Wide Sets, ZFCU, and the Iterative Conception

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


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.”



External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

The iterative conception of set.Thomas Forster - 2008 - Review of Symbolic Logic 1 (1):97-110.
The Iterative Conception of Set: a (Bi-)Modal Axiomatisation.J. P. Studd - 2013 - Journal of Philosophical Logic 42 (5):1-29.
On the iterative explanation of the paradoxes.Christopher Menzel - 1986 - Philosophical Studies 49 (1):37 - 61.
Categoricity theorems and conceptions of set.Gabriel Uzquiano - 2002 - Journal of Philosophical Logic 31 (2):181-196.
Proper classes via the iterative conception of set.Mark F. Sharlow - 1987 - Journal of Symbolic Logic 52 (3):636-650.
Iterative set theory.M. D. Potter - 1994 - Philosophical Quarterly 44 (171):178-193.
Plural Quantification and the Iterative Concept of Set.Stephen Pollard - 1985 - Philosophy Research Archives 11:579-587.
How to be a minimalist about sets.Luca Incurvati - 2012 - Philosophical Studies 159 (1):69-87.
Boolos on the justification of set theory.Alexander Paseau - 2007 - Philosophia Mathematica 15 (1):30-53.
New V, ZF and Abstraction.Stewart Shapiro & Alan Weir - 1999 - Philosophia Mathematica 7 (3):293-321.
Logically possible machines.Eric Steinhart - 2002 - Minds and Machines 12 (2):259-280.
The Graph Conception of Set.Luca Incurvati - 2014 - Journal of Philosophical Logic 43 (1):181-208.
The dense linear ordering principle.David Pincus - 1997 - Journal of Symbolic Logic 62 (2):438-456.


Added to PP

1,332 (#8,084)

6 months
109 (#31,679)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Christopher Menzel
Texas A&M University

Citations of this work

Pure Logic and Higher-order Metaphysics.Christopher Menzel - 2023 - In Peter Fritz & Nicholas K. Jones (eds.), Higher-order Metaphysics. Oxford University Press.
A Purely Recombinatorial Puzzle.Fritz Peter - 2017 - Noûs 51 (3):547-564.
Modality and Paradox.Gabriel Uzquiano - 2015 - Philosophy Compass 10 (4):284-300.

View all 15 citations / Add more citations

References found in this work

No references found.

Add more references