\{\omega_{1}\})$-formula from the previous section.

• •

  The least set of $\omega_{1}$-branches in $W$ through elements of $\vec{S}^{1}$ that code $x$ at the $\omega$-block of trees in $\vec{S}^{1}$ starting at $\eta$. Specifically, we collect:

  $$\{b_{\beta}\subset S^{1}_{\beta}:\beta\in E_{x,\eta}\cup F_{x,\eta}\}$$

In $L[X]$, we can decode $x$ by examining the $\omega$-block of $\vec{S}^{1}$-trees starting at $\eta$ and determining which tree has an $\omega_{1}$-branch in $L[X]$:

• $(\ast)$

  $n\in x$ if and only if $S^{1}_{\omega\cdot\eta+2n+1}$ has an $\omega_{1}$-branch, and $n\notin x$ if and only if $S^{1}_{\omega\cdot\eta+2n}$ has an $\omega_{1}$-branch.

Indeed, if $n\notin x$, we added a branch through $S^{1}_{\omega\eta+2n}$. If $S^{1}_{\omega\eta+2n}$ is Suslin in $L[X]$, then we must have added an $\omega_{1}$-branch through $S^{1}_{\omega\eta+2n+1}$, since we always add an $\omega_{1}$-branch through either $S^{1}_{\omega\eta+2n+1}$ or $S^{1}_{\omega\eta+2n}$. Adding branches through some $S^{1}_{\alpha}$’s will not affect whether some $S^{1}_{\beta}$ is Suslin in $L[X]$, as $\vec{S}^{1}$ is independent.

We can now apply an argument similar to David’s trick (see [2] or [3]). We rewrite the information of $X\subset\omega_{1}$ as a set $Y\subset\omega_{1}$. Any transitive, $\aleph_{1}$-sized model $M$ of $\mathsf{ZF}^{-}$ containing $X$ will be able to decode all the information from $X$. Therefore, if we code the model $(M,\in)$ containing $X$ as a set $X_{M}\subset\omega_{1}$, then for any uncountable $\beta$ such that $L_{\beta}[X_{M}]\models\mathsf{ZF}^{-}$ and $X_{M}\in L_{\beta}[X_{M}]$, we have:

In particular, there will be an $\aleph_{1}$-sized ordinal $\beta$, and we can fix a club $C\subset\omega_{1}$ and a sequence $(M_{\alpha}:\alpha\in C)$ of countable elementary submodels of $L_{\beta}[X_{M}]$ such that:

We now define $Y\subset\omega_{1}$ to code the pair $(C,X_{M})$, where the odd entries of $Y$ code $X_{M}$, and $E(Y)$ denotes the set of even entries of $Y$. Let $\{c_{\alpha}:\alpha<\omega_{1}\}$ be the enumeration of $C$, and we define the following conditions:

• -

  $E(Y)\cap\omega$ codes a well-ordering of type $c_{0}$.

• -

  $E(Y)\cap[\omega,c_{0})=\emptyset$.

• -

  For each $\beta$, $E(Y)\cap[c_{\beta},c_{\beta}+\omega)$ codes a well-ordering of type $c_{\beta+1}$.

• -

  For each $\beta$, $E(Y)\cap[c_{\beta}+\omega,c_{\beta+1})=\emptyset$.

Finally, we obtain:

• $({\ast}{\ast})$

  For any countable transitive model $M$ of $\mathsf{ZF}^{-}$ such that $\omega_{1}^{M}=(\omega_{1}^{L})^{M}$ and $Y\cap\omega_{1}^{M}\in M$, $M$ can construct its version of the universe $L[Y\cap\omega_{1}^{N}]$, and the latter will see that there is an $\aleph_{1}^{M}$-sized transitive model $N\in L[Y\cap\omega_{1}^{N}]$ which models $(\ast)$ for $x$ at some $\omega$-block of $\vec{S}^{1}$-trees.

This establishes a local version of the property $(\ast)$.

In the next step, we use almost disjoint forcing $\mathbb{A}_{D}(Y)$ relative to to some canonically defined almost disjoint family of reals $D\in L$ to code the set $Y$ into one real $r$. This forcing depends only on the subset of $\omega_{1}$ being coded, so $\mathbb{A}_{D}(Y)$ will be independent of the surrounding universe, as long as it has the correct $\omega_{1}$ and contains $Y$.

We finally obtain a real $r$ such that

• $({\ast}{\ast}{\ast})$

  For any countable, transitive model $M$ of $\mathsf{ZF}^{-}$ such that $\omega_{1}^{M}=(\omega_{1}^{L})^{M}$ and $r\in M$, $M$ can construct its version of $L[r]$ which in turn thinks that there is a transitive $\mathsf{ZF}^{-}$-model $N$ of size $\aleph_{1}^{M}$ such that $N$ believes $(\ast)$ for $x$.

Note that $({\ast}{\ast}{\ast})$ is a $\Pi^{1}_{2}$-formula in the parameters $r$ and $x$. We will often suppress the reals $r,x$ when referring to $({\ast}{\ast}{\ast})$ as they will be clear from the context. We say in the above situation that the real $x$ is written into $\vec{S}^{1}$ at some $\eta$, or that $x$ is coded into $\vec{S^{1}}$ (at $\eta$) and $r$ witnesses that $x$ is coded. Likewise a forcing $\operatorname{Code}(x,i)$ is defined for coding the real $x$ into $\vec{S^{i}}$.

The projective and local statement $({\ast}{\ast}{\ast})$, if true, will determine how certain inner models of the surrounding universe will look like with respect to branches through $\vec{S}^{i}$. That is to say, if we assume that $({\ast}{\ast}{\ast})$ holds for a real $x$ and is the truth of it is witnessed by a real $r$. Then $r$ also witnesses the truth of $({\ast}{\ast}{\ast})$ for any transitive $\mathsf{ZF}^{-}$-model $M$ which contains $r$ (i.e. we can drop the assumption on the countability of $M$). Indeed if we assume that there would be an uncountable, transitive $M$, $r\in M$, which witnesses that $({\ast}{\ast}{\ast})$ is false. Then by Löwenheim-Skolem, there would be a countable $N\prec M$, $r\in N$ which we can transitively collapse to obtain the transitive $\bar{N}$. But $\bar{N}$ would witness that $({\ast}{\ast}{\ast})$ is not true for every countable, transitive model, which is a contradiction.

Consequently, the real $r$ carries enough information that the universe $L[r]$ will see that certain trees from $\vec{S}^{i}$ have branches in that

and

Indeed, the universe $L[r]$ will see that there is a transitive $\mathsf{ZF}^{-}$-model $N$ which believes $(\ast)$, the latter being coded into $r$. But by upwards $\Sigma_{1}$-absoluteness, and the fact that $N$ can compute $\vec{S}^{i}$ correctly, if $N$ thinks that some tree in $\vec{S^{1}}$ has a branch, then $L[r]$ must think so as well.