1. Introduction and preliminaries

In the following, we tried very hard to make the present article as self-contained as possible. For notions and notation which remain unexplained, the reader may refer to [Reference Jech23, Reference Kanamori24], or [Reference Kunen25]. Set-theoretic forcing is treated here just as in [Reference Kunen25] with the exception that ${\mathbb {P}}$ -names for a poset ${\mathbb {P}}$ are represented with an under-tilde, e.g., as . We adopt the (fake but consistently interpretable) narration that generic filters “exist” though otherwise we remain in the ZFC narrative so that all classes mentioned here are (meta-mathematically) definable classes.

The main theorem of Viale [Reference Viale32] states the following.

Theorem 1.1 [(M. Viale, Theorem 1.4 in [Reference Viale32])].

Assume that MM $^{++}$ holds, and there are class many Woodin cardinals. Then, for any stationary preserving poset  ${\mathbb {P}}$ with $\,\|\hspace {-.35ex}\textsf {--}_{\,{\mathbb {P}}\,}{"}\,\textsf {BMM}\text{"}$ , we have

$$ \begin{align*} {\mathcal{H}}(\aleph_2)^{\mathsf{V}}\prec_{\Sigma_2}{\mathcal{H}}(\aleph_2)^{{\mathsf{V}}[{\mathbb{G}}]}\quad \textit{for} \ ({\mathsf{V}},{\mathbb{P}})\text{-}\textit{generic}\ {\mathbb{G}}. \end{align*} $$

Here MM $^{++}$ is the following strengthening of the Martin’s Axiom (MM):

1. (MM ⁺⁺): For any stationary preserving ${\mathbb {P}}$ , any family ${\mathcal {D}}$ of dense subsets of ${\mathbb {P}}$ with $\mathopen {|\,}{\mathcal {D}}\mathclose {\,|}<\aleph _2$ , and any family ${\mathcal S}$ of ${\mathbb {P}}$ -names of stationary subsets of $\omega _1$ with $\mathopen {|\,}{\mathcal S}\mathclose {\,|}<\aleph _2$ , there is a ${\mathcal {D}}$ -generic filter ${\mathbb {G}}$ on ${\mathbb {P}}$ such that is a stationary subset of $\omega _1$ for all .

BMM is the Bounded Martin’s Maximum, a weakening of MM which is an instance of Bounded Forcing Axioms: for a class ${\mathcal P}$ of posets closed under forcing equivalence and a cardinal $\kappa $ , the Bounded Forcing Axiom for ${\mathcal P}$ and ${<}\,\kappa $ is the axiom stating:

1. (BFA _{< κ} (𝒫)): For any complete BooleanFootnote ² ${\mathbb {P}}\in {\mathcal P}$ , and a family ${\mathcal {D}}$ of maximal antichains in ${\mathbb {P}}$ such that $\mathopen {|\,}{\mathcal {D}}\mathclose {\,|}<\kappa $ and $\mathopen {|\,}I\mathclose {\,|}<\kappa $ for all $I\in {\mathcal {D}}$ , there is a ${\mathcal {D}}$ -generic filter ${\mathbb {G}}$ on ${\mathbb {P}}$ .

The Bounded Martin’s Maximum (BMM) is $\textsf {BFA}_{{<}\,\aleph _2}(\text {stationary pres. posets})$ . Bounded Forcing Axioms were introduced by Goldstern and Shelah [Reference Goldstern and Shelah18] answering a problem asked by the first author of the present paper in [Reference Fuchino10].

In Theorem 1.1, the condition “ $\,\|\hspace {-.35ex}\textsf {--}_{\,{\mathbb {P}}\,}{"}\,\textsf {BMM}\,\text{"}$ ” cannot be simply dropped. For example, the formula saying that there is a set which is the power set of $\omega $ is $\Sigma _2$ . Since $\neg \textsf {CH}$ holds in $\mathsf {V}$ under MM, if ${\mathbb {P}}$ forces CH then ${\mathcal {H}}(\aleph _2)^{\mathsf {V}}\not \prec _{\Sigma _2}{\mathcal {H}}(\aleph _2)^{\mathsf {V}[{\mathbb {G}}]}$ for $(\mathsf {V},{\mathbb {P}})$ -generic ${\mathbb {G}}$ .

