Elementary Embeddings and Extensions

Origin. Tarski and Vaught, "Arithmetical extensions of relational systems" (1957), which introduced elementary substructures and the Tarski–Vaught test; Robinson's model-completeness and diagram method (1956); Frayne, Morel, and Scott (1962) for ultrapower embeddings. Structure preservation. Elementary equivalence. Chains and limits. Foundation of model-theoretic algebra.

Models. Maps preserving first-order truth. Elementary substructures. Chains with unions. Tarski-Vaught test. Model-theoretic constructions.

Formalism.

Elementary embedding: j: M → N elementary iff for all φ and ā ∈ M: M ⊨ φ(ā) iff N ⊨ φ(j(ā)). Preserves all first-order properties. Strong preservation.

Elementary equivalence: M ≡ N iff M, N satisfy same sentences. Same theory. May not be isomorphic. First-order indistinguishable.

Elementary substructure: M ≺ N: M ⊆ N and inclusion elementary. M ⊨ φ(ā) iff N ⊨ φ(ā) for ā ∈ M. Downward transfer.

Tarski-Vaught test: M ≺ N iff for all φ(x, ȳ) and ā ∈ M: if N ⊨ ∃x.φ(x, ā), then some b ∈ M: N ⊨ φ(b, ā). Witness in substructure. Practical criterion.

Elementary chain: M₀ ≺ M₁ ≺ M₂ ≺ ... Increasing chain. Union: M_ω = ⋃_n M_n. M_n ≺ M_ω for all n.

Löwenheim-Skolem: Downward: if M infinite and |M| > |L|, exists M' ≺ M with |M'| = |L|. Upward: if M infinite, for all κ ≥ |M|, exists M' ≻ M with |M'| = κ. Cardinality control.

Applications: Model existence. Omitting types. Prime models. Constructing models.

Symbols.

Symbol	Unicode	Meaning
≺	U+227A	elementary substructure
≡	U+2261	elementary equivalence
j: M → N	—	elementary embedding
Th(M)	—	theory of M

Metatheory. Elementary preservation. Chains. Löwenheim-Skolem. Tarski-Vaught.

Applies to. Model theory. Algebra. Set theory. Model constructions.

Limitations. First-order only. Large cardinals for stronger. Technical prerequisites. Abstract.