## Set theory seminar -Forcing axioms and inner models -Intermezzo

This posting complements a series of talks given at the Set Theory Seminar at BSU from September 12 to October 24, 2008. Here is a list of links to the talks in this series:

• First talk, September 12, 2008.
• Second talk, September 19, 2008.
• Third talk, September 26, 2008.
• Fourth talk, October 3, 2008.
• Fifth talk, October 10, 2008.
• Sixth talk, October 17, 2008.
• Seventh talk, October 24, 2008.

[Version of October 31.]

I’ll use this post to provide some notes about consistency strength of the different natural hierarchies that forcing axioms and their bounded versions suggest. This entry will be updated with some frequency until I more or less feel I don’t have more to add. Feel free to email me additions, suggestions and corrections, or to post them in the comments. In fact, please do.

• The consistency strength of ${\sf MM}$ is bounded above by a supercompact cardinal. See Foreman, Magidor, Shelah, Martin’s maximum, saturated ideals, and nonregular ultrafilters. I. Ann. of Math. (2) 127 (1) (1988), 1-47.
• ${\sf PFA}$ implies that there is a non-domestic premouse. This in turn implies that there is a sharp for an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. See Jensen, Schimmerling, Schindler, Steel, Stacking mice. The Journal of Symbolic Logic, to appear.

This is significantly stronger than ${\sf AD}^{L({\mathbb R})}$ in any forcing extension and that the existence of inner models of ${\sf AD}_{\mathbb R}$ containing all the reals.

• ${\sf BPFA}$ is equiconsistent with a reflecting cardinal.

Definition: Let $\omega_1\le\kappa\le\lambda$ be cardinals, with $\kappa$ regular. Then $\kappa$ is $H_\lambda$-reflecting iff for any $a\in H_\lambda$ and any $\varphi$, if there is some regular $\theta$ such that $H_\theta\models\varphi(a)$ then there are stationarily many $N\prec H_\lambda$ with $|N|<\kappa$ and $a\in N$ such that there is some $\theta'<\kappa$ such that $H_{\theta'}\models\varphi(\pi_N(a))$, where $\pi_N:N\to\tilde N$ is the transitive collapse of $N$.

This notion is due to Miyamoto. We say that $\kappa$ is reflecting iff it is $H_\kappa$-reflecting; this relativizes down to $L$; in fact, $\kappa$ being $H_{\kappa^+}$-reflecting relativizes to $L$. A cardinal $\kappa$ is supercompact iff it is $H_\lambda$-reflecting for all $\lambda\ge\kappa$.

The proof of the equiconsistency of ${\sf BPFA}$ and reflecting cardinals goes back to Goldstern, Shelah, The bounded proper forcing axiom. The Journal of Symbolic Logic, 60 (1) March 1995, 58-73.

Definition: For any cardinal $\kappa\ge\omega_1$, let ${\sf PFA}_T(\kappa)$ denote the following fragment of ${\sf PFA}$: Given any proper ${\mathbb P}$ and any sequence $(D_\alpha:\alpha<\omega_1)$ of predense subsets of ${\mathbb P}$ with $|D_\alpha|\le\kappa$ for each $\alpha$, there is a filter $G\subseteq{\mathbb P}$ meeting all the $D_\alpha$.

The subscript $T$ is for Todorcevic, who seems to have been first to study these fragments systematically. See Todorcevic, Localized reflection and fragments of ${\sf PFA}$. In DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol 58, 2002, 135-148. Notice ${\sf PFA}_T(\omega_1)$ is ${\sf BPFA}$.

