Journal of Symbolic Logic

Non-well-foundedness of well-orderable power sets

T. E. Forster and J. K. Truss

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Abstract

Tarski [Tarski] showed that for any set X, its set w(X) of well-orderable subsets has cardinality strictly greater than that of X, even in the absence of the axiom of choice. We construct a Fraenkel-Mostowski model in which there is an infinite strictly descending sequence under the relation |w(X)| = |Y|. This contrasts with the corresponding situation for power sets, where use of Hartogs’ ℵ-function easily establishes that there can be no infinite descending sequence under the relation |𝒫(X)| = |Y|.

Article information

Source
J. Symbolic Logic, Volume 68, Issue 3 (2003), 879- 884.

Dates
First available in Project Euclid: 17 July 2003

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1058448446

Digital Object Identifier
doi:10.2178/jsl/1058448446

Mathematical Reviews number (MathSciNet)
MR2004e:03083

Zentralblatt MATH identifier
1055.03029

Subjects
Primary: 03E25: Axiom of choice and related propositions 03E10: Ordinal and cardinal numbers

Keywords
well-orderable well-founded cardinal number

Citation

Forster, T. E.; Truss, J. K. Non-well-foundedness of well-orderable power sets. J. Symbolic Logic 68 (2003), no. 3, 879-- 884. doi:10.2178/jsl/1058448446. https://projecteuclid.org/euclid.jsl/1058448446


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References

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