We show that a conclusion similar to and more general than that of Theorem 1.1 follows from each of $({\mathcal P},{\mathcal {H}}(\kappa))_{\Sigma _2}$ -RcA $^+$ which is a fragment of Recurrence Axiom (a reformulation of Maximality Principle introduced in [Reference Fuchino and Usuba15]) for ${\mathcal P}$ and ${\mathcal {H}}(\kappa)$ , see Section 2 below, and the existence of the tightly ${\mathcal P}$ -Laver-gen. large cardinal (Theorems 4.1 and 5.7).

The notion of Laver-generic large cardinal is introduced in Fuchino et al., [Reference Fuchino, Rodrigues and Sakai13]. The definition we give here is the slightly modified version in later papers such as in Fuchino [Reference Fuchino11].

For an iterable class ${\mathcal P}$ of posets (i.e., class ${\mathcal P}$ of posets satisfying (1.2) and (1.3) below) a cardinal $\kappa $ is said to be (tightly, resp.) ${\mathcal P}$ -Laver-gen. supercompact if, for any $\lambda>\kappa $ and ${\mathbb {P}}\in {\mathcal P}$ , there is a ${\mathbb {P}}$ -name with , such that for -generic $\mathbb {H}$ , there are $j, M\subseteq \mathsf {V}[\mathbb {H}]$ such that $j:\mathsf {V}\stackrel {\prec \hspace {0.8ex}}{\rightarrow }_{\kappa }M$ ,Footnote ³ $j(\kappa)>\lambda $ , ${\mathbb {P}},\mathbb {H}, j{}^{\,{\prime }{\prime }}\lambda \in M$ (and is of size $\leq j(\kappa)$ , resp.).

This definition can be adopted to many other large cardinal notions other than supercompactness. The reader may refer to [Reference Fuchino11] for definitions of other variants of Laver-generic large cardinal. Defined as above, it is not obvious at first glance that the Laver-genericity is formalizable in the language ${{\mathcal L}}_{\in }$ of ZFC. That it is actually the case, is shown in Fuchino and Sakai [Reference Fuchino and Sakai14].

A tightly ${\mathcal P}$ -Laver-generic large cardinal, if it exists, is unique and decided to be $\kappa _{\mathfrak {r}\mathfrak {e}\mathfrak {f}\mathfrak {l}\,}:=\max (\{\aleph _2,2^{\aleph _0}\})$ for all known reasonable non-trivial instances of ${\mathcal P}$ with a strong enough large cardinal notion (see [Reference Fuchino, Rodrigues and Sakai13], or [Reference Fuchino11, Reference Fuchino12]). This is the reason why we often simply talk about the tightly ${\mathcal P}$ -Laver-generic large cardinal.

While under “ ${\mathcal P}=$ all semi-proper posets”, our results are not much more than slight variants of Viale’s (but without relying on the stationary tower forcing technique), for other classes of posets, for example, “ ${\mathcal P}=$ all proper posets” or “ ${\mathcal P}=$ all ccc posets”, they are not at all covered by Viale’s result in [Reference Viale32] nor by its proof.

In the following we shall always assume that the classes ${\mathcal P}$ of posets we consider are normal, that is,

1. (1.1) ${\mathcal P}$ is closed with respect to forcing equivalence, and $\{{\mathord {\mathbb {1}}}\}\in {\mathcal P}$ .

In particular, we assume that for any ${\mathbb {P}}_0\in {\mathcal P}$ there is a complete Boolean³ ${\mathbb {P}}\in {\mathcal P}$ which is forcing equivalent to ${\mathbb {P}}_0$ . In some cases like the case “ ${\mathbb {P}}=$ all $\sigma $ -closed posets” where the original class of posets is not normal we just replace ${\mathcal P}$ with its closure with respect to forcing equivalence without mention.

In some cases (like in the definition of Laver-genericity above) it is natural to consider (normal) classes of posets which are closed with respect to two-step iteration. A class ${\mathcal P}$ of posets is called iterable if

1. (1.2) ${\mathcal P}$ is closed with respect to restriction. That is, for ${\mathbb {P}}\in {\mathcal P}$ and , we always have , and

2. (1.3) For any ${\mathbb {P}}\in {\mathcal P}$ , and any ${\mathbb {P}}$ -name of a poset with , we have .

