last times: **Theorem.** Suppose $\delta$ is measurable Woodin, $\mathbb{P}$ a forcing notion, $|\mathbb{P}|<\delta$, $\psi(x)$ is $\Sigma^{2}_{1}$ formula, $a\in V$ a real, then $V\vDash \psi(a)\iff V^{\mathbb{P}}\vDash \psi(a)$ **provided** $V,V^{\mathbb{P}}\vDash \mathrm{CH}$. >[!note] Remark >Every $\Sigma_{1}$ statement about $H_{\omega_{2}}$ can be expressed as a $\sigma^{2}_{1}$ statement about reals. >So equivalently the conclusion is that if $\psi$ is $\Sigma_{1}$ then $H_{\omega_{2}}^{V}\vDash \psi(a)\iff H_{\omega_{2}}^{V^{\mathbb{P}}}\vDash \psi(a)$ ($a$ a real) >[!note] Notation >$\Sigma^{2}_{2}$ formulas are of the form $\psi(z)=\exists X \forall Y \Phi(X,Y,z)$ Want: a large cardinal assumption that would give $V\vDash \psi(a)\iff V^{\mathbb{P}}\vDash \psi(a)$ for small forcings. >[!note] Remark > we can add the predicate $\mathrm{NS}(A)$ saying that $A$ is a non-stationary subset of $\omega_{1}$. >[!note] Remark >$\diamondsuit_{\omega_{1}}$ is a $\Sigma^{2}_{2}$ statement about reals. >So to get absoluteness we need models satisfying $\diamondsuit_{\omega_{1}}$ >[!question] Conjecture. > Under an appropriate large cardinal $\delta$, we get: $\mathbb{P}$ a forcing notion, $|\mathbb{P}|<\delta$, $\psi(x)$ is $\Sigma^{2}_{2}$ formula, $a\in V$ a real, then $V\vDash \psi(a)\iff V^{\mathbb{P}}\vDash \psi(a)$ **provided** $V,V^{\mathbb{P}}\vDash \diamondsuit_{\omega_{1}}$. ## Problems/why is this interesting Inner models for large cardinals so far all have $\Sigma^{2}_{2}$ well-order of the reals. Suppose $V$ is such an inner model, $V\vDash \diamondsuit_{\omega_{1}}$. Force a single Cohen real. So in the extension there is no definable well-order of the reals and satisfies $\diamondsuit_{\omega_{1}}$. So the large cardinal in question must be more than what inner model theory can deal with currently. Abraham-Shelah: Over every model of theory there is an extension satisfying $\mathrm{CH}$ and that there is $\Sigma_{2}^{2}$ well-order of the reals. So we can't start with failure of such well-ordering. Woodin's $\Omega$-conjecture implies that $\Sigma_{2}^{2}(\mathrm{NS}_{\omega_{1}})$ generic absoluteness **fails**. # How would we try to prove the conjecture? We found $\bar{\delta}<\bar{\eta}<\delta$ and force with a Woodin tower to get a generic $j:V\to M$ , $j(\omega_{1})=\delta$, all the reals of $V[G]$ are in $M$. We find in $V[G]$ a generic $H\subseteq \mathbb{P}$ and a generic embedding $j^{*}:V[H]\to N$ such that $N$ has the same reals as $V[G]$ and $M$. In $V[H]$ there is a witness for $\psi$, so there is one in $N$, which is also a witness in $M$, so there is by elementarity one in $V$. What about $\Sigma_{2}^{2}$? We would need that $N,M$ agree on *sets* of reals, which is harder. Consider a $\Pi_{1}^{2}$ statement $\forall Y(\Phi(X,Y,z))$. Assume $\mathrm{CH}$ We can define a tree of size $\omega_{1}$ which has no cofinal branch iff the $\forall Y(\Phi(X,Y,z))$. How to construct: Fix an enumeration of the reals. elements are $(Z,Y)$ such that $Y\subseteq Z$, $Z$ an initial segment of the enumeration such that $Z\vDash \neg \Phi(X\cap Z,Y,a)$ . $(Z,Y)<(Z',Y')$ iff $(Z,Y,X\cap Z)\prec (Z',Y',X\cap Z')$ So we want to arrange that in addition to agreeing on reals, every $\omega_{1}$ tree $T\in N$, $N,M,V[G]$ agree on whether it has a