# The Arithmetic of Linear Orders

Garrett Irvin

## Institution:

California Institute of Technology

## Time:

Monday, April 22, 2024 - 4:00pm to 5:20pm

## Location:

RH 440R

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.

# Extensions of the Axiom of Determinacy and the ABCD Conjecture

Nam Trang

## Institution:

University of North Texas

## Time:

Monday, April 15, 2024 - 4:00pm to 5:20pm

## Location:

RH 440 R

The axiom AD^+, a structural strengthening of the Axiom of Determinacy (AD), was introduced by Hugh Woodin in the 1980's. AD^+ resolves many basic structural questions unsettled by AD. However, there are still many basic questions not answered by AD^+. One such class of questions concerns comparing cardinalities of sets under AD^+: given any two sets X and Y, how can we compare |X| and |Y|? One concrete instance of this is the following conjecture.

Conjecture (the ABCD conjecture): suppose \alpha,\beta,\gamma,\delta are infinite cardinals such that \beta \leq \alpha and \delta\leq \gamma. Then |\alpha^\beta| \leq |\gamma^\delta| if and only if \alpha\leq \gamma and \beta \leq \delta.

The ABCD Conjecture is false under ZFC. It is open whether AD^+ implies the conjecture holds, but many instances of the conjecture have been established (by work of Woodin, Chan-Jackson-Trang etc). We introduce a structural strengthening of the axiom AD^+, called AD^{++}. AD^{++} implies the ABCD Conjecture and appears to have other interesting consequences not known to follow from AD^+. We do not know if AD^+ implies AD^{++} but some special cases have been proved. We will define these notions and discuss some of the partial results mentioned above. This is ongoing joint work with W. Chan and S. Jackson.

# Lossless expansion and measure hyperfiniteness

Jan Grebik

UCLA

## Time:

Monday, April 1, 2024 - 4:00pm to 5:30pm

## Location:

RH 440 R

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.

# Combinatorics of Very Large Cardinals

Julian Eshkol

UC Irvine

## Time:

Monday, March 11, 2024 - 4:00pm to 5:20pm

## Location:

RH 340 R

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.

# The combinatorics of Large Cardinals

Julian Eshkol

UC Irvine

## Time:

Monday, March 4, 2024 - 4:00pm to 5:50pm

## Location:

RH 340N

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.

# Ineffability

Julian Eshkol

UC Irvine

## Time:

Monday, February 26, 2024 - 4:00pm to 5:20pm

## Location:

RH 340

Ineffability

This is the first of series of seminars that surveys the results of ineffability and its use in forcing extensions.

The first talk will be about the results in Magidor's Thesis where the fundamental notions were introduced.

# Algorithmic Randomness

Michael Hehman

UC Irvine

## Time:

Tuesday, February 20, 2024 - 11:00am to 12:30pm

## Location:

440R

NOTE: Tuesday meeting

This is the last lecture in an introductory survey of the theory of algorithmic randomness. The primary question we wish to answer is: what does it mean for a set of natural numbers, or equivalently an infinite binary sequence, to be random? We will focus on three intuitive paradigms of randomness: (i) a random sequence should be hard to describe, (ii) a random sequence should have no rare properties, and (iii) a random sequence should be unpredictable, in the sense that we should not be able to make large amounts of money by betting on the next bit of the sequence. Using ideas from computability theory, we will make each of these three intuitive notions of randomness precise and show that the three define the same class of sets.

# Algorithmic Randomness part III

Michael Hehman

UC Irvine

## Time:

Monday, February 12, 2024 - 4:00pm to 5:30pm

## Location:

RH 340 N

This is the third lecture in an introductory survey of the theory of algorithmic randomness. The primary question we wish to answer is: what does it mean for a set of natural numbers, or equivalently an infinite binary sequence, to be random? We will focus on three intuitive paradigms of randomness: (i) a random sequence should be hard to describe, (ii) a random sequence should have no rare properties, and (iii) a random sequence should be unpredictable, in the sense that we should not be able to make large amounts of money by betting on the next bit of the sequence. Using ideas from computability theory, we will make each of these three intuitive notions of randomness precise and show that the three define the same class of sets.

# Algorithmic Randomness Part 2

Michael Hehmann

UC Irvine

## Time:

Monday, January 29, 2024 - 4:00pm to 5:30pm

## Location:

RH 340N

We give an introductory survey of the theory of algorithmic randomness. The primary question we wish to answer is: what does it mean for a set of natural numbers, or equivalently an infinite binary sequence, to be random? We will focus on three intuitive paradigms of randomness: (i) a random sequence should be hard to describe, (ii) a random sequence should have no rare properties, and (iii) a random sequence should be unpredictable, in the sense that we should not be able to make large amounts of money by betting on the next bit of the sequence. Using ideas from computability theory, we will make each of these three intuitive notions of randomness precise and show that the three define the same class of sets.

# Algorithmic Randomness

Michael Hehmann

UC Irvine

## Time:

Monday, January 22, 2024 - 4:00pm to 5:30pm

## Location:

RH 340 N

We give an introductory survey of the theory of algorithmic randomness. The primary question we wish to answer is: what does it mean for a set of natural numbers, or equivalently an infinite binary sequence, to be random? We will focus on three intuitive paradigms of randomness: (i) a random sequence should be hard to describe, (ii) a random sequence should have no rare properties, and (iii) a random sequence should be unpredictable, in the sense that we should not be able to make large amounts of money by betting on the next bit of the sequence. Using ideas from computability theory, we will make each of these three intuitive notions of randomness precise and show that the three define the same class of sets.