## World Logic Day 2022 / Cross-Alps Logic Seminars special session

Menachem Magidor
Hebrew University of Jerusalem

COSA

QUANDO

DOVE

REFERENTE

### Sets of reals are not created equal: regularity properties of subsets of the reals and other Polish spaces

A “pathological set” can be a non measurable set, a set which does not have the property of Baire (namely it is not a Borel set modulo a first category set).

A subset $A \subseteq P^{\omega}(\mathbb{N})$(=the infinite subsets of natural numbers) can be considered to be ”pathological” if it is a counterexample to infinitary Ramsey theorem. Namely there does not exist an infinite set of natural numbers such that all its subsets are in our sets or all its infinite subsets are not in the set.

A subset of the Baire space $A\subseteq \mathbb{N}^{\mathbb{N}}$ can be considered to be “pathological” if the infinite game $G_A$ is not determined. The game $G_A$ is an infinite game where two players alternate picking natural numbers, forming an infinite sequence, namely a member of $\mathbb{N}^{\mathbb{N}}$. The first player wins the round if the resulting sequence is in $A$. The game is determined if one of the players has a winning strategy.

A prevailing paradigm in Descriptive Set Theory is that sets that have a “simple description” should not be pathological. Evidence for this maxim is the fact that Borel sets are not pathological in any of the senses described above.In these talks we shall present a notion of “super regularity” for subsets of a Polish space, the family of universally Baire sets. This family of sets generalizes the family of Borel sets and forms a $\sigma$-algebra. We shall survey some of the regularity properties of universally Baire sets , such as their measurability with respect to any regular Borel measure, the fact that they have an infinitary Ramsey property etc. Some of these theorems will require assuming some strong axioms of infinity. Most of the talk should be accessible to a general Mathematical audience, but in the second part we shall survey some newer results.

Here are the coordinates of the event (note that they are different from the usual CALS seminars):

https://unito.webex.com/unito/j.php?MTID=m856b66bb4c2386ef593821382ddb34cd
Numero riunione: 2733 365 4796