Douglas Blue

Clarke-Doane and Ash (2024) argues that mathematics and philosophy are on a par as a priori disciplines. In particular, each fails to be objective. Should this be so, it is unclear that philosophy can do anything for set theory or that new axioms can ever be rationally justified. Blue (2024) explicates a methodology for rationally justifying new axioms. I will argue against (Clarke-Doane and Ash 2024) by buttressing (Blue 2024), describing how it accounts for the case for Definable Determinacy and how it might extend to a potential case for Baire category principles. Along the way, I will describe a not-yet-refuted scenario for positively answering Todorčević’s question “In order to have the true structure theory of $$\wp (\omega _1)$$ do we really need to retreat to an inner model of the universe of sets?” (Todorcevic 2024, Question 4.7).

The generic multiverse was introduced in [74] and [81] to explicate the portion of mathematics which is immune to our independence techniques. It consists, roughly speaking, of all universes of sets obtainable from a given universe by forcing extension. Usuba recently showed that the generic multiverse contains a unique definable universe, assuming strong large cardinal hypotheses. On the basis of this theorem, a non-pluralist about set theory could dismiss the generic multiverse as irrelevant to what set theory is really about, namely that unique definable universe. Whatever one’s attitude towards the generic multiverse, we argue that certain impure proofs ensure its ongoing relevance to the foundations of set theory. The proofs use forcing-fragile theories and absoluteness to prove ${\mathrm {ZFC}}$ theorems about simple “concrete” objects.

We isolate two abstract determinacy theorems for games of length $\omega_1$ from work of Neeman and use them to conclude, from large-cardinal assumptions and an iterability hypothesis in the region of measurable Woodin cardinals thatif the Continuum Hypothesis holds, then all games of length $\omega_1$ which are provably $\Delta_1$ -definable from a universally Baire parameter are determined;all games of length $\omega_1$ with payoff constructible relative to the play are determined; andif the Continuum Hypothesis holds, then there is a model of ${\mathsf{ZFC}}$ containing all reals in which all games of length $\omega_1$ definable from real and ordinal parameters are determined.