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Date of Final Oral Examination (Defense)
Type of Culminating Activity
Thesis - Boise State University Access Only
Master of Science in Mathematics
Andres E. Caicedo, Ph.D.
The study of games, and the determinacy thereof, has become incredibly important in modern day set theory. Modern work on the inner model program grew out of the wonderful connection between determinacy, descriptive set theory, large cardinals, and the structure of L-like models that can accommodate these large cardinals. In this document, we seek to give an exposition of the Martin-Harrington theorem, which can be seen as the beginning of this program. The theorem itself states that for a real x, II11 (x) determinacy is equivalent to the existence of x#. To reduce prerequisites needed to understand the result, our presentation of sharps is classical rather than in terms of premise.
This theorem only occupies the last twenty or so pages of this document, while the rest of this document is spent giving content to the above statement. In particular, we work through Martin's inductive proof of Borel determinacy, attempt to exhibit the power of determinacy hypotheses, and give a glimpse of the connection between descriptive set theory and constructibility. Through all of this, one can hopefully see the connecting thread of determinacy leading to the beautiful theorem of Martin and Harrington.
Ahmed, Shehzad, "Pi-1-1-determinacy and Sharps" (2014). Boise State University Theses and Dissertations. 812.