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

Theorem (Viale). Assume and let be an inner model where is regular and such that . Then .

This allows us to conclude, via the results shown last time, that if holds in and computes cardinals correctly, then it also computes correctly ordinals of cofinality .

An elaboration of this argument is expected to show that, at least if we strengthen the assumption of to , then computes correctly ordinals of cofinality .

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

Go to the intermezzo for a discussion of consistency strengths.

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2 Responses to Set theory seminar -Forcing axioms and inner models VII

The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]

As other answers point out, yes, one needs choice. The popular/natural examples of models of ZF+DC where all sets of reals are measurable are models of determinacy, and Solovay's model. They are related in deep ways, actually, through large cardinals. (Under enough large cardinals, $L({\mathbb R})$ of $V$ is a model of determinacy and (something stronge […]

Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

The result was proved by Kenneth J. Falconer. The reference is MR0629593 (82m:05031). Falconer, K. J. The realization of distances in measurable subsets covering $R^n$. J. Combin. Theory Ser. A 31 (1981), no. 2, 184–189. The argument is relatively simple, you need a decent understanding of the Lebesgue density theorem, and some basic properties of Lebesgue m […]

Yes, there is an $\aleph_2$ and an $\aleph_3$, and there are alephs beyond all the $\aleph_n$. A suitable version of Cantor's diagonal proof is perfectly general and shows that, for any set $X$, $|X|

Given a class $S$, to say that it can be proper means that it is consistent (with the axioms under consideration) that $S$ is a proper class, that is, there is a model $M$ of these axioms such that the interpretation $S^M$ of $S$ in $M$ is a proper class in the sense of $M$. It does not mean that $S$ is always a proper class. In fact, it could also be consis […]

As the other answers point out, the question is imprecise because of its use of the undefined notion of "the standard model" of set theory. Indeed, if I were to encounter this phrase, I would think of two possible interpretations: The author actually meant "the minimal standard model of set theory", that is, $L_\Omega$ where $\Omega$ is e […]

I assume you want the measures to be $0$ on singletons. If this is the intention, it is impossible for both measures to coexist, for the reason that you identify: $m_1$ would force $\kappa$ to be (atomlessly) real-valued measurable, which implies that $\kappa\le\mathfrak c=|\mathbb R|$, while $m_2$ would force $\kappa$ to be a measurable cardinal, and theref […]

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