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University of North Texas

1.
Khafizov, Farid T.
Descriptions and Computation of *Ultrapowers* in L(R).

Degree: 1995, University of North Texas

URL: https://digital.library.unt.edu/ark:/67531/metadc277867/

The results from this dissertation are an exact computation of ultrapowers by measures on cardinals \aleph_{n}, n∈ w, in L(\IR), and a proof that ordinals in L(\IR) below δ_{sp}{5}{1} represented by descriptions and the identity function with respect to sequences of measures are cardinals. An introduction to the subject with the basic definitions and well known facts is presented in chapter I. In chapter II, we define a class of measures on the \aleph_{n}, n∈ω, in L(\IR) and derive a formula for an exact computation of the ultrapowers of cardinals by these measures. In chapter III, we give the definitions of descriptions and the lowering operator. Then we prove that ordinals represented by descriptions and the identity function are cardinals. This result combined with the fact that every cardinal <δ_{sp}{5}{1} in L(\IR) is represented by a description (J1), gives a characterization of cardinals in L(\IR) below δ_{sp}{5}{1}. Concrete examples of formal computations are shown in chapter IV.
*Advisors/Committee Members: Jackson, Steve, 1957-, Mauldin, R. Daniel, Lewis, Paul Weldon, Brand, Neal E., Jacob, Roy Thomas.*

Subjects/Keywords: ultrapowers; mathematics; Cardinal numbers.; Set theory.

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

APA (6^{th} Edition):

Khafizov, F. T. (1995). Descriptions and Computation of Ultrapowers in L(R). (Thesis). University of North Texas. Retrieved from https://digital.library.unt.edu/ark:/67531/metadc277867/

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):

Khafizov, Farid T. “Descriptions and Computation of Ultrapowers in L(R).” 1995. Thesis, University of North Texas. Accessed August 05, 2020. https://digital.library.unt.edu/ark:/67531/metadc277867/.

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

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Khafizov, Farid T. “Descriptions and Computation of Ultrapowers in L(R).” 1995. Web. 05 Aug 2020.

Vancouver:

Khafizov FT. Descriptions and Computation of Ultrapowers in L(R). [Internet] [Thesis]. University of North Texas; 1995. [cited 2020 Aug 05]. Available from: https://digital.library.unt.edu/ark:/67531/metadc277867/.

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

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Khafizov FT. Descriptions and Computation of Ultrapowers in L(R). [Thesis]. University of North Texas; 1995. Available from: https://digital.library.unt.edu/ark:/67531/metadc277867/

Not specified: Masters Thesis or Doctoral Dissertation

2. Trujillo, Timothy Onofre. Topological Ramsey Spaces, Associated Ultrafilters, and Their Applications to the Tukey Theory of Ultrafilters and Dedekind Cuts of Nonstandard Arithmetic.

Degree: PhD, Mathematics, 2014, U of Denver

URL: https://digitalcommons.du.edu/etd/661

This dissertation makes contributions to the areas of combinatorial set theory, the model theory of arithmetic, and the Tukey theory of ultrafilters. The main results are broken into three parts.
In the first part, we identify some new partition relations among finite trees and use them to answer an open question of Dobrinen; namely, "for n < omega, are the notions of Ramsey for Rn and selective for Rn equivalent?" We show that for each n < omega, it is consistent with ZFC that there exists a selective for Rn ultrafilter which is not Ramsey for Rn.
In the second part, we extend results of Blass concerning Dedekind cuts associated to ultrafilter mappings from p-point and weakly-Ramsey ultrafilters to ultrafilter mappings from Ramsey for R1 ultrafilters. Blass associates to each ultrafilter U on a countable set X and each function g with domain X a Dedekind cut in the model of arithmetic given by the ultrapower omega^ran(g)/g(U). We characterize, under the continuum hypothesis, the cuts obtainable from an ultrafilter mapping from a Ramsey for R1 ultrafilter. We also show that the only cut obtainable for ultrafilter mappings between p-points, which are Tukey reducible to a given Ramsey for R1 ultrafilter, is the standard cut consisting of equivalence classes of constant sequences. These results imply new existence theorems for various special kinds of ultrafilters.
In final part of the dissertation, we extend results of Dobrinen and Todorcevic concerning the canonical Ramsey theory of R1 to the space H2 given by forming the product of the space R1 with itself. These results imply new existence theorems for initial Tukey structures of nonprincipal ultrafilters. These results shed light on the following open question of Dobrinen concerning the Tukey theory of ultrafilters, "what are the possible initial Tukey structures for ultrafilters on a countable base set?" In particular, we show for the first time that it is consistent with ZFC that the four-element Boolean algebra appears as an initial Tukey structure.
*Advisors/Committee Members: Natasha Dobrinen, Ph.D..*

