Oct

16

Wed

**Assessing the Reverse Mathematical Strength of Gratzer-Schmidt Theorem**

Gratzer-Schmidt theorem in lattice theory states that all complete and compactly generated lattices are isomorphic to the congruence lattice of an algebra. There has been an effort to assess the strength of this theorem in the reverse mathematical setting. I will discuss my recent progress on this topic and its potential implications.

Oct

30

Wed

I will speak about the recent paper “Condensable models of set theory” by Ali Enayat. The abstract can be found here: https://arxiv.org/abs/1910.04029

Jan

31

Fri

The first meeting of the logic seminar will be **today** at 2:30–3:20 in Keller 314. Our speaker will be Jack Yoon, who will give an introductory lecture on reverse mathematics. An abstract for his talk is below.

I will introduce the basics of reverse mathematics and begin Hunter’s paper on higher order reverse topology, which can be found here: https://www.math.wisc.edu/logic/theses/hunter.pdf

Reverse mathematics is a study of foundations of mathematics by assessing the “strength” of the theorems from ordinary mathematics. Rather than starting from given axioms to prove a theorem, it asks a reverse question “which axioms are necessary to prove the theorem?”. Traditionally, reverse mathematics has played out within the second order arithmetic, but further progress has been made on higher order systems as well. For example, Hunter’s paper above branches out to higher order systems to study the theorems of topology.

Feb

14

Fri

The logic seminar today will be given by David Webb. A title and abstract are below.

Title: On The Levin-V’yugin Degrees

Abstract: I will define and discuss the Levin-V’yugin degrees, a measure algebra defined on collections of reals closed under Turing equivalence. Roughly speaking, in this ordering collections A and B have that A<B if for any probabilistic algorithm, the probability that it produces an element of A that is not in B is 0. Time permitting, I will prove that the computable reals and the random reals each form an atom in this Boolean algebra, and discuss other degrees and their positions in the lattice.

The paper this talk is based on is here: https://arxiv.org/pdf/1907.07815.pdf

## University of Hawaiʻi at Mānoa