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Determinacy in the Low Levels of the Projective Hierarchy

Cotton, Michael R.
http://rave.ohiolink.edu/etdc/view?acc_num=miami1343245802

Abstract Details

2012, Master of Arts, Miami University, Mathematics.
We give an expository introduction to determinacy for sets of reals. If we are to assume the Axiom of Choice, then not all sets of reals can be determined. So we instead establish determinacy for sets according to their levels in the usual hierarchies of complexity. At a certain point ZFC is no longer strong enough, and we must begin to assume large cardinals. The primary results of this paper are Martin's theorems that Borel sets are determined, homogeneously Suslin sets are determined, and if a measurable cardinal κ exists then the complements of analytic sets are κ-homogeneously Suslin. We provide the necessary preliminaries, prove Borel determinacy, introduce measurable cardinals and ultrapowers, and then prove analytic determinacy from the existence of a measurable cardinal. Finally, we give a very brief overview of some further results which provide higher levels of determinacy relative to larger cardinals.
Paul B. Larson, PhD (Advisor)
Dennis K. Burke, PhD (Committee Member)
Tetsuya Ishiu, PhD (Committee Member)
44 p.
Logic; Mathematics
determinacy; descriptive set theory; large cardinal; measurable cardinal; set theory; projective; Borel; ZFC; homogeneously Suslin; ultrapower; Ramsey cardinal; analytic determinacy; analytic; Borel hierarchy; projective hierarchy; sets; Borel determinacy

Recommended Citations

Citations

  • Cotton, M. R. (2012). Determinacy in the Low Levels of the Projective Hierarchy [Master's thesis, Miami University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=miami1343245802

    APA Style (7th edition)

  • Cotton, Michael. Determinacy in the Low Levels of the Projective Hierarchy. 2012. Miami University, Master's thesis. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=miami1343245802.

    MLA Style (8th edition)

  • Cotton, Michael. "Determinacy in the Low Levels of the Projective Hierarchy." Master's thesis, Miami University, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=miami1343245802

    Chicago Manual of Style (17th edition)

miami1343245802
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© 2012, all rights reserved.
This open access ETD is published by Miami University and OhioLINK.