branch. # Magidor-Malitz quantifier [[Magidor-Malitz quantifiers]] Suppose $\phi(x,y)$ defines a tree order. then $Q^{2}xy\phi(x,y) \iff$ there is an uncountable branch in the tree. $\exists X\forall Y(\Phi(X,Y,z))$ is equivalent to: $\exists \mathcal{A}$ satisfying some fragment of ZF s.t. 1. $\mathbb{R}\subseteq A$ 2. $\mathcal{A}$ is wellfounded, $\omega_{1}^{\mathcal{A}}=\omega_{1}$ 3. $\mathcal{A}\vDash \exists X \forall Y(\Phi(X,Y,a))$ 4. $\mathcal{A}\vDash T^{a}_{\Phi}$ has a branch >[!info] Definition >Say $\mathcal{A}$ is a *very nice model* if >1. $\mathbb{R}\subseteq A$ >2. $\mathcal{A}$ is wellfounded, $\omega_{1}^{\mathcal{A}}=\omega_{1}$ >3. $|A|=\omega_{1}$ >4. $\mathcal{A}$ agrees with $V$ about sentences of $\mathcal{L}(Q^{2,\mathrm{MM}})$. So a $\Sigma^{2}_{2}$ statement is equivalent to having a very nice model satisfying some theory countable Magidor-Malitz theory $T$. But this is also in itself a $\Sigma_{2}^{2}$ statement. **Conjecture $(*)$.** ($\diamondsuit$) Under an appropriate large cardinal $\delta$, we get: $\mathbb{P}$ a forcing notion, $|\mathbb{P}|<\delta$, $\psi(x)$ is $\Sigma^{2}_{2}$ formula, $a\in V$ a real, then $V,V^{\mathbb{P}}$ agree on existence of very nice models for fixed theories. **Theorem.** Under $\diamondsuit$ if $T$ is a countable theory in MM and there is a forcing adding a model for $T$ then there is in $V$ a model of $T$. *Remark:* these won't necessarily be very nice models. That would require sharps. >[!info] Definition >Say $\mathcal{A}$ is an *almost nice model* if >1. $\mathbb{R}\subseteq A$ >2. $\mathcal{A}$ is wellfounded, $\omega_{1}^{\mathcal{A}}=\omega_{1}$ **Theorem.** Suppose $\diamondsuit$ holds, $\delta$ Woodin and there is measurable $\mu>\delta$. Suppose $T$ is a theory in MM-logic (+ fragment of ZFC), $\mathbb{P}$ with $|\mathbb{P}|<\delta$. Then if $V^{\mathbb{P}}\vDash$ there is an almost nice model of $T$ , then $V\vDash$ an almost nice model of $T$. >[!note] Remark >Can assume the model is correct about stationary sets, and about the $\mathrm{MM}^{*}$ logic where you require a *stationary* homogeneous set. *Proof sketch.* Take $M\prec H_{\theta}$ in $V$, $M$ countable containing $\mathbb{P},T,\delta,\mu, \dots$. $\bar{M}$ the transitive collapse $\pi:M\to \bar{M}$ the collapse. $\bar{M}$ knows that if we force with $\mathbb{\bar{P}}$ we get an almost nice model. Let $G$ be generic over $\bar{M}$ and $\bar{A}$ the model. The idea is to push $\omega_{1}^{M}$ to the real $\omega_{1}$ and $\bar{A}$ to a real example in $V$. Define a sequence of countable models $\left< M_{\beta}\mid \beta<\omega_{1} \right>$ and embeddings starting at $\bar{M}[G]$. At limit stages take direct limits. At successor stages $M_{\alpha}$ where we assume $\delta_{\alpha}$ is the image of $\bar{\delta}$ take $H_{\alpha}\subseteq \mathbb{Q}_{<\delta_{\alpha}^{M_{\alpha}}}$ generic over $M_{\alpha}$ such that (!) and $M_{\alpha+1}$ is the generic ultrapower. We use the measurable to show that this iteration can be embedded into a well-founded iteration, so it is well founded at the limit. Claim that $A^{*}$, the image of $\bar{A}$, is really a model of $T$ in $V$. Need to use $\diamondsuit$ to form a condition (!) on the generic that we choose at successor steps in order to block the formation of problematic sets.