Notre Dame Journal of Formal Logic

Skolem Redux

W. D. Hart

Abstract

Hume's Principle requires the existence of the finite cardinals and their cardinal, but these are the only cardinals the Principle requires. Were the Principle an analysis of the concept of cardinal number, it would already be peculiar that it requires the existence of any cardinals; an analysis of bachelor is not expected to yield unmarried men. But that it requires the existence of some cardinals, the countable ones, but not others, the uncountable, makes it seem invidious; it is as if an analysis of people required that there be men but not women, or whites but not blacks. If we deprive the Principle of existential commitments, it will cease to yield Dedekind's axioms for the natural numbers and so fail a good test of material adequacy. But since there are cardinals no second-order theory guarantees, neither can the Principle be beefed up to require all cardinals.

Article information

Source
Notre Dame J. Formal Logic, Volume 41, Number 4 (2000), 399-414.

Dates
First available in Project Euclid: 26 November 2002

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1038336883

Digital Object Identifier
doi:10.1305/ndjfl/1038336883

Mathematical Reviews number (MathSciNet)
MR1963489

Zentralblatt MATH identifier
1014.03503

Subjects
Primary: 00A30: Philosophy of mathematics [See also 03A05]
Secondary: 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30} 03E55: Large cardinals

Keywords
cardinal number Hume's Principle Skolem's Paradox

Citation

Hart, W. D. Skolem Redux. Notre Dame J. Formal Logic 41 (2000), no. 4, 399--414. doi:10.1305/ndjfl/1038336883. https://projecteuclid.org/euclid.ndjfl/1038336883


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