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

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, […]

Yes. This is obvious if there are no such cardinals. (I assume that the natural numbers of the universe of sets are the true natural numbers. Otherwise, the answer is no, and there is not much else to do.) Assume now that there are such cardinals, and that "large cardinal axiom" is something reasonable (so, provably in $\mathsf{ZFC}$, the relevant […]

Please send an email to mathrev@ams.org, explaining the issue. (This is our all-purpose email address; any mistakes you discover, not just regarding references, you can let us know there.) Give us some time, I promise we'll get to it. However, if it seems as if the request somehow fell through the cracks, you can always contact one of your friendly edit […]

The characterization mentioned by Mohammad in his answer really dates back to Lev Bukovský in the early 70s, and, as Ralf and Fabiana recognize in their note, has nothing to do with $L$ or with reals (in their note, they indicate that after proving their result, they realized they had essentially rediscovered Bukovský's theorem). See MR0332477 (48 #1080 […]

This is a nice problem, it turned out more interesting than I first thought. Suppose first that $E$ is a measure zero set, and let's show that there is such a sequence of intervals. Since $E$ is measure zero, for each $\epsilon>0$ there is a sequence $\mathcal I_\epsilon$ of open intervals, the sum of whose lengths adds up to less than $\epsilon$. Co […]

The paper MR1029909 (91b:03090). Mekler, Alan H.; Shelah, Saharon. The consistency strength of "every stationary set reflects". Israel J. Math. 67 (1989), no. 3, 353–366, that you mention in the question actually provides the relevant references and explains the key idea of the argument. Note first that $\kappa$ is assumed regular. They refer to MR […]

Start with Conway's base 13 function $c $ (whose range on any interval is all of $\mathbb R $), which is everywhere discontinuous, and note that if $f $ only takes values $0$ and $1$, then $c+f $ is again everywhere discontinuous (since its range on any interval is unbounded). Now note that there are $2^\mathfrak c $ such functions $f $: the characteris […]

Yes, there are such sets. To describe an example, let's start with simpler tasks. If we just want $P\ne\emptyset$ with $P^1=\emptyset$, take $P$ to be a singleton. If we want $P^1\ne\emptyset$ and $P^2=\emptyset$, take $P$ to be a strictly increasing sequence together with its limit $a$. Then $P^1=\{a\}$. If we want $P^2\ne\emptyset$ and $P^3=\emptyset$ […]

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 […]

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!