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University of Illinois – Chicago

1. Sahota, Davender S. Borel Complexity of the Isomorphism Relation for O-minimal Theories.

Degree: 2013, University of Illinois – Chicago

URL: http://hdl.handle.net/10027/10171

In 1988, Mayer published a strong form of Vaught's Conjecture for o-minimal theories. She showed Vaught's Conjecture holds, and characterized the number of countable models of an o-minimal theory T if T has fewer than continuum many countable models. Friedman and Stanley have shown that several elementary classes are Borel complete. In this thesis we address the class of countable models of an o-minimal theory T when T has continuum many countable models.
Our main result gives a model theoretic dichotomy describing the Borel complexity of isomorphism on the class of countable models of T. The first case is if T has no simple types, isomorphism is Borel on the class of countable models of T. In the second case, T has a simple type over a finite set A, and there is a finite set B containing A such that the class of countable models of the completion of T over B is Borel complete.
*Advisors/Committee Members: Marker, David E. (advisor), Baldwin, John T. (committee member), Goldbring, Isaac (committee member), Rosendal, Christian (committee member), Laskowski, Michael C. (committee member).*

Subjects/Keywords: Model Theory; Descriptive Set Theory; O-minimal; Borel complete; Vaught's Conjecture

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Sahota, D. S. (2013). Borel Complexity of the Isomorphism Relation for O-minimal Theories. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/10171

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Sahota, Davender S. “Borel Complexity of the Isomorphism Relation for O-minimal Theories.” 2013. Thesis, University of Illinois – Chicago. Accessed August 08, 2020. http://hdl.handle.net/10027/10171.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Sahota, Davender S. “Borel Complexity of the Isomorphism Relation for O-minimal Theories.” 2013. Web. 08 Aug 2020.

Vancouver:

Sahota DS. Borel Complexity of the Isomorphism Relation for O-minimal Theories. [Internet] [Thesis]. University of Illinois – Chicago; 2013. [cited 2020 Aug 08]. Available from: http://hdl.handle.net/10027/10171.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Sahota DS. Borel Complexity of the Isomorphism Relation for O-minimal Theories. [Thesis]. University of Illinois – Chicago; 2013. Available from: http://hdl.handle.net/10027/10171

Not specified: Masters Thesis or Doctoral Dissertation

2. Drueck, Fred R. Limit Models, Superlimit Models, and Two Cardinal Problems in Abstract Elementary Classes.

Degree: 2013, University of Illinois – Chicago

URL: http://hdl.handle.net/10027/9996

This dissertation examines three main topics, the topic of
defining "superstability" for abstract elementary classes (AECs),
uniqueness of limit models, and two cardinal models in abstract
elementary classes. In particular we further generalize an
analogue of Vaught's theorem which constructs an uncountable two
cardinal model starting from the existence of a countable
Vaughtian pair in an elementary class to the AEC context
originally published by Lessmann, who in turn built upon the work
of Grossberg and VanDieren, Shelah, and others. We also give
various sufficient conditions on countable models, as well as a
condition on models of size kappa that, assuming that a simplified
morass,ñ allows us to construct a (kappa^++,kappa)-model. We
discuss how this work in AECs to some degree parallels the proof
of Jensen's Gap-2 transfer theorem for elementary classes. We
also discuss difficulties inherent in proving a true gap-2
transfer theorem for AECs. Additionally, we discuss, progress
that has been made toward proving the uniqueness of limit models
assuming various "superstability-like" assumptions (much of the
work described is due to Shelah, Villaveces, Grossberg, and
VanDieren). One small original result is contributed to this
discussion.
*Advisors/Committee Members: Baldwin, John T. (advisor), Marker, David (committee member), Takloo-Bighash, Ramin (committee member), VanDieren, Monica (committee member), Scow, Lynn (committee member).*

Subjects/Keywords: Limit Models; Superlimit Models; Two Cardinal Problems; two cardinal models; two cardinal; 2 cardinal; 2 cardinal problems; 2 cardinal model; gap-2 transfer; gap-2; Abstract Elementary Classes; mathematical logic; uniqueness of limit models; morasses; lessmann

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Drueck, F. R. (2013). Limit Models, Superlimit Models, and Two Cardinal Problems in Abstract Elementary Classes. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/9996

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Drueck, Fred R. “Limit Models, Superlimit Models, and Two Cardinal Problems in Abstract Elementary Classes.” 2013. Thesis, University of Illinois – Chicago. Accessed August 08, 2020. http://hdl.handle.net/10027/9996.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Drueck, Fred R. “Limit Models, Superlimit Models, and Two Cardinal Problems in Abstract Elementary Classes.” 2013. Web. 08 Aug 2020.

Vancouver:

Drueck FR. Limit Models, Superlimit Models, and Two Cardinal Problems in Abstract Elementary Classes. [Internet] [Thesis]. University of Illinois – Chicago; 2013. [cited 2020 Aug 08]. Available from: http://hdl.handle.net/10027/9996.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Drueck FR. Limit Models, Superlimit Models, and Two Cardinal Problems in Abstract Elementary Classes. [Thesis]. University of Illinois – Chicago; 2013. Available from: http://hdl.handle.net/10027/9996

Not specified: Masters Thesis or Doctoral Dissertation