Wide Sets, ZFCU, and the Iterative Conception

Journal of Philosophy 111 (2):57-83 (2014)
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Abstract

The iterative conception of set is typically considered to provide the intuitive underpinnings for ZFCU (ZFC+Urelements). It is an easy theorem of ZFCU that all sets have a definite cardinality. But the iterative conception seems to be entirely consistent with the existence of “wide” sets, sets (of, in particular, urelements) that are larger than any cardinal. This paper diagnoses the source of the apparent disconnect here and proposes modifications of the Replacement and Powerset axioms so as to allow for the existence of wide sets. Drawing upon Cantor’s notion of the absolute infinite, the paper argues that the modifications are warranted and preserve a robust iterative conception of set. The resulting theory is proved consistent relative to ZFC + “there exists an inaccessible cardinal number.”

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Christopher Menzel
Texas A&M University

Citations of this work

Modality and Paradox.Gabriel Uzquiano - 2015 - Philosophy Compass 10 (4):284-300.
A Purely Recombinatorial Puzzle.Fritz Peter - 2017 - Noûs 51 (3):547-564.
Maximality Principles in Set Theory.Luca Incurvati - 2017 - Philosophia Mathematica 25 (2):159-193.

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