Abstract: In this talk, I consider two important areas of research in contemporary set theory: (1) work on “choiceless” large cardinals (plausibly consistent with ZF but known to be inconsistent with ZFC) over the last decade by Joan Bagaria, Gabriel Goldberg, Peter Koellner, Farmer Schlutzenberg, and Hugh Woodin, and (2) very recent results on “hodless” large cardinals (plausibly consistent with ZFC but known to be inconsistent with ZFC + V = HOD) by Juan Pablo Aguilera, Bagaria, Philipp Lücke, and Goldberg.
What is the philosophical significance of these results? On one view, the ‘new large cardinals’ demonstrate that V = HOD and choice are restrictive principles in the same sense as V = L. I argue against this thesis: there is a reasonable case for the restrictiveness of V = HOD, but it does not extend to choice. I suggest that set-theoretic realists should view theories with hodless large cardinal axioms as genuine contenders for descriptions of the universe; choiceless theories, in contrast, are ultimately to be understood instrumentally.
In this talk, I consider two important areas of research in contemporary set theory: (1) work on “choiceless” large cardinals (plausibly consistent with ZF but known to be inconsistent with ZFC) over the last decade by Joan Bagaria, Gabriel Goldberg, Peter Koellner, Farmer Schlutzenberg, and Hugh Woodin, and (2) very recent results on “hodless” large cardinals (plausibly consistent with ZFC but known to be inconsistent with ZFC + V = HOD) by Juan Pablo Aguilera, Bagaria, Philipp Lücke, and Goldberg.
What is the philosophical significance of these results? On one view, the ‘new large cardinals’ demonstrate that V = HOD and choice are restrictive principles in the same sense as V = L. I argue against this thesis: there is a reasonable case for the restrictiveness of V = HOD, but it does not extend to choice. I suggest that set-theoretic realists should view theories with hodless large cardinal axioms as genuine contenders for descriptions of the universe; choiceless theories, in contrast, are ultimately to be understood instrumentally.
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Philosophy of Mathematics Seminar Convenors: Daniel Isaacson and Beau Mount
