I have been asked to be part of the Scientific committee for the sixth Young set theory workshop, to be held in Piamonte, Italy, during a week in Spring 2013. The other members of the committee are Matteo Viale (chair), Asgard Tornquist, Sean Cox, and Andrew Brooke-Taylor.

James Cummings – Large cardinals: PCF-theory and its interactions with large cardinals, forcing and -like combinatorial principles. The tutorial will focus on applications of Shelah’s PCF-theory outside cardinal arithmetic, including constructions for Jónsson algebras, strong covering lemmas, and constructions for almost-free, non-free objects.

Sy David Friedman – Forcing and combinatorial set theory: Descriptive set theory on . Assuming GCH, most of the basic notions of classical descriptive set theory generalize easily from Baire space to generalized Baire space , for an uncountable regular cardinal . But many of the theorems do not. The tutorial will discuss regularity properties and the Borel reducibility of deﬁnable equivalence relations in the generalized setting.

Su Gao – Descriptive Set Theory: Borel markers in the study of countable Borel equivalence relations. The tutorial will give an introduction to the theory of Borel markers in the study of countable Borel equivalence relations. Borel markers are important tools used to attach structures to the classes of countable Borel equivalence relations. They play a prominent role in the study of hyperﬁnite and treeable equivalence relations, and have applications in other topics such as the computation of Borel chromatic numbers.

John Steel – Inner Model Theory: Iteration Trees. The tutorial will cover the basic theory of iteration trees, and some of its applications. It starts at a basic level, deﬁning ultrapowers of models of ZFC and their properties, and tries to keep the presentation accessible to anyone who has taken a graduate-level course in set theory. (In particular, no background in inner model theory will be assumed.)

I will add more details as they are decided, we expect to invite five post doctoral researchers to give one hour talks. For now, here are the links to the previous workshops:

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3 Responses to Upcoming: 6th Young set theory workshop

(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 […]

I learned of this problem through Su Gao, who heard of it years ago while a post-doc at Caltech. David Gale introduced this game in the 70s, I believe. I am only aware of two references in print: Richard K. Guy. Unsolved problems in combinatorial games. In Games of No Chance, (R. J. Nowakowski ed.) MSRI Publications 29, Cambridge University Press, 1996, pp. […]

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Suppose $M$ is an inner model (of $\mathsf{ZF}$) with the same reals as $V$, and let $A\subseteq \mathbb R$ be a set of reals in $M$. Suppose further that $A$ is determined in $M$. Under these assumptions, $A$ is also determined in $V$. The point is that since winning strategies are coded by reals, and any possible run of the game for $A$ is coded by a real, […]

The two concepts are different. For example, $\omega$, the first infinite ordinal, is the standard example of an inductive set according to the first definition, but is not inductive in the second sense. In fact, no set can be inductive in both senses (any such putative set would contain all ordinals). In the context of set theory, the usual use of the term […]

I will show that for any positive integers $n,\ell,k$ there is an $M$ so large that for all positive integers $i$, if $i/M\le \ell$, then the difference $$ \left(\frac iM\right)^n-\left(\frac{i-1}M\right)^n $$ is less than $1/k$. Let's prove this first, and then argue that the result follows from it. Note that $$ (i+1)^n-i^n=\sum_{k=0}^{n-1}\binom nk i^ […]

I think it is cleaner to argue without induction. If $n$ is a positive integer and $n\ge 8$, then $7n$ is both less than $n^2$ and a multiple of $n$, so at most $n^2-n$ and therefore $7n+1$ is at most $n^2-n+1

Let PRA be the theory of Primitive recursive arithmetic. This is a subtheory of PA, and it suffices to prove the incompleteness theorem. It is perhaps not the easiest theory to work with, but the point is that a proof of incompleteness can be carried out in a significantly weaker system than the theories to which incompleteness actually applies. It is someti […]

Since this page is the first google hit for “Young Set Theory 2013”, I thought I’d link here to the webpage for the conference that is now up:

http://www2.dm.unito.it/paginepersonali/viale/YST2013/yst2013-home.html

Also note that registration is open until the end of February.

Thanks Andrew, that reminds me I need to post an update. Why don’t you post a link on G+ meanwhile?

Sorry – that’s a good idea but I only just saw it, and I see that you beat me to it!