11,580
pages

Let $$L$$ be a formal language. An elementary embedding $$N \to M$$ for $$L$$-structures $$N$$ and $$M$$ is a homomorphism $$h \colon N \to M$$ of $$L$$-structures such that $$\phi(a_1,\ldots,a_n)$$ is true in $$N$$ if and only if $$\phi(h(a_1),\ldots,h(a_n))$$ is true in $$M$$ for any $$n \in \mathbb{N}$$, any $$L$$-formula $$\phi(x_1,\ldots,x_n)$$ with $$n$$ free variables $$x_1,\ldots,x_n$$, and any parameter $$(a_1,\ldots,a_n) \in N^n$$.[1] More generally, for a set $$\Sigma$$ of $$L$$-formulae, a $$\Sigma$$-elementary embedding $$N \to M$$ is a homomorphism $$h \colon N \to M$$ of $$L$$-structures such that $$\phi(a_1,\ldots,a_n)$$ is true in $$N$$ if and only if $$\phi(h(a_1),\ldots,h(a_n))$$ is true in $$M$$ for any any $$n \in \mathbb{N}$$, any $$L$$-formula $$\phi(x_1,\ldots,x_n) \in \Sigma$$ with $$n$$-free variables $$x_1,\ldots,x_n$$, and any parameter $$(a_1,\ldots,a_n) \in N^n$$. Kanamori has also formalized elementary embeddings for structures that are inner models based off of proper classes[2].

## Caution

This formulation of an elementary embedding makes sense only when the satisfaction of $$L$$-formulae in $$\Sigma$$ (with parameters in $$N$$) at $$N$$ and $$M$$ are formalised. For example, when $$L_\in$$ is the language of first order set theory, then the satisfaction of $$L_\in$$-formulae (without any restrictions) at $$V$$ is not formalisable in ZFC set theory as long as it is consistent by Tarski's undefinability theorem, and hence we need to be very careful when we consider an elementary embedding into $$V$$.

When we deal with an $$L_\in$$-structure which is a set, then the usual satisfaction relation works. When we deal with an $$L_\in$$-structure which is a definable class and consider a segment of Levy hierarchy, we can use Levy's truth predicate in order to formalise the satisfaction.

## Example

One of the simplest examples of an elementary embedding is the identity map $$\textrm{id}_N \colon N \to N$$. We say an elementary embedding is non-trivial if it is not an identity map.

## Elementary Substructure

If $$N$$ is an $$L$$-substructure of $$M$$, then the inclusion map $$N \hookrightarrow M$$ forms a homomorphism of $$L$$-structures. We say that $$N$$ is an elementary substructure (resp. $$\Sigma$$-elementary substructure) of $$M$$ if $$N$$ is an $$L$$-substructure of $$M$$ such that the inclusion map is an elementary embedding (resp. $$\Sigma$$-elementary embedding).