Subjects/Keywords: Dedekind cuts; Selective; Topological Ramsey Spaces; Tukey theory; Ultrafilters; Ultrapowers; Applied Mathematics; Physical Sciences and Mathematics

Record Details Similar Records

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

APA (6^{th} Edition):

Trujillo, T. O. (2014). Topological Ramsey Spaces, Associated Ultrafilters, and Their Applications to the Tukey Theory of Ultrafilters and Dedekind Cuts of Nonstandard Arithmetic. (Doctoral Dissertation). U of Denver. Retrieved from https://digitalcommons.du.edu/etd/661

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

Trujillo, Timothy Onofre. “Topological Ramsey Spaces, Associated Ultrafilters, and Their Applications to the Tukey Theory of Ultrafilters and Dedekind Cuts of Nonstandard Arithmetic.” 2014. Doctoral Dissertation, U of Denver. Accessed August 05, 2020. https://digitalcommons.du.edu/etd/661.

MLA Handbook (7^{th} Edition):

Trujillo, Timothy Onofre. “Topological Ramsey Spaces, Associated Ultrafilters, and Their Applications to the Tukey Theory of Ultrafilters and Dedekind Cuts of Nonstandard Arithmetic.” 2014. Web. 05 Aug 2020.

Vancouver:

Trujillo TO. Topological Ramsey Spaces, Associated Ultrafilters, and Their Applications to the Tukey Theory of Ultrafilters and Dedekind Cuts of Nonstandard Arithmetic. [Internet] [Doctoral dissertation]. U of Denver; 2014. [cited 2020 Aug 05]. Available from: https://digitalcommons.du.edu/etd/661.

Council of Science Editors:

Trujillo TO. Topological Ramsey Spaces, Associated Ultrafilters, and Their Applications to the Tukey Theory of Ultrafilters and Dedekind Cuts of Nonstandard Arithmetic. [Doctoral Dissertation]. U of Denver; 2014. Available from: https://digitalcommons.du.edu/etd/661

University of Florida

3. Leaning, Jeffrey Scott, 1971-. Disassociated indiscernibles.

Degree: PhD, Mathematics, 1999, University of Florida

URL: https://ufdc.ufl.edu/AA00017710

Subjects/Keywords: Academic degrees; Critical points; Indiscernibles; Logical theorems; Mathematical sequences; Mathematical set theory; Mathematical theorems; Mathematics; Ultrafilters; Ultrapowers

Record Details Similar Records

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

APA (6^{th} Edition):

Leaning, Jeffrey Scott, 1. (1999). Disassociated indiscernibles. (Doctoral Dissertation). University of Florida. Retrieved from https://ufdc.ufl.edu/AA00017710

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

Leaning, Jeffrey Scott, 1971-. “Disassociated indiscernibles.” 1999. Doctoral Dissertation, University of Florida. Accessed August 05, 2020. https://ufdc.ufl.edu/AA00017710.

MLA Handbook (7^{th} Edition):

Leaning, Jeffrey Scott, 1971-. “Disassociated indiscernibles.” 1999. Web. 05 Aug 2020.

Vancouver:

Leaning, Jeffrey Scott 1. Disassociated indiscernibles. [Internet] [Doctoral dissertation]. University of Florida; 1999. [cited 2020 Aug 05]. Available from: https://ufdc.ufl.edu/AA00017710.

Council of Science Editors:

Leaning, Jeffrey Scott 1. Disassociated indiscernibles. [Doctoral Dissertation]. University of Florida; 1999. Available from: https://ufdc.ufl.edu/AA00017710