Journal of Symbolic Logic

Non-well-foundedness of well-orderable power sets

T. E. Forster and J. K. Truss

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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

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

First available in Project Euclid: 17 July 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

well-orderable well-founded cardinal number


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.

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