Algebra Seminar talk
2025-03-07
Sakaé Fuchino
Extendibility and Laver-generic large cardinal axioms
Abstract:
We introduce the (super-$C^{(\infty)}$-)Laver-generic large cardinal axioms for extendibility (LgLCAs for extendible, for short) , and show that most of the previously known consequences of the (super-$C^{(\infty)}$-)LgLCAs for ultrahugeness already follow from this family of axioms.
For example, the Resurrection Axiom for the class of posets $\mathcal{P}$ with the parameter set $\mathcal{H}(\kappa_{refl})$, and the Maximality Principle for $\mathcal{P}$ and $\mathcal{H}(\kappa_{refl})$ are among the consequences of the super-$C^{(\infty)}$-$\mathcal{P}$-LgLCA for extendible.
The consistency strength of super-$C^{(\infty)}$-LgLCAs for extendibility (in connection with a transfinitely iterable class $\mathcal{P}$ of posets) can be bound by the consistency of the second-order Vopěnka principle which in turn is bound by that of an almost-huge cardinal.
On the other hand, (super-$C^{(\infty)}$-)LgLCAs for hyperhuge are known to be equiconsistent with a (super-$C^{(\infty)}$-)hyperhuge cardinal by a theorem of Fuchino and Usuba.
In contrast to these results where extremely strong large cardinal properties are involved, most 'mathematical' applications are decided on the level of LgLCAs for supercompact (which is also very strong but not so much as extendible or hyperhuge). For example the reflection of non-freeness of an algebra down to a subalgebra of cardinality less than continuum follows from $\mathcal{P}$-LgLCA for supercompact where $\mathcal{P}$ is the class of all ccc posets. The reflection of non-metrizability of a topological space of character less than continuum down to a subspace of cardinality less than continuum is a theorem under $\mathcal{P}$-LgLCA for supercompact where $\mathcal{P}$ is now the class of Cohen forcings.