Viale’s Absoluteness Theorem 1.1 is a result built upon the following Theorem 1.2. We shall use the following notation for the formulation of the Theorem. For an ordinal $\alpha $ , let ${\alpha ^{(+)}}:=\sup (\{\mathopen {|\,}\beta \mathclose {\,|}^+\,:\,\beta <\alpha \})$ . Note that $\alpha ^{(+)}=\alpha $ if $\alpha $ is a cardinal. Otherwise, we have $\alpha ^{(+)}=\mathopen {|\,}\alpha \mathclose {\,|}^+$ .

In Bagaria [Reference Bagaria3] the following theorem contains the extra assumption that, translated into the context of the following formulation, $\kappa $ is a successor of a cardinal of uncountable cofinality. However we can eliminate this assumption by slightly modifying the proof in [Reference Bagaria3].

The following is one of many nice applications of Theorem 1.2.

Suppose that ${\mathcal R}$ is a definable class (proper or set). We shall say that a class ${\mathcal P}$ of posets is provably correct for ${\mathcal R}$ if the following is provable in ZFC:

1. (1.5) for any ${\mathbb {P}}\in {\mathcal P}$ and $a (\in \mathsf {V})$ , $a\in {\mathcal R} \Leftrightarrow \,\|\hspace {-.35ex}\textsf {--}_{\,{\mathbb {P}}\,}{"}\,a\in {\mathcal R}\text{"}$ .

Thus if ${\mathcal P}$ is provably correct for ${\mathcal R}$ and ${\mathbb {G}}$ is a $(\mathsf {V},{\mathbb {P}})$ -generic for a ${\mathbb {P}}\in {\mathcal P}$ then $(\mathsf {V},\in ,{\mathcal R}^{\mathsf {V}})$ is a (class) substructure of $(\mathsf {V}[{\mathbb {G}}],\in ,{\mathcal R}^{\mathsf {V}[{\mathbb {G}}]})$ .

Let $I_{\mathsf {NS}}$ denote the non stationary ideal over $\omega _1$ . Thus

• $I_{\textsf {NS}}:=\{X\subseteq \omega _1\,:\,X \text { is non stationary}\}$ .

Let ${\mathcal R}$ be (the ${{\mathcal L}}_{\in }$ -definition of) a class. Let be the language which extends ${{\mathcal L}}_{\in }$ with a new unary predicate symbol where is interpreted as $a\in {\mathcal R}^M$ in an $\in $ -structure M. In the following, we shall often identify the (definition) of the class ${\mathcal R}$ with the symbol of ${\mathcal R}$ , and simply write ${\mathcal R}$ and ${\mathcal L}_{\in ,{\mathcal R}}$ instead of and . This also applies when we are talking about $I_{\textsf {NS}}$ and ${\mathcal L}_{\in ,I_{\textsf {NS}}}$ .

The following Lemma 1.5 can be proved in the same way as with the corresponding lemma for ${{\mathcal L}}_{\in }$ formulas.

The following Theorem 1.8 is an extension of Bagaria’s Absoluteness Theorem 1.2. A special case of this theorem (the case where ${\mathcal P}={}$ the stationary preserving posets) is also attributed to Bagaria in [Reference Woodin33]. Though Theorem 1.8 in its generality must have been known, we included it here since we could not find any proof in the literature.

We consider the following “plus”-version of Bounded Forcing Axioms: For a (normal) class of posets ${\mathcal P}$ ,

• ( ${\sf BFA}^{+{<}\,\kappa}_{{<}\,\kappa}({\mathcal P})$ ): For any complete Boolean ${\mathbb {P}}\in {\mathcal P}$ , a family ${\mathcal {D}}$ of maximal antichains in ${\mathbb {P}}$ such that $\mathopen {|\,}{\mathcal {D}}\mathclose {\,|}<\kappa $ and $\mathopen {|\,}I\mathclose {\,|}<\kappa $ for all $I\in {\mathcal {D}}$ , and for a set ${\mathcal S}$ of ${\mathcal P}$ -names of cardinality $<\kappa $ such that each is a ${\mathbb {P}}$ -name of a stationary subset of $\omega _1$ , there is a ${\mathcal {D}}$ -generic filter ${\mathbb {G}}$ on ${\mathbb {P}}$ such that are stationary subsets of $\omega _1$ .