• ${\sf PFA}_T(\omega_2)$ is equiconsistent with an $H_{\kappa^+}$-reflecting cardinal $\kappa$.
• ${\sf PFA}_T(\lambda)$ (for $\lambda\ge\omega_2$) implies $\lnot\square(\kappa)$ for all regular cardinals $\kappa\in[\omega_2,\lambda]$. For $\lambda>\omega_3$, this already implies the lower bound mentioned above for ${\sf PFA}$. For $\lambda=\omega_3$, this implies ${\sf AD}^{L({\mathbb R})}$. See for example Schimmerling, Coherent sequences and threads. Adv. Math. 216 (1) (2007), 89-117.
• ${\sf PFA}_T(\omega_{1+\alpha})$ is consistent relative to a $H_{\kappa^{+\alpha}}$-reflecting cardinal $\alpha$. This follows from the results in Miyamoto, Localized reflecting cardinals and weak segments of ${\sf PFA}$. Preprint, 1996. The same holds with ${\sf SPFA}_T$ in place of ${\sf PFA}_T.$
• On the other hand, ${\sf MM}_T(\lambda)$ seems significantly stronger than ${\sf SPFA}_T(\lambda)$. For example, ${\sf BMM}$ implies that for every set $X$ there is an inner model that contains $X$ and has a strong cardinal, see Schindler, Bounded Martin’s Maximum and strong cardinals, in: Set theory. Centre de recerca Matematica, Barcelona 2003-4 (Bagaria, Todorcevic, eds.), Basel 2006, 401-406.
• An upper bound for the consistency strength of ${\sf BMM}$ is $\omega+1$ Woodin cardinals. This is a result of Woodin, and it is buried somewhere in The Axiom of Determinacy, Forcing Axioms, and the Nonstationary ideal, De Gruyter, 1999; see for instance the arguments on ${\sf MM}({\mathfrak c})$ on section 9.2. You can see a proof in Schindler’s talk The role of absoluteness and correctness, Barcelona, 2003 (pages 76-80).

Definition: For any cardinal $\kappa\ge\omega_1$, let ${\sf PFA}(\kappa)$ denote ${\sf PFA}$ restricted to forcings of size at most $\kappa$.

This notation is due to Woodin. I believe ${\sf PFA}_T(\kappa)$ is a more useful notion, so one should probably call that one ${\sf PFA}(\kappa)$ and this one ${\sf PFA}_W(\kappa)$, or something like that.

• ${\sf PFA}(\omega)$ is just Martin’s axiom for Cohen forcing. It is equiconsistent with ${\sf ZFC}$.
• A bit more interestingly, ${\sf PFA}(\omega_1)$ is also equiconsistent with ${\sf ZFC}$. I heard the rumor that ${\sf PFA}(\omega_1)$ implies ${\mathfrak c}=\omega_2$, but I remain skeptical. [Update: the claim was withdrawn.]
• Clearly, ${\sf PFA}(\omega_2)+{\mathfrak c}=\omega_2$ follows from ${\sf PFA}_T(\omega_2)$, so it can be forced over $L$ and has consistency strength at most an $H_{\kappa^+}$-reflecting cardinal $\kappa$ (also called a $\Sigma^1_2$-indescribable cardinal). ${\sf PFA}({\mathfrak c})$ settles the size of the continuum, see Velickovic, Forcing axioms and stationary sets. Adv. Math. 94 (2) (1992), 256-284. This fragment has consistency strength at least a weakly compact (or $\Pi^1_1$-indescribable) cardinal.
• Foreman and Larson showed that ${\sf MM}(\omega_2)$ implies ${\mathfrak c}=\omega_2$, but their arguments do not even seem to adapt to ${\sf SPFA}(\omega_2).$ See Foreman, Larson, Small posets and the continuum. Unpublished note, 2004.
• Woodin has investigated ${\sf MM}({\mathfrak c})$ in his book mentioned above, where he shows that it implies the existence of sharps for all subsets of $\omega_2$ (see section 9.5); this is the first step of an induction showing that ${\sf MM}({\mathfrak c})$ implies ${\sf PD}$. Later work of Steel and Zoble has extended this to show that it in fact implies ${\sf AD}^{L({\mathbb R})}$, see Determinacy from strong reflection, preprint, September 16, 2008. Steel and Zoble conjecture that ${\sf MM}({\mathfrak c})$ has consistency strength at least $\omega^2$ Woodin cardinals. Woodin also establishes (in section 9.2 of the book mentioned above) that ${\sf ZF}+{\sf AD}_{\mathbb R}+\Theta\mbox{ is regular}$ is an upper bound for the consistency strength of ${\sf MM}({\mathfrak c})$. Recent work of  Sargsyan shows that this determinacy hypothesis has strength below a Woodin limit of Woodin cardinals.
• ${\sf PFA}({\mathfrak c}^+)$ implies ${\sf AD}^{L({\mathbb R})}$.
• ${\sf PFA}({\mathfrak c}^{++})$ implies the lower bound mentioned above of an inner model with a proper class of Woodin cardinals and a proper class of strong cardinals.

