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Iterating logarithms and well-foundedness
Abstract
We prove that there is no sequence { Xn }n∈ω such that for all n, there is an injection from 𝒫(Xn+1) to Xn 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(·,·) such that for each n∈ω there are X and Y with Φ(n,X), Φ(n+1,Y), and 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.
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BibTeX
@misc{figueroa2026iterating,
title = {{Iterating logarithms and well-foundedness}},
author = {Gervasio Figueroa and Joel Kuperman and Pedro Sánchez Terraf},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04963},
eprint = {2611.04963},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04963},
}
RIS
TY - UNPB
TI - Iterating logarithms and well-foundedness
AU - Gervasio Figueroa
AU - Joel Kuperman
AU - Pedro Sánchez Terraf
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04963
UR - https://arxiv.org/abs/2611.04963
AB - We prove that there is no sequence { X_n }_n∈ω such that for all n, there is an injection from 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(·,·) such that for each n∈ω there are X and Y with Φ(n,X), Φ(n+1,Y), and 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.
ER -