What David Teaches Goliath: An Inevitable Extension of ZFC
Abstract
We examine the metamathematical significance of the Levy scheme Λ, which asserts, for each natural number n, the existence of an n-Mahlo cardinal κ such that Vκ≺ΣnV. Several initially unrelated constructions single out ZFC+Λ: comparisons with the model theory of arithmetic and finite set theory; class theories in which the class of ordinals has the tree property; set theories with a proper class of indiscernibles; and interpretations of ordinary set theory in Cantorian parts of models of the extension \mathsf{NFUA} of Quine--Jensen set theory NFU. Taken together, the results surveyed here provide a precise sense in which ZFC+Λ is an inevitable extension of ZFC if one attempts to generalize certain metamathematical results about the microcosm of hereditarily finite sets to the entire macrocosm of sets.
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BibTeX
@misc{enayat2026what,
title = {{What David Teaches Goliath: An Inevitable Extension of ZFC}},
author = {Ali Enayat},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19283},
eprint = {2610.19283},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19283},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - What David Teaches Goliath: An Inevitable Extension of ZFC
AU - Ali Enayat
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.19283
UR - https://arxiv.org/abs/2610.19283
AB - We examine the metamathematical significance of the Levy scheme Λ, which asserts, for each natural number n, the existence of an n-Mahlo cardinal κ such that V_κ≺_Σ_nV. Several initially unrelated constructions single out ZFC+Λ: comparisons with the model theory of arithmetic and finite set theory; class theories in which the class of ordinals has the tree property; set theories with a proper class of indiscernibles; and interpretations of ordinary set theory in Cantorian parts of models of the extension \mathsf{NFUA} of Quine--Jensen set theory NFU. Taken together, the results surveyed here provide a precise sense in which ZFC+Λ is an inevitable extension of ZFC if one attempts to generalize certain metamathematical results about the microcosm of hereditarily finite sets to the entire macrocosm of sets.
ER -