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).
See also[]
Sources[]
- ↑ A. Kanamori, The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, Springer Monographs in Mathematics, 2008.
- ↑ A. Kanamori, The Higher Infinite (p.44)
Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation · Absolute infinity
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function) · \(\omega_1^\mathfrak{Ch}\) (Omega one of chess) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\gamma,\zeta,\Sigma\) (Infinite time Turing machine ordinals) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Hardy hierarchy · Slow-growing hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Weak Buchholz's function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Weak Buchholz's function · Extended Buchholz's function · Jäger-Buchholz function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function · Arai's psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal
Classes: \(\textrm{Card}\) · \(\textrm{On}\) · \(V\) · \(L\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)