(a) $\Rightarrow $ (b): Assume that $\textsf {BFA}^{+{<}\,\kappa }_{{<}\,\kappa }({\mathcal P})$ holds, and let ${\mathbb {P}}\in {\mathcal P}$ . Without loss of generality, we may assume that ${\mathbb {P}}$ is completely Boolean with ${\mathbb {P}}={\mathbb {B}}\setminus \{{\mathord {\mathbb {0}}}_{\mathbb {B}}\}$ . Suppose $a\in {\mathcal {H}}(\kappa )$ and $\varphi $ is a $\Sigma _1$ -formula in ${\mathcal L}_{\in ,I_{\textsf {NS}}}$ . If $\varphi (a)$ holds in $\mathsf {V}$ , then we also have $\,\|\hspace {-.35ex}\textsf {--}_{\,{\mathbb {P}}\,}{"}\,\varphi (a)\text{"}$ by Lemma 1.5.

Suppose now that $\varphi =\exists y\,\psi (x,y)$ for a bounded formula $\psi $ in ${\mathcal L}_{\in ,I_{\textsf {NS}}}$ , and $\,\|\hspace {-.35ex}\textsf {--}_{\,{\mathbb {P}}\,}{"}\,\varphi (a)\text{"}$ . Without loss of generality, we may assume that $a\subseteq \mu $ for some cardinal $\mu <\kappa $ (this is because a can be reconstructed from , and can be coded by a subset $a^*$ of ). The formula $\varphi (a)$ can be then replaced by another formula saying:

• $\exists x\,(\,x \text { is the set "}a \text {" reconstructed from the transitive set coded by }a^*\text {and }\varphi (x)\text { holds})$ .

Note that this formula is $\Sigma _1$ in ${\mathcal L}_{\in ,I_{\textsf {NS}}}$ with the parameter $a^*$ if $\varphi $ is $\Sigma _1$ in ${\mathcal L}_{\in ,I_{\textsf {NS}}}$ .

We may also assume that a is not an ordinal (if necessary, we can replace a with a subset of $\mu $ with some redundant complexity to make $a\not \in \text {On}$ ).

Let be a ${\mathbb {P}}$ -name such that . Let ${\mathbb {G}}$ be a $(\mathsf {V},{\mathbb {P}})$ -generic filter and we work in $\mathsf {V}[{\mathbb {G}}]$ . Letting , we have $\psi (a,b)$ .

Working further in $\mathsf {V}[{\mathbb {G}}]$ , let $\lambda $ be large enough such that $V_\lambda $ satisfies a large enough fragment of ZFC, a, $b\in V_\lambda $ , and $V_\lambda \models \psi (a,b)$ . Let $M\prec V_\lambda $ be such that $\mu \subseteq M$ , a, $b\in M$ , and $\mathopen {|\,}M\mathclose {\,|}=\mu <\kappa $ . Note that we have $\langle M,\in , I_{\textsf {NS}}\cap M\rangle \prec \langle V_\lambda ,\in ,I_{\textsf {NS}}\rangle $ since $I_{\textsf {NS}}$ is definable in $\langle V_\lambda ,\in \rangle $ . Let $m:M\stackrel {\cong \hspace {0.8ex}}{\rightarrow }M_0$ be the Mostowski collapse of M and let $\nu =\text {On}\cap M_0$ . Note that we have $m\restriction \mu \cup \{a\}\cup (I_{\textsf {NS}}\cap M)=\mathop {id\,}_{\mu \cup \{a\}\cup (I_{\textsf {NS}}\cap M)}$ and hence $I_{\textsf {NS}}\cap M=I_{\textsf {NS}}\cap M_0$ .

