Two particularly popular, completely virtual logic seminars:
The regular "virtual" meeting time is Mondays at 3:30PM (Central Time) during the spring semester 2021 (with occasional meetings Tuesdays at 4PM for joint meetings with the Midwest Model Theory Seminar).
We have a joint seminar with other Midwestern logic groups (the Midwest Computability Seminar) every other week, and local seminars on the other Mondays, with occasional joint meetings on Tuesdays with the Midwest Model Theory Seminar.
The recurring Zoom link for the Midwest Computability Seminar is:
https://notredame.zoom.us/j/99754332165?pwd=RytjK1RFZU5KWnZxZ3VFK0g4YTMyQT09
Meeting ID: 997 5433 2165, Passcode: midwest
The recurring Webex link for the Midwest Model Theory Seminar is:
Meeting number: 126 772 4675, Password: (the password is stable, but ask me if you don't have it)
The recurring Zoom link for the local UW logic seminar is:
https://uwmadison.zoom.us/j/99860137362?pwd=RzRIRlY3QmJGTklBK3EvWkZ5dTladz09
Meeting ID: 998 6013 7362, Passcode: 247845
The talk should be generally accessible to university logic students, requiring little beyond familiarity with the incompleteness theorem and some elementary ideas from computability theory.
Discussion and commentary can be made on the speaker's web site at http://jdh.hamkins.org/natural-instances-of-nonlinearity-in-the-hierarchy-of-consistency-strength-uwm-logic-seminar-january-2021/.
Greenberg et al. also determined the distance from a dimension 1 sequence to the nearest dimension t sequence. But they left open the general problem of reducing dimension, which is made difficult by the fact that the information in a dimension s sequence can be coded (at least somewhat) redundantly. Goh, Miller, Soskova, and Westrick recently gave a complete solution.
I will talk about both the results in the 2018 paper and the more recent work. In particular, I will discuss some of the coding theory behind these results. No previous knowledge of coding theory is assumed.
In the 1980's, Slaman and Steel proved that the second part of Martin's Conjecture holds for order-preserving Borel functions. In joint work with Benny Siskind, we prove the complementary result that (assuming analytic determinacy) the first part of the conjecture also holds for order-preserving Borel functions (and under AD, for all order-preserving functions). Our methods also yield several other new results, including an equivalence between the first part of Martin's Conjecture and a statement about the Rudin-Keisler order on ultrafilters on the Turing degrees.
In my talk, I will give an overview of Martin's Conjecture and then describe our new results.
into
and within
set
operations in order to construct sets of arbitrary intrinsic density from any
Martin-Löf random. We then show that these operations are useful more
generally for working with other notions of density as well, in particular,
for viewing Church and MWC stochasticity as a form of density.
As we show, (1) the number of relatively random closed sets that can have a non-empty intersection varies depending on the choice of underlying probability measure on the space of closed subsets of Cantor space - this number being the degree of intersectability of a given family of random closed sets - and (2) the effective dimension of a sequence X is inversely proportional to the minimum degree of intersectability of a family of random closed sets, at least one of which contains X as a member. Put more simply, a sequence of lower dimension can only be in random closed sets with more branching, which are thus more intersectable, whereas higher dimension sequences can be in random closed sets with less branching, which are thus less intersectable, and the relationship between these two quantities (that is, effective dimension and degree of intersectability) can be given explicitly.
Complexity profiles are a way of measuring, for two structures A generic Muchnik reducible to B, which subsets of A can be defined using B. The complexity profile of A on itself is the natural analog of considering the relatively intrinsically Σksets in A.
Using complexity profiles, I will compare three generic muchnik degrees: Cantor space < Baire space < the Borel-complete degree. In particular, I will describe some dichotomy theorems regarding simple expansions of these and describe how to build degrees strictly between them. (Joint work with Joseph S. Miller, Noah Schweber, and Mariya Soskova.)
The usual base theory used in reverse mathematics is RCA0, which is intended to correspond roughly to the idea of "computable mathematics". The main two axioms of RCA0 are: comprehension for computable properties of natural numbers and mathematical induction for c.e. properties. A weaker theory, in which induction for c.e. properties is replaced by induction for computable properties, has also been introduced, but it has received much less attention. In the reverse mathematics literature, this weaker theory is known as RCA*0.
In this talk, I will discuss some results concerning the reverse mathematics of combinatorial principles over RCA*0. We will focus mostly on Ramsey's theorem and some of its well-known special cases: the chain-antichain principle CAC, the ascending-descending chain principle ADS, and the cohesiveness principle COH.
The results I will talk about are part of a larger project joint with Marta Fiori Carones, Katarzyna Kowalik, Tin Lok Wong, and Keita Yokoyama.
The monograph "Continuous Model Theory" by Chang and Keisler, Annals of Mathematics Studies (1966), studied structures with truth values in [0,1], with formulas that had continuous functions as connectives, sup and inf as quantifiers, and equality. In 2008, Ben Yaacov, Berenstein, Henson, and Usvyatsov introduced the model theory of metric structures, where equality is replaced by a metric, and all functions and predicates are required to be uniformly continuous. This has led to an explosion of research with results that closely parallel first-order model theory, with many applications to analysis. In my forthcoming paper "Model Theory for Real-valued Structures", the "Expansion Theorem" allows one to extend many model-theoretic results about metric structures to general [0,1]-valued structures - the structures in the 1966 monograph but without equality.
My paper "Ultrapowers Which are Not Saturated", J. Symbolic Logic 32 (1967), 23-46, introduced a pre-ordering M ≤ N on all first-order structures, that holds if every regular ultrafilter that saturates N saturates M, and suggested using it to classify structures. In the last decade, in a remarkable series of papers, Malliaris and Shelah showed that that pre-ordering gives a rich classification of simple first-order structures. Here, we lay the groundwork for using the analogous pre-ordering to classify [0,1]-valued and metric structures.
This work is joint with André Nies.
Joint with Arno Pauly and Liang Yu.