Iterating logarithms and well-foundedness

Abstract

We prove that there is no sequence $\{ X_n \}_{n\inω}$ such that for all $n$, there is an injection from $\mathcal{P}(X_{n+1})$ to $X_n$ under a restricted set of axioms that conforms to the "working mathematician's" intuition (excluding Choice, Replacement, and Foundation). We also show such a sequence cannot be defined using a formula either. More precisely, it is inconsistent with Zermelo set theory minus Infinity to have a formula $Φ(\cdot,\cdot)$ such that for each $n\inω$ there are $X$ and $Y$ with $Φ(n,X)$, $Φ(n+1,Y)$, and $\mathcal{P}(Y)$ being injectable into $X$. We finally show that the theorem fails if we allow infinitely many definitions, one for each term of the sequence, by using Hamkins, Linetsky, and Reitz's pointwise definable extensions of countable models of ZFC.