Ehrenfeucht's lemma in set theory

TL;DR

This paper explores the validity of Ehrenfeucht's lemma in set theory models, showing it holds in models where V=HOD but can fail otherwise, and introduces generalized principles and algebraic analogues.

Contribution

It extends Ehrenfeucht's lemma to set theory, analyzes its limitations in models with V≠HOD, and introduces parametric and algebraic generalizations.

Findings

Ehrenfeucht's lemma holds in models with V=HOD.

The lemma can fail in models with V≠HOD and after Cohen forcing.

Algebraicity and definability do not always coincide in models of set theory.

Abstract

Ehrenfeucht's lemma (1973) asserts that whenever one element of a model of Peano arithmetic is definable from another, then 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 can fail, however, in models of set theory with $V\neq HOD$, and it necessarily fails in the forcing extension to add a generic 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 whenever an object $b$ is definable from some $a\in A$ using parameters in $P$, with $b\neq a$, then the types of $a$ and $b$ over $Q$ are different. 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 $a$ (as in Hamkins, Leahy arXiv:1305.5953). Ehrenfeucht's lemma holds for the ordinal-algebraic sets, we prove, if and only if the ordinal-algebraic and ordinal-definable sets coincide. Using similar analysis, we answer two open questions posed by Hamkins and Leahy, by 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.