This is the first in a series of lectures going through notes entitled "Naive Descriptive Set Theory" that are available on ArXiV.  

In the last 20 years the field has had many applications to areas in Analysis and Dynamical Systems The lectures are intended to be an opportunity to learn the subject matter, and will be interspersed with research lectures during the quarter.

No background beyond basic elements of the 210 sequence are required. 

The purpose of this talk is to introduce a new forcing method from a joint project with Daniel Iosub called "Shuffling", and explain how it is used to obtain new consistency results involving different square principles.  Given a poset P, the shuffling method aims to form a related poset that captures certain essential generic sets added by P while avoiding other undesirable ones. After introducing the method, I will explain its connection with square principles and how it is used to answer questions by Jensen, Cummings and Friedman about the failure of the global square principle, and questions about the points at which squares fail. 

There are two natural arithmetic operations on the class of linear orders: the sum + and lexicographic product x. These operations generalize the sum and product of ordinals. 

The arithmetic laws obeyed by the sum were uncovered in the pre-forcing days of set theory and are surprisingly nice. For example, while the left cancellation law A + X \cong B + X => A \cong B is not true in general, its failure can be completely characterized: a linear order X fails to cancel in some such isomorphism if and only if there is a non-empty order R such that R + X \cong X. Right cancellation is symmetrically characterized. 

Tarski and Aronszajn characterized the commuting pairs of linear orders, i.e. the pairs X and Y such that X + Y \cong Y + X.

Lindenbaum showed that X + X \cong Y + Y implies X \cong Y for linear orders X and Y. More generally, the finite cancellation law nX \cong nY => X \cong Y holds. Lindenbaum showed that the sum even satisfies the Euclidean algorithm! 

On the other hand, the arithmetic of the lexicographic product is much less well understood. The lone totally general classical result is due to Morel, who characterized when the right cancellation law A x X \cong B x X => A \cong B holds. Morel showed that an order X fails to cancel in some such isomorphism if and only if there is a non-singleton order R such that R x X \cong X, in analogy with the additive case. 

In this talk we focus on the question of whether Morel’s cancellation theorem is true on the left. We’ll show that, while the literal left-sided version of Morel’s theorem is false, an appropriately reformulated version is true. Our results suggest that a complete characterization of left cancellation in lexicographic products is possible. We’ll also discuss how our work might help in proving multiplicative versions of Tarski’s, Aronszajn’s, and Lindenbaum’s additive laws.

This is joint work with Eric Paul.  

Abstract: The notions of measure hyperfiniteness and measure reducibility of countable Borel equivalence relations are variants of the usual notions of hyperfiniteness and Borel reducibility. Conley and Miller proved that every basis for the countable Borel equivalence relations strictly above E_0 under measure reducibility is uncountable and asked whether there is a "measure successor of E_0"—i.e. a countable Borel equivalence relation E such that E is not measure reducible to E_0 and any F which is measure reducible to E is either equivalent to E or measure reducible to E_0. In an ongoing work with Patrick Lutz, we have isolated a combinatorial condition on a Borel group action (a strong form of expansion that we call "lossless expansion" after a similar property which is studied in computer science and finite combinatorics) which implies that the associated orbit equivalence relation is a measure successor of E_0. We have also found several examples of group actions which are plausible candidates for satisfying this condition. In this talk, I will explain the context for Conley and Miller's question, the condition that we have isolated and discuss some of the candidate examples we have identified.

All of this is joint work with Patrick Lutz.

At and above the level of measurability, large cardinal notions are typically characterized by the existence of certain elementary embeddings of the universe into an inner model. We may contrast this with smaller large cardinal notions, whose characterizations tend to be strictly combinatorial. In this series of talks, we survey results from Magidor's thesis, in which he shows that the large notion of supercompactness can also be viewed combinatorially, and in this light supercompactness is seen to be a natural strengthening of ineffability. We will also survey modern results which show how these strong combinatorial principles can be forced to hold at small successor cardinals.