November 26
Zachiri Mckenzie,
University of Chester
Well-founded models of fragments of Collection
Let $\mathsf{M}$ be the weak set theory (with powersets) axiomatised by: $\textsf{Extensionality}$, $\textsf{Pair}$, $\textsf{Union}$, $\textsf{Infinity}$, $\textsf{Powerset}$, transitive containment ($\textsf{TCo}$), $\Delta_0\textsf{-Separation}$ and $\textsf{Set-Foundation}$. In this talk I will discuss the relationship between two alternative versions of the set-theoretic collection scheme: Collection and Strong Collection. Both of these schemes yield $\mathsf{ZF}$ when added to $\mathsf{M}$, but when restricted the $\Pi_n$-formulae (denoted $\Pi_n\textsf{-Collection}$ and $\textsf{Strong } \Pi_n\textsf{-Collection}$) these alternative versions of set-theoretic collection differ. In particular, over the theory $\mathsf{M}$, $\textsf{Strong }\Pi_n\textsf{-Collection}$ is equivalent to $\Pi_n\textsf{-Collection}+\Sigma_{n+1}\textsf{-Separation}$. And, $\mathsf{M}+\textsf{Strong }\Pi_n\textsf{-Collection}$ proves the consistency of $\mathsf{M}+\Pi_n\textsf{-Collection}$. In this talk I will show that, despite this difference in consistency strength, every countable well-founded model of $\mathsf{M}+\Pi_n\textsf{-Collection}$ satisfies $\textsf{Strong } \Pi_n\textsf{-Collection}$. If time permits I will outline how this argument can be refined to show that $\mathsf{M}+\Pi_n\textsf{-Collection}+\Pi_{n+1}\textsf{-Foundation}$ proves $\Sigma_{n+1}\textsf{-Separation}$.