Compare the following with the notion of indescribability defined above.

Definition: Let $\lambda\le\theta$ be cardinals. We say that $\lambda$ is $(\theta,\Sigma^2_1)$subcompact, or that $\null[\lambda,\theta]$ is a $\Sigma^2_1$indescribable interval of cardinals, iff for all $Q\subseteq H_\theta$ and formulas $\varphi(x)$, if there is some $B\subseteq H_{\theta^+}$ such that $(H_{\theta^+},\in,B)\models\varphi(Q),$ then there is some $\eta$ with $\kappa\le\eta<\lambda$, and sets $P\subseteq H_\eta$ and $A\subseteq H_{\eta^+}$ such that $(H_{\eta^+},\in,A)\models\varphi(P)$, and there is an elementary embedding $\pi : (H_\eta,P) \to(H_\theta,Q)$ with critical point $\kappa$ such that $\pi(\kappa)=\lambda$.

This notion is due to Neeman and Schimmerling, see their joint paper Hierarchies of forcing axioms I, The Journal of Symbolic Logic, 73 (2008), 343-362, and Neeman, Hierarchies of forcing axioms II, The Journal of Symbolic Logic, 73 (2008), 522-542.

Supercompactness of $\lambda$ is equivalent to $\null[\lambda,\theta]$ being $\Sigma^2_1$-indescribable for all $\theta\ge\lambda$.

Recall that a poset ${\mathbb P}$ is $\theta$linked iff is the union of $\theta$ compatible sets. This of course holds if $|{\mathbb P}|=\theta$, and it of course implies that ${\mathbb P}$ is $\theta^+$-cc.

Definition: ${\sf PFA}(\lambda\mbox{-linked})$ is the assertion that ${\sf PFA}$ holds restricted to $\lambda$-linked posets.

• ${\sf PFA}({\mathfrak c}\mbox{-linked})$ and ${\sf SPFA}({\mathfrak c}\mbox{-linked})$ are equiconsistent with the existence of a $\lambda$ such that $\null[\lambda,\lambda]$ is a $\Sigma^2_1$-indescribable interval of cardinals. We also say that $\lambda$ is $\Sigma^2_1$-indescribable, and this notion relativizes to $L$.
• More generally, the existence of a $\kappa$ such that $\null[\kappa,\kappa^{+\tau}]$ is $\Sigma^2_1$-indescribable is an upper bound for the consistency strength of ${\mathfrak c}=\omega_2+{\sf PFA}(\omega_{2+\tau}\mbox{-linked})$. For $\tau=1$ Neeman has shown that this is quite close to an equiconsistency. It is possible that the equiconsistency holds for all $\tau$, and once inner model theory is developed at the appropriate level, we will be able to prove this. For $\tau=1$, this upper bound falls beyond the upper bound for ${\sf MM}({\mathfrak c})$ mentioned above.
• One can generalize the notions above to include $\Sigma^m_n$-indescribability  for all $m,n$ (see the Neeman, Schimmerling paper for details). A cardinal $\lambda$ with $\null [\lambda,\lambda^+]$ a $\Sigma^2_2$-indescribable interval is an upper bound for the consistency strength of ${\sf SPFA}({\mathfrak c}^++\mbox{-cc})$.

Index of talks:

• First talk, September 12, 2008.
• Second talk, September 19, 2008.
• Third talk, September 26, 2008.
• Fourth talk, October 3, 2008.
• Fifth talk, October 10, 2008.
• Sixth talk, October 17, 2008.
• Seventh talk, October 24, 2008.

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