Large Cardinals and Projective Determinacy
Undergraduate thesis, Harvard University, 2017.
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Abstract
This thesis comes out of two streams of motivation: purely mathematical considerations, and considerations relating to the history and philosophy of mathematics. Both aspects are reflected in the two very different, but closely interconnected halves of the present work.
The mathematical goal of this thesis is to provide a thorough and accessible exposition of a series of important results which, through the axiom of determinacy, link descriptive set theory—loosely speaking, the study of well-behaved subsets of the real line—to large cardinal axioms—that is, axioms which extend the “height” of the universe of sets. This portion of the thesis, Part II, culminates in a proof that \(\mathbf{I}_0\) implies projective determinacy, which has never before appeared in print.
The non-mathematical aim of this thesis is to try to understand these results in their broader intellectual context. The historical portion of this thesis, Part I, examines the mathematical currents that coalesced into descriptive set theory in the late 19th and early 20th centuries. This historical development is illustrated through two case studies: first, the confluence of factors which led to the adoption of the infinite into mathematics as a legitimate object of study; and second, the French and Russian analysts who, at the turn of the 20th century, navigated between the revolutionary theory of sets first developed by Georg Cantor in the last decades of the 19th century and the traditions of mathematical analysis stretching back to the 17th century.
While very different in nature, it is the author’s sincere hope that it will be evident how Part I and Part II are rooted in a common collection of interests and concerns and form a unified intellectual project.