## 580 -Cardinal arithmetic (11)

March 12, 2009

4. Strongly compact cardinals and ${{\sf SCH}}$

Definition 1 A cardinal ${\kappa}$ is strongly compact iff it is uncountable, and any ${\kappa}$-complete filter (over any set ${I}$) can be extended to a ${\kappa}$-complete ultrafilter over ${I.}$

The notion of strong compactness has its origin in infinitary logic, and was formulated by Tarski as a natural generalization of the compactness of first order logic. Many distinct characterizations have been found.

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

October 24, 2008

I concluded my series of talks by showing the following theorem of Viale:

Theorem (Viale). Assume ${\sf CP}(\kappa^+)$ and let $M\subseteq V$ be an inner model where $\kappa$ is regular and such that $(\kappa^+)^M=\kappa^+$. Then ${\rm cf}(\kappa)\ne\omega$.

This allows us to conclude, via the results shown last time, that if ${\sf PFA}$ holds in $V$ and $M$ computes cardinals correctly, then it also computes correctly ordinals of cofinality $\omega$.

An elaboration of this argument is expected to show that, at least  if we strengthen the assumption of ${\sf PFA}$ to ${\sf MM}$, then $M$ computes correctly ordinals of cofinality $\omega_1$

Under an additional assumption, Viale has shown this:  If ${\sf MM}$ holds in $V$$\kappa$ is a strong limit cardinal, $(\kappa^+)^M=\kappa^+$, and in $M$ we have that $\kappa$ is regular, then in $V$, the cofinality of $\kappa$ cannot be $\omega_1$. The new assumption on $\kappa$ allows us to use a result of Dzamonja and Shelah, On squares, outside guessing of clubs and $I_{, Fund. Math. 148 (1995), 165-198, in place of the structure imposed by ${\sf CP}(\kappa^+)$. It is still open if the corresponding covering statement ${\sf CP}(\kappa^+,\omega_1)$ follows from ${\sf MM}$, which would eliminate the need for this the strong limit requirement.

• Go to the intermezzo for a discussion of consistency strengths.

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

October 17, 2008

I presented a sketch of a nice proof due to Todorcevic that ${\sf PFA}$ implies the P-ideal dichotomy ${\sf PID}$. I then introduced Viale’s covering property ${\sf CP}$ and showed that it follows from ${\sf PID}$. Next time I will indicate how it can be used to provide a proof of part 1 of the following theorem:

Theorem (Viale). Assume $M\subseteq V$ is an inner model.

1. If ${\sf PFA}$ holds in $V$ and $M$ computes cardinals correctly, then it also computes correctly ordinals of cofinality $\omega$.
2. If ${\sf MM}$ holds in $V$, $\kappa$ is a strong limit cardinal, $(\kappa^+)^M=\kappa^+$, and in $M$ we have that $\kappa$ is regular, then in $V$, the cofinality of $\kappa$ cannot be $\omega_1$

It follows from this result and the last theorem from last time that if $V$ is a model of ${\sf MM}$ and a forcing extension of an inner model $M$ by a cardinal preserving forcing, then ${\sf ORD}^{\omega_1}\subset M$.

In fact, the argument from last time shows that we can weaken the assumption that $V$ is a forcing extension to the assumption that for all $\kappa$ there is a regular cardinal $\lambda\ge\kappa$ such that  in $M$ we have a partition $S^\lambda_\omega=\sqcup_{\alpha<\kappa}S_\alpha$ where each $S_\alpha$ is stationary in $V$

It is possible that this assumption actually follows from ${\sf MM}$ in $V$. However, something is required for it: In Gitik, Neeman, Sinapova, A cardinal preserving extension making the set of points of countable $V$ cofinality nonstationary, Archive for Mathematical Logic, vol. 46 (2007), 451-456, it is shown that (assuming large cardinals) one can find a (proper class) forcing extension of $V$ that preserves cardinals, does not add reals, and (for some cardinal $\kappa$) the set of points of countable $V$-cofinality in $\lambda$ is nonstationary for every regular $\lambda\ge\kappa^+$. Obviously, this situation is incompatible with ${\sf PFA}$ in $V$, by Viale’s result.