Elementary embedding

Elementary embedding

In model theory, an elementary embedding is a special case of an embedding that preserves all first-order formulas.

Definition

Given models M and N in the same language L, a function

: f:M o N

is called an elementary embedding if f(M) is an elementary substructure of N and f is a model isomorphism between M and f(M).

Elementary embeddings are the most important maps in model theory. Elementary embeddings whose domain is "V" (the universe of set theory) play an important role in the theory of large cardinals (see also critical point).


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