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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

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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. Dworetzky, Samuel. The Classification Problem for Models of ZFC.

Degree: 2017, Boise State University

URL: https://scholarworks.boisestate.edu/td/1253

Models of ZFC are ubiquitous in modern day set theoretic research. There are many different constructions that produce countable models of ZFC via techniques such as forcing, ultraproducts, and compactness. The models that these techniques produce have many different characteristics; thus it is natural to ask whether or not models of ZFC are classifiable. We will answer this question by showing that models of ZFC are unclassifiable and have maximal complexity. The notions of complexity used in this thesis will be phrased in the language of Borel complexity theory.
In particular, we will show that the class of countable models of ZFC is Borel complete. Most of the models in the construction as it turns out are ill-founded. Thus, we also investigate the sub problem of identifying the complexity for well-founded models. We give partial results for the well-founded case by identifying lower bounds on the complexity for these models in the Borel complexity hierarchy.

Subjects/Keywords: Models of ZFC; Models of PA; Borel Complete; Set Theory

…*Borel* *complete*. Formally, we
have the following definition:
Definition 1.2.2. The class of… …countable models of a theory T is *Borel* *Complete*
if and only if for any other class of countable… …class (XLO , ∼
=LO ) of countable linear orders under isomorphism
is *Borel* *complete*… …than
*Borel* *complete*. Other complexities that will arise in this thesis are =, E0 , and Eω1… …introduction, models of PA are known to be *Borel* *complete*. In
Chapter 2, we will examine the argument…

Record Details Similar Records

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

APA (6^{th} Edition):

Dworetzky, S. (2017). The Classification Problem for Models of ZFC. (Thesis). Boise State University. Retrieved from https://scholarworks.boisestate.edu/td/1253

Not specified: Masters Thesis or Doctoral Dissertation

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

Dworetzky, Samuel. “The Classification Problem for Models of ZFC.” 2017. Thesis, Boise State University. Accessed August 08, 2020. https://scholarworks.boisestate.edu/td/1253.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Dworetzky, Samuel. “The Classification Problem for Models of ZFC.” 2017. Web. 08 Aug 2020.

Vancouver:

Dworetzky S. The Classification Problem for Models of ZFC. [Internet] [Thesis]. Boise State University; 2017. [cited 2020 Aug 08]. Available from: https://scholarworks.boisestate.edu/td/1253.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Dworetzky S. The Classification Problem for Models of ZFC. [Thesis]. Boise State University; 2017. Available from: https://scholarworks.boisestate.edu/td/1253

Not specified: Masters Thesis or Doctoral Dissertation

3.
Braga, Bruno.
On the *Borel* complexity of some classes of Banach
spaces.

Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2013, Kent State University

URL: http://rave.ohiolink.edu/etdc/view?acc_num=kent1377115201

In this dissertation I mainly study several classes of
Banach spaces and I try to compute, of at least to obtain a
lower/upper bound, to its Borel complexity. Also, using those
results, we show some non-universality results for some of those
classes of Banach spaces.
*Advisors/Committee Members: Diestel, Joe (Advisor).*

Subjects/Keywords: Mathematics; Effros-Borel structure, Banach spaces, Banach-Saks property, Radon-Nikodym property, complete continuous property, weak compact operators, unconditional converging operators, local structure

…bounded operators, are non *Borel*. The
first one is actually *complete* coanalytic. In both of… …x28;resp. *complete* coanalytic) if for all standard *Borel* space Y and
all B ⊂ Y analytic… …X → Y }, for X ∈ SB), which is well known to be non *Borel* (*complete*… …Our goal for this dissertation is to study the *Borel* complexity of certain classes of Banach… …general techniques to compute the *Borel* complexity of classes of Banach
spaces. Therefore, we do…

Record Details Similar Records

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

APA (6^{th} Edition):

Braga, B. (2013). On the Borel complexity of some classes of Banach spaces. (Doctoral Dissertation). Kent State University. Retrieved from http://rave.ohiolink.edu/etdc/view?acc_num=kent1377115201

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

Braga, Bruno. “On the Borel complexity of some classes of Banach spaces.” 2013. Doctoral Dissertation, Kent State University. Accessed August 08, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=kent1377115201.

MLA Handbook (7^{th} Edition):

Braga, Bruno. “On the Borel complexity of some classes of Banach spaces.” 2013. Web. 08 Aug 2020.

Vancouver:

Braga B. On the Borel complexity of some classes of Banach spaces. [Internet] [Doctoral dissertation]. Kent State University; 2013. [cited 2020 Aug 08]. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=kent1377115201.

Council of Science Editors:

Braga B. On the Borel complexity of some classes of Banach spaces. [Doctoral Dissertation]. Kent State University; 2013. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=kent1377115201