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.

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.

This is a very interesting question (and I really want to see what other answers you receive). I do not know of any general metatheorems ensuring that what you ask (in particular, about consistency strength) is the case, at least under reasonable conditions. However, arguments establishing the proof theoretic ordinal of a theory $T$ usually entail this. You […]

This is false; take a look at https://en.wikipedia.org/wiki/Analytic_set for a quick introduction. For details, look at Kechris's book on Classical Descriptive Set Theory. There you will find also some information on the history of this result, how it was originally thought to be true, and how the discovery of counterexamples led to the creation of desc […]

This is open. In $L(\mathbb R)$ the answer is yes. Hugh has several proofs of this, and it remains one of the few unpublished results in the area. The latest version of the statement (that I know of) is the claim in your parenthetical remark at the end. This gives determinacy in $L(\mathbb R)$ using, for example, a reflection argument. (I mentioned this a wh […]

A classical reference is Hypothèse du Continu by Waclaw Sierpiński (1934), available through the Virtual Library of Science as part of the series Mathematical Monographs of the Institute of Mathematics of the Polish Academy of Sciences. Sierpiński discusses equivalences and consequences. The statements covered include examples from set theory, combinatorics, […]

There is a new journal of the European Mathematical Society that seems perfect for these articles: EMS Surveys in Mathematical Sciences. The description at the link reads: The EMS Surveys in Mathematical Sciences is dedicated to publishing authoritative surveys and high-level expositions in all areas of mathematical sciences. It is a peer-reviewed periodical […]

You may be interested in the following paper: Lorenz Halbeisen, and Norbert Hungerbühler. The cardinality of Hamel bases of Banach spaces, East-West Journal of Mathematics, 2, (2000) 153-159. There, Lorenz and Norbert prove a few results about the size of Hamel bases of arbitrary infinite dimensional Banach spaces. In particular, they show: Lemma 3.4. If $K\ […]

You just need to show that $\sum_{\alpha\in F}\alpha^k=0$ for $k=0,1,\dots,q-2$. This is clear for $k=0$ (understanding $0^0$ as $1$). But $\alpha^q-\alpha=0$ for all $\alpha$ so $\alpha^{q-1}-1=0$ for all $\alpha\ne0$, and the result follows from the Newton identities.

Nice question. Let me first point out that the Riemann Hypothesis and $\mathsf{P}$-vs-$\mathsf{NP}$ are much simpler than $\Pi^1_2$: The former is $\Pi^0_1$, see this MO question, and the assertion that $\mathsf{P}=\mathsf{NP}$ is a $\Pi^0_2$ statement ("for every code for a machine of such and such kind there is a code for a machine of such other kind […]

For brevity's sake, say that a theory $T$ is nice if $T$ is a consistent theory that can interpret Peano Arithmetic and admits a recursively enumerable set of axioms. For any such $T$, the statement "$T$ is consistent" can be coded as an arithmetic statement (saying that no number codes a proof of a contradiction from the axioms of $T$). What […]