Andreas Voellmer's PhD Thesis
In the aftermath of Kurt Godel and Paul Cohen's discovery that many important mathematical theorems are undecidable from the basic axioms, there has been a proliferation of distinct foundational models for mathematics. In the eyes of some thinkers, this explosion of alternative models dealt a blow to the idea that mathematical truth is uniquely determined-- the truth or falsehood of an open problem might simply depend on which model we choose to work in. One major goal of contemporary set theory is to re-unify the foundations of mathematics, by showing that a single 'core model' is nested inside all these prima facie incompatible models. My PhD research was focused on this goal: specifically, Neeman & Steel's definition of "Plus-One Premice" (https://math.berkeley.edu/~steel/papers/plusone.pdf) was a first step towards building a more robust core model, and my work has yielded substantial clarification and revision of this theory.
Many of the challenges in this work arise from subtle technical differences between the higher-level "language of premice" which Neeman and Steel originally used for their definitions, and the lower-level "language of coherent structures" which is needed for more precise analysis. As my research continued, the complications that arose in translating between these languages became unmanageable. Taking a step back, I realized that by changing a few key definitions in Neeman & Steel's original work, the translations could be simplified while retaining all the virtues of the original definitions. Steel now agrees that my revisions yield a strictly superior version of plus-one premice, and advocates their adoption in all future work on this core model.
The main theorem in my dissertation is a partial characterization of Square-Kappa for plus-one premice. Square-Kappa is a structural property of certain models of set theory which is useful for two reasons:
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It is a powerful tool which enables many further constructions in the model;
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It is viewed as a "test question" to gauge how well we understand the model-- if we can prove Square-Kappa, then we must have a highly detailed theory of how this model works.
My dissertation accomplishes these goals for Plus-One Premice under certain "smallness" assumptions.