Notre Dame Journal of Formal Logic

Ehrenfeucht’s Lemma in Set Theory

Gunter Fuchs, Victoria Gitman, and Joel David Hamkins

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Ehrenfeucht’s lemma asserts that whenever one element of a model of Peano arithmetic is definable from another, they satisfy different types. We consider here the analogue of Ehrenfeucht’s lemma for models of set theory. The original argument applies directly to the ordinal-definable elements of any model of set theory, and, in particular, Ehrenfeucht’s lemma holds fully for models of set theory satisfying V=HOD. We show that the lemma fails in the forcing extension of the universe by adding a Cohen real. We go on to formulate a scheme of natural parametric generalizations of Ehrenfeucht’s lemma, namely, the principles of the form EL(A,P,Q), which asserts that P-definability from A implies Q-discernibility. We also consider various analogues of Ehrenfeucht’s lemma obtained by using algebraicity in place of definability, where a set b is algebraic in a if it is a member of a finite set definable from a. Ehrenfeucht’s lemma holds for the ordinal-algebraic sets, we prove, if and only if the ordinal-algebraic and ordinal-definable sets coincide. Using a similar analysis, we answer two open questions posed earlier by the third author and C. Leahy, showing that (i) algebraicity and definability need not coincide in models of set theory and (ii) the internal and external notions of being ordinal algebraic need not coincide.

Article information

Notre Dame J. Formal Logic, Volume 59, Number 3 (2018), 355-370.

Received: 8 January 2015
Accepted: 15 October 2015
First available in Project Euclid: 20 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E45: Inner models, including constructibility, ordinal definability, and core models
Secondary: 03E47: Other notions of set-theoretic definability 03C55: Set-theoretic model theory 03C62: Models of arithmetic and set theory [See also 03Hxx]

Ehrenfeucht’s lemma Leibniz–Mycielski axiom algebraicity ordinal definability


Fuchs, Gunter; Gitman, Victoria; Hamkins, Joel David. Ehrenfeucht’s Lemma in Set Theory. Notre Dame J. Formal Logic 59 (2018), no. 3, 355--370. doi:10.1215/00294527-2018-0007.

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