Douglas Blue

Cantor’s Continuum Hypothesis (CH) asks what the cardinality of the real line is. It is a mathematically precise problem. Yet, because it cannot be solved using the standard axiomatic framework for mathematics, it is seen by many as a philosophical problem. I want to show what philosophy can contribute to solving the problem by returning it to mathematics, and I will do that by elucidating what it is to be a solution to a mathematical problem. I will argue that CH is an ordinary mathematical problem like the Riemann Hypothesis. I will do this in the course of countering two objections to an alleged solution to CH. This alleged solution, I argue, embodies the general form any solution to CH must take. From this we will see what a solution to CH must be like. It is just like what a solution to the Riemann Hypothesis must be like.

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.