The Hyperuniverse Program, founded by Sy D. Friedman, intends to produce new second-order axioms of set theory which appropriately formalize "the universe is maximal" in one of a few ways. A height-maximality principle is one intended to formalize "the class of ordinals is as long as possible" (1, p.9). First-order axioms are not considered, because by the reflection principle, they do not effectively enforce this (1, p.14). As a first attempt for a height-maximality principle, the extended reflection axiom $\textrm{ERA}$ is introduced, then strengthened to obtain the principle of $\sharp$-generation - the latter "stands out as the correct formalization of the principle of height maximality" (1, p.17).
I have questions about the undertaking of the program:
- Assume second-order reflection holds in $V$, which follows from $\textrm{ERA}$ and therefore from $\sharp$-generation. If $\phi$ is a second-order formula claimed to be an optimal height-maximality principle, if we assume $\phi$ then there exists $\alpha$ where $(V_\alpha,\in,V_{\alpha+1})\vDash\phi$. However, existence of a set-sized model satisfying $\phi$ is enough to rule $\phi$ out from being the optimal maximality principle - all first-order $\phi$ are ruled out because they have set models, leading to the immediately stronger height-maximality principle "$\phi$ and there is a set model of $\phi$". (I would argue the existence of a set-sized model of $\phi$ alone is enough, it shows $\phi$ fails to formalize "$\textrm{Ord}$ is as tall as possible".)
- Assume that some $\phi$ is an accepted formalization of height-maximality, e.g. the axiom of $\sharp$-generation. Let $\phi'$ be the axiom $\forall\alpha\exists(\beta>\alpha)((V_\beta,\in,V_{\beta+1})\vDash\phi)$. Is $\phi'$ ruled out on the grounds of "general properties of $V$"? Would $\phi'$ be a strictly stronger height-maximality principle, as it enforces existence of many models where, when the universe is chopped at that model, the claimed height-maximality principle $\phi$ holds?
Sources:
- Friedman, "Evidence for Set-Theoretic Truth and the Hyperuniverse Programme"
- Friedman, Ternullo, "Maximality Principles in the Hyperuniverse Programme"