Let $\mathfrak {M}:=\langle \nu +\mu ,E,I,f,g\rangle $ be the structure in the language such that there is an isomorphism

1. (1.6) $i:\langle M_0, \in , I_{\textsf {NS}}\cap M_0, rank, g_0\rangle \stackrel {\cong \hspace {0.8ex}}{\rightarrow }\langle \nu +\mu , E,I,f,g\rangle $

   such that $i\restriction {\nu }=\mathop {id\,}_{\nu }$ , $i(a)=\nu $ , and $i(m(b))=\nu +1,$

where $rank$ is the rank function restricted to $M_0$ and $g_0:M_0\rightarrow M_0$ is a mapping such that $g_0\restriction \mu $ is an enumeration of ( $=( \text {the set of all stationary subsets of } \omega _1)^{M_0}$ ) and $g_0{}^{\,{\prime }{\prime }}{(M_0\setminus \mu )}=\{\emptyset \}$ .

Clearly, $\langle \nu +\mu ,E, I\rangle \models \psi ^*(\nu ,\nu +1)),$ where $\psi ^*$ is the formula obtained from $\psi $ by replacing symbols and in $\psi $ by and .

Note that we have

1. (1.7) $\langle \nu +\mu , E,I,f,g\rangle \models \forall x\subseteq \omega _1\ (I(x)\lor \exists \alpha <\kappa \,(g(\alpha )=x))$ .

Let , , , , be ${\mathbb {P}}$ -names of $\mathfrak {M}$ , E, I, f and $g,$ respectively. By replacing ${\mathbb {P}}$ with for some if necessary, we may assume that

1. (1.8) all the properties of $\langle \nu +\mu , E, I,f,g\rangle $ used below are forced (as a statement on ) by ${\mathord {\mathbb {1}}}_{\mathbb {P}}$ .

In $\mathsf {V}$ , let ${\mathcal {D}}$ be the family of dense sets in ${\mathbb {P}}$ generated by the following predense sets in ${\mathbb {P}}$ , each of size $\leq \mu <\kappa $ . Note that since ${\mathbb {P}}$ is assumed to be completely Boolean, each predense subset of ${\mathcal P}$ can be replaced by a maximal antichain of at most the same size.

1. (1.9) , and , for all $\alpha \in \nu +\mu $ .

2. (1.10) , for all $\Sigma _0$ -formulas $\theta $ in ${\mathcal L}$ and $a_0,{} [0]\hspace {0.04ex}{{.}{.}{.}\hspace {0.1ex},\,} [0] a_{k-1}\in \nu +\mu $ .

3. (1.11) , for all $\Sigma _0$ -formulas $\eta $ , $\theta $ in ${\mathcal L}$ and $a_0,{} [0]\hspace {0.04ex}{{.}{.}{.}\hspace {0.1ex},\,} [0] a_{k-1}\in \nu +\mu $ .

4. (1.12) , for all $\Sigma _1$ -formulas $\eta =\eta (x,x_0,{} [0]\hspace {0.04ex}{{.}{.}{.}\hspace {0.1ex},\,} [0] x_{k-1})$ in ${\mathcal L}$ and c, $a_0,{} [0]\hspace {0.04ex}{{.}{.}{.}\hspace {0.1ex},\,} [0] a_{k-1}\in \nu +\mu $ .

To see that each of the sets in (1.9) is a maximal antichain in ${\mathbb {P}}$ of size $\leq \mu $ , suppose that $\alpha \in \nu +\mu $ and . Then there is which decides by (1.8), if follows that for some $\beta \in \nu $ . For the sets corresponding to the argument is the same.

It is clear that elements of each of the sets in (1.9) are pairwise incompatible, and these sets are of size $\leq \mu <\kappa $ .

The sets in (1.11) and (1.12) are not necessarily maximal antichains but it can be proved similarly that each of them sums up to ${\mathord {\mathbb {1}}}_{\mathbb {B}}$ and is of size $\leq \mu <\kappa $ .

By (1.8), we have that

1. (1.13) .

Now, in $\mathsf {V}$ , let ${\mathbb {G}}$ be ${\mathcal {D}}$ -generic filter such that

1. (1.14) is a stationary subset of $\omega _1$ for all $\alpha <\mu $ .

${\mathbb {G}}$ exists by $\textsf {BFA}^{+{<}\,\kappa }_{{<}\,\kappa }({\mathcal P})$ , by (1.13), and since ${\mathcal {D}}$ is a family of dense sets generated by small sets with $\mathopen {|\,}{\mathcal {D}}\mathclose {\,|}<\kappa $ .

