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Hope Duncan

Publications and source records attributed to Hope Duncan.

2 recordsLinked to original sources
arXiv2026

Large cardinals, small violations of choice

We study the relationship between large cardinals and Blass' Small Violation of Choice ($\mathsf{SVC}$), which states that the Axiom of Choice holds in a set-generic extension of the universe. Our main theorem is that in the presence of a supercompact cardinal, either the Axiom of Choice fails significantly, or else $\mathsf{SVC}$ holds in $L(\mathcal P(\mathrm{Ord}))$. We show that a supercompact is not strong enough to reflect $\mathsf{SVC}$, while an extendible cardinal is. If we assume Usuba's LoÅ› Property holds for the supercompact, then we can outright derive $\mathsf{SVC}$ in the universe.

math.LO↗
arXiv2026

A note on the modal logic of symmetric extensions

Taking symmetric extensions can be considered as a generalisation of forcing, which produces a richer multiverse of models with and without the axiom of choice. We can study the structure of this multiverse using modal logic. In particular, we define the concept of of choice-switches, and show any independent system of choice-switches is not itself independent from any standard example of an independent system of buttons.

math.LO↗