covering property | Teaching blog

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.

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1 Comment | 580: Topics in set theory | Tagged: Alfred Tarski, compactness, covering property, fine measure, Jussi Ketonen, Matteo Viale, regular ultrafilter, Robert Solovay, sch, strongly compact cardinal, ultrapower, weakly normal | Permalink
Posted by andrescaicedo

Set theory seminar -Forcing axioms and inner models VII

October 24, 2008

Go to previous talk.

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 $I_{<f} [\lambda]$ , 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.

2 Comments | Set theory seminar | Tagged: covering property, mm, pfa | Permalink
Posted by andrescaicedo

Set theory seminar -Forcing axioms and inner models VI

October 17, 2008

Go to previous talk.

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.

If ${\sf PFA}$ holds in $V$ and $M$ computes cardinals correctly, then it also computes correctly ordinals of cofinality $\omega$ .
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.

Go to next talk.
Go to the intermezzo for a discussion of consistency strengths.

2 Comments | Set theory seminar | Tagged: covering property, mm, pfa, pid | Permalink
Posted by andrescaicedo

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In a comment to François's answer, I point out that the least $\alpha$ such that $L_\alpha$ is a model of $\mathsf{ZFC}$ is countable. In what follows, "model" means "transitive model of $\mathsf{ZFC}$." If $M$ is transitive, then $L^M=L_\beta\models\mathsf{ZFC}$ for $\beta=\mathsf{ORD}\cap M$, so the least height of a model is count […]

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Ricky: I think I see how to answer the problem under a stronger assumption. Rather than $\mathsf{AD}$, work in $$ {\sf AD}^+ + V=L({\mathcal P}({\mathbb R})). $$ This is a bit unsatisfying, since it is very possible the question can be answered assuming only $\mathsf{AD}$. In any case, $\mathsf{AD}^+$ is potentially harmless, since it may be that $\mathsf{AD […]

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Andres Caicedo

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Suppose that $A$ is infinite and Dedekind finite. Then $\mathfrak m=|A\cup\mathbb N|$ satisfies that $|A|

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There is a hierarchy of fast-growing functions $f_\alpha:\mathbb N\to\mathbb N$ indexed by ordinals $\alpha

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[Edit: I address first the original version of the question, where the requirement that $\Psi$ is transitive was not stated. The last paragraph addresses this case.] As stated, this is false, and it requires too many wrinkles to fix appropriately: (Assume that) $\mathsf{ZF}$ is consistent, and that $\Psi$ is a model of $\mathsf{ZF}+\lnot\mathrm{Con}(\mathsf{ […]

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There is still much to explore in the area of partition calculus. The notation $$\alpha\to(\beta,\gamma)^2$$ means that if one colors the $2$-element subsets of $\alpha$ with two colors, red and blue, then there is an $H\subseteq\alpha$, all of whose $2$-sized subsets have the same color and, either the color is red and $H$ has order type $\beta$, or the col […]

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580 -Cardinal arithmetic (11)

Set theory seminar -Forcing axioms and inner models VII

Set theory seminar -Forcing axioms and inner models VI

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