Let

• .

where

• ,

• ,

• , and

• .

By Claim 1.8.1, (3), we can take the Mostowski collapse of the structure . Since by (1.6), (1.8), and Claim 1.8.1, (2), we have $m^*(\nu )=a$ and hence $\langle M_2,\in , I^*\rangle \models \psi (a,m^*(\nu +1)),$ where the predicate $I_{\textsf {NS}}$ is interpreted as $I^*$ . Thus $\langle M_2,\in , I^*\rangle \models \varphi (a)$ . By (1.7), (1.8), (1.14), and Claim 1.8.1, (2), we have $I^*=I_{\textsf {NS}}\cap M_2$ .

Since $\varphi $ is $\Sigma _1$ , it follows that $V\models \varphi (a)$ by Lemma 1.5, (2).

(b) $\Rightarrow $ (a): Assume that (b) holds, and suppose that ${\mathbb {P}}\in {\mathcal P}$ is complete Boolean, ${\mathcal {D}}$ is a set of maximal antichains each of size $<\kappa $ with $\mathopen {|\,}{\mathcal {D}}\mathclose {\,|}<\kappa $ , and ${\mathcal S}$ is a set of ${\mathbb {P}}$ -names of stationary subsets of $\omega _1$ with $\mathopen {|\,}{\mathcal S}\mathclose {\,|}<\kappa $ .

By replacing elements of ${\mathcal S}$ by equivalent ${\mathbb {P}}$ -names which are sufficiently nice, we may assume that each element of ${\mathcal S}$ is nice ${\mathbb {P}}$ -name of size $\aleph _1$ (this is possible since we assumed that ${\mathbb {P}}$ is completely Boolean).

Let $X=\bigcup {\mathcal {D}}$ then $\mathopen {|\,}X\mathclose {\,|}<\kappa $ . Let $\mu :=\max \{\mathopen {|\,}X\mathclose {\,|},\mathopen {|\,}{\mathcal S}\mathclose {\,|}\}$ . Let $\lambda $ be sufficiently large with $V_\lambda \prec _{\Sigma _n}\mathsf {V}$ for sufficiently large n. Let $M\prec V_\lambda $ be such that $\mathopen {|\,}M\mathclose {\,|}=\mu $ , (1.16) ${\mathbb {P}}$ , ${\mathcal {D}}$ , X, ${\mathcal S}\in M$ , and $\mu +1\subseteq M$ . Note that (1.16) implies ${\mathcal {D}},{\mathcal S}\subseteq M$ and .

Let $m:M\stackrel {\cong \hspace {0.8ex}}{\rightarrow }M_0$ be the Mostowski collapse and $\langle {\mathbb {P}}_0,\leq _{{\mathbb {P}}_0}\rangle :=m(\langle {\mathbb {P}},\leq _{\mathbb {P}}\rangle )$ . Let .

Since $(\mathsf {V},{\mathbb {P}})$ -generic filter ${\mathbb {G}}$ generates an $(M_0,{\mathbb {P}}_0)$ -generic filter, we have

• $\,\|\hspace {-.35ex}{\mathsf {-}}_{\,{\mathbb {P}}\,}{"}\, \begin {array}[t]{@{}l} \text {there is a }(M_0,{\mathbb {P}}_0)\text {-generic filter which realizes each element of }{\mathcal S}_0\\ \text { to be a stationary subset of }\omega _1\text{"}. \end {array} $

By the assumption (b), it follows that

• $\mathsf {V}\models \!{"\,} \begin {array}[t]{@{}l} \text {there is a }(M_0,{\mathbb {P}}_0)\text {-generic filter which realizes each element of }{\mathcal S}_0\\ \text { to be a stationary subset of }\omega _1\text{"}. \end {array} $

Let ${\mathbb {G}}_0$ be such a filter. Then $m^{-1}{}^{\,{\prime }{\prime }}{{\mathbb {G}}_0}$ generates a ${\mathcal {D}}$ -generic filter ${\mathbb {G}}_1$ on ${\mathbb {P}}$ which realizes each element of ${\mathcal S}$ to be a stationary subset of $\omega _1$ .