The 15th SLALM (Latin American Symposium in Mathematical Logic) was held in Bogotá, June 4-8, 2012. There were also three tutorials preceding the main event, on May 30-June 2. I give one of the tutorials, on Determinacy and Inner model theory, 10:40-12 each day, at the Universidad Nacional. Here is the abstract:

Since the invention of forcing, we know of many statements that are independent of the usual axioms of set theory, and even more that we know are consistent with the axioms (but we do not yet know whether they are actually provable).

These proofs of consistency typically make use of large cardinal assumptions. Inner model theory is the most powerful technique we have developed to analyze the structure of large cardinals. It also allows us to show that the use of large cardinals is in many cases indispensable. For years, the main tool in the development of this area was fine structure theory.

Determinacy (in suitable inner models) is a consequence of large cardinals, and recent work has revealed deep interconnections between determinacy assumptions and the existence of inner models with large cardinals, thus showing that descriptive set theory is also a key tool.

The development of these connections started in earnest with Woodin’s core model induction technique, and has led to what we now call Descriptive inner model theory.

The goal of the mini-course is to give a rough overview of these developments.

I have written a set of notes based on these talks, and will be making it available soon.

In addition, I gave one of the invited talks during the set theory session: Forcing with over models of strong versions of determinacy. Here is the abstract:

Hugh Woodin introduced , a definable poset, and showed that, when forcing with it over (in the presence of determinacy), one recovers choice, and obtains a model of many combinatorial assertions for which simultaneous consistency was not known by traditional forcing techniques. can be applied to larger models of determinacy. As part of joint work with Larson, Sargsyan, Schindler, Steel, and Zeman, we show how this allows us to calibrate the strength of different square principles.

This is of course related to the paper I discussed recently.

43.614000-116.202000

Advertisements

Like this:

LikeLoading...

Related

This entry was posted on Monday, May 21st, 2012 at 3:30 pm and is filed under Conferences. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

Perhaps the following may clarify the comments: for any ordinal $\delta$, there is a Boolean-valued extension of the universe of sets where $2^{\aleph_0}>\aleph_\delta$ holds. If you rather talk of models than Boolean-valued extensions, what this says is that we can force while preserving all ordinals, and in fact all initial ordinals, and make the contin […]

I do not know of any active set theorists who think large cardinals are inconsistent. At least, within the realm of cardinals we have seriously studied. [Reinhardt suggested an ultimate axiom of the form "there is a non-trivial elementary embedding $j:V\to V$". Though some serious set theorists found it of possible interest immediately following it […]

There is a fantastic (and not too well-known) result of Shelah stating that $L({\mathcal P}(\lambda))$ is a model of choice whenever $\lambda$ is a singular strong limit of uncountable cofinality. This is a consequence of a more general theorem that can be found in 4.6/6.7 of "Set Theory without choice: not everything on cofinality is possible", Ar […]

In set theory, definitely the notion of a Woodin cardinal. First, it is not an entirely straightforward notion to guess. Significant large cardinals were up to that point defined as critical points of certain elementary embeddings. This is not the case here: Woodin cardinals need not be measurable. If $\kappa$ is Woodin, then $V_\kappa$ is a model of set the […]

The first example that came to mind was MR0270881 (42 #5764) van der Waerden, B. L. How the proof of Baudet's conjecture was found. 1971 Studies in Pure Mathematics (Presented to Richard Rado) pp. 251–260 Academic Press, London. There, van der Waerden describes some of the history as well as his proof of his well-known theorem. Another example: MR224589 […]

Yes, it is consistent to have such cardinals. In fact, it is consistent relative to an inaccessible cardinal that $\omega\to(\omega)^\omega_2$. This is a famous result of Mathias, in MR0491197 (58 #10462). Mathias, A. R. D. Happy families. Ann. Math. Logic 12 (1977), no. 1, 59–111. (It is still open whether the inaccessible cardinal is required.) The result […]

The inductive definition of forcing (by complexity of formulas) gives in particular that $p$ forces $\lnot\psi$ if and only if no extension of $p$ forces $\psi$. That is exactly what is being claimed. As for why this general fact about forcing of negations holds, it is either immediate from the fact that a statement holds in a generic extension if and only i […]

The principle $\lozenge$ (diamond) is in a sense the right set-theoretic version of the continuum hypothesis, as it presents it instead as a reflection principle. Formally, it asserts that there is a diamond sequence, that is, a sequence $(A_\alpha:\alpha

Unfortunately, Maddy is being imprecise in her use of terminology and the surrounding explanation. The mention of Borel in page 496 is a good hint that the notion she is discussing is that of being strong measure zero, as suggested in the comments. A set of reals is (or has) measure zero if and only if for any $\epsilon>0$ it can be covered by countably m […]

Note that $\alpha\mapsto\|c_\alpha^\lambda\|_S$ is strictly increasing (trivially): After all, $$\{\delta\in S\mid c_\beta^\lambda(\delta)\ge c_\alpha^\lambda(\delta)\}=\{\delta\in S\mid\beta\ge \alpha\}=\emptyset$$ if $\beta