Elementary embedding

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 \((a_1,\ldots,a_n) \in N^n\).A. Kanamori, The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, Springer Monographs in Mathematics, 2008. 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 classesA. Kanamori, The Higher Infinite (p.44).
== 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 , 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).
==See also==
*Stable ordinal
*Rank-into-rank cardinal
==Sources==
Category:Set theory
Category:Model theory