Wed, 20 Jan 2021

16:00 - 17:30
Virtual

Iteration, reflection, and singular cardinals

Dima Sinapova
(University of Illinois at Chicago)
Abstract

Two classical results of Magidor are: 

(1) from large cardinals it is consistent to have reflection at $\aleph_{\omega+1}$, and 

(2) from large cardinals it is consistent to have the failure of SCH at $\aleph_\omega$.

These principles are at odds with each other. The former is a compactness type principle. (Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object.) In contrast, failure of SCH is an instance of incompactness. The natural question is whether we can have both of these simultaneously. We show the answer is yes.

We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH.  Then we obtain this situation at $\aleph_\omega$. This is joint work with Alejandro Poveda and Assaf Rinot.

Tue, 15 Jan 2019
16:00
L5

On strongly minimal Steiner systems Zilber’s Conjecture, Universal Algebra, and Combinatorics

John Baldwin
(University of Illinois at Chicago)
Abstract

With Gianluca Paolini (in preparation), we constructed, using a variant on the Hrushovski dimension function, for every k ≥ 3, 2^µ families of strongly minimal Steiner k systems. We study the mathematical properties of these counterexamples to Zilber’s trichotomy conjecture rather than thinking of them as merely exotic examples. In particular the long study of finite Steiner systems in reflected in results that depend on the block size k. A quasigroup is a structure with a binary operation such that for each equation xy = z the values of two of the variables determines a unique value for the third. The new Steiner 3-systems are bi-interpretable with strongly minimal Steiner quasigroups. For k > 3, we show the pure k-Steiner systems have ‘essentially unary definable closure’ and do not interpret a quasigroup. But we show that for q a prime power the Steiner q systems can be interpreted into specific sorts of quasigroups, block algebras. We extend the notion of an (a, b)-cycle graph arising in the study of finite and infinite Stein triple systems (e.g Cameron-Webb) by introducing what we call the (a, b)-path graph of a block algebra. We exhibit theories of strongly minimal block algebras where all (a, b)-paths are infinite and others in which all are finite only in the prime model. We show how to obtain combinatorial properties (e.g. 2-transitivity) by the either varying the basic collection of finite partial Steiner systems or modifying the µ function which ensures strong minimality

Thu, 17 Jan 2019
11:00
L6

Philosophical implications of the paradigm shift in model theory

John Baldwin
(University of Illinois at Chicago)
Abstract



Traditionally, logic was thought of as `principles of right reason'. Early twentieth century philosophy of mathematics focused on the problem of a general foundation for all mathematics. In contrast, the last 70 years have seen model theory develop as the study and comparison of formal theories for studying specific areas of mathematics. While this shift began in work of Tarski, Robinson, Henkin, Vaught, and Morley, the decisive step came with Shelah's stability theory. After this paradigm shift there is a systematic search for a short set of syntactic conditions which divide first order theories into disjoint classes such that models of different theories in the same class have similar mathematical properties. This classification of theories makes more precise the idea of a `tame structure'. Thus, logic (specifically model theory) becomes a tool for organizing and doing mathematics with consequences for combinatorics, diophantine geometry, differential equations and other fields. I will present an account of the last 70 years in model theory that illustrates this shift. This reports material in my recent book published by Cambridge: Formalization without Foundationalism: Model Theory and the Philosophy of Mathematical Practice.

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