MidWest Model Theory Day

Tuesday, April 5th, 2016 at UIC

Speakers: Steffen Lempp, Gabriel Conant, H. Jerome Keisler

Schedule: All talks are about an hour long, in SEO 636. There will also be coffee&cookies in 636.
It is probably easiest to park in the university parking lot on Morgan St. between Roosevelt and Taylor.
Let us know (@math.wisc.edu">andrewsmath.wisc.edu) if you are planning to come to lunch and/or dinner so we can make approximately correct reservations!

Titles & Abstracts:

Steffen Lempp
Title: Spectra of computable models of disintegrated strongly minimal theories

Abstract: The investigation of which countable models of a given first-order theory are computable (i.e., have a copy on the natural numbers with a computable atomic diagram) goes back to the dawn of modern logic. The problem has been most thoroughly investigated in the case of uncountably (but not totally) categorical theories T, since there, by Baldwin and Lachlan, the countable models form an elementary chain of length ω+1:
M0M1M2 ≺ ... ≺ Mω
In 1978, Goncharov showed an example in which some but not all countable models are computable. To simplify presenting his and similar results, we define the spectrum of computable models of T as
SCM(T) = {α ≤ ω : Mα is computable},
so Goncharov's result can be phrased as saying that {0} is a possible spectrum. Over the next four decades, a fairly small number of other possible spectra have been found, but there are almost no general results restricting which sets can be spectra (other than the obvious upper bound Σ0ω+3). Also, thus far, no differences have been found between uncountably categorical and the more restrictive strongly minimal theories, so we restrict ourselves to the latter case.

The first strong negative result on spectra is due to Andrews and A. Medvedev (2014) that for a strongly minimal disintegrated theory T in a finite language, the only possible spectra are ∅, {0} and [0, ω], in which case one can effectively (but non-uniformly) reduce to the case of a binary relational language.

In this talk, will present on-going joint work with Andrews vastly extending this result. In particular, we are able to show:

  1. There are exactly seven possible spectra for strongly minimal disintegrated theories in a (possibly infinite) binary relational language.
  2. There are exactly ten possible spectra for strongly minimal disintegrated theories in a relational language of bounded arity in which each relation has Morley rank at most 1.
  3. The only additional possible spectra for strongly minimal disintegrated theories in a relational language of unbounded arity in which each relation has Morley rank at most 1 are of the form [0, α) or [0, α) ∪ {ω}.
  4. There are at most eighteen possible spectra for strongly minimal disintegrated theories in a ternary relational language.

Gabriel Conant
Title: Stable groups and expansions of the integers

Abstract: Motivated by a question of Marker on stable expansions of the group of integers, we give a characterization of superstable groups of finite U-rank, which uses model theoretic weight together with chain conditions on definable subgroups. Combined with recent work of Palacin and Sklinos, we conclude that the group of integers has no proper stable expansions of finite weight, and thus none of finite dp-rank. We finish with a brief discussion of other results on expansions of the integers, including open questions about the unstable, finite dp-rank case. This is joint work with Anand Pillay.

H. Jerome Keisler
Title:Randomizations of Scattered Sentences

Abstract: In 1970, Morley introduced the notion of an infinitary sentence being scattered. He showed the number of (isomorphism types of) countable models is at most aleph one for a scattered sentence, and is continuum for a non-scattered sentence. The absolute form of Vaught's conjecture says that a scattered sentence has at most countably many countable models. Generalizing previous work of Ben Yaacov and the author, we introduce here the notion of a separable randomization of an infinitary sentence, which is a separable continuous structure whose elements are random elements of countable models of the sentence. We improve a result by Andrews and the author, showing that an infinitary sentence with countably many countable models has few separable randomizations, that is, every separable randomization is isomorphic to a very simple structure called a basic randomization. We also show that an infinitary sentence with few separable randomizations is scattered. Hence if the absolute Vaught conjecture holds, then a sentence has few separable randomizations if and only if it has at most countably many countable models, and also if and only if it is scattered. Moreover, assuming Martin's axiom for aleph one, we show that every scattered sentence has few separable randomizations.

Other MWMTDs:
April 5th, 2016
October 28th, 2014
October 22nd, 2013
April 18th, 2013
October 23rd, 2012
April 26th, 2012
October 11th, 2011
April 7th, 2011
October 26th, 2010