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	Name	Meaning
ā	U+0101	A with macron	A finite tuple of parameters from M
ȳ	U+0233	Y with macron	A tuple of variables
κ	U+03BA	Kappa	Target cardinality in upward Löwenheim–Skolem
φ	U+03C6	Phi	Metavariable over first-order formulas
ω	U+03C9	Omega	Length of an elementary chain
→	U+2192	Rightwards arrow	The map j: M → N
∃	U+2203	Existential quantifier	The witness clause of the Tarski–Vaught test
∈	U+2208	Element of	Membership of a parameter in M
≡	U+2261	Elementary equivalence	Same first-order theory
≥	U+2265	Greater than or equal	Cardinality bound κ ≥ \|M\|
≺	U+227A	Elementary substructure	M ≺ N: inclusion is elementary
≻	U+227B	Elementary extension	Converse of ≺
⊆	U+2286	Substructure	Inclusion, which alone does not give ≺
⊨	U+22A8	Satisfaction	M ⊨ φ(ā)
⋃	U+22C3	N-ary union	Union of an elementary chain
j	—	Elementary embedding	j: M → N preserving all first-order formulas
V	—	Set-theoretic universe	Source of j: V → M; no nontrivial j: V → V
M, N	—	Structures	Source and target of an embedding
Th(M)	—	Theory of M	The set of sentences true in M
crit(j)	—	Critical point	Least ordinal moved by j
M^I/U	—	Ultrapower	Reduced power by an ultrafilter U
\|M\|	—	Cardinality	Size of the universe of M

Metatheory. Tarski and Vaught (1957) supplied both working tools: the test that reduces M ≺ N to a witness condition on existential formulas, and the elementary chain theorem, that the union of an elementary chain is an elementary extension of every member — the latter is what makes transfinite constructions of models legitimate. Robinson's elementary diagram lemma turns embeddings into satisfaction: M embeds elementarily in N exactly when N expands to a model of the elementary diagram of M, which is how compactness produces embeddings. Łoś's theorem makes the canonical map into an ultrapower elementary, and the Keisler–Shelah theorem is the exact converse for elementary equivalence: M ≡ N iff M and N have isomorphic ultrapowers, proved by Keisler (1961) under GCH and by Shelah (1971) in ZFC. Robinson's model completeness is the statement that every embedding between models of T is elementary, which holds for algebraically closed and for real closed fields. In set theory, Scott (1961) showed a nontrivial j: V → M with critical point κ exists exactly when κ is measurable.

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

Limitations. Only first-order truth transfers, and that is much less than it sounds: a countable elementary submodel of the ordered field of real numbers is a real closed field but is not Dedekind complete, so completeness, well-foundedness, cardinality, and every other second-order property can differ across an elementary embedding. Substructure does not imply elementary substructure — ⊆ gives ≺ only when the theory is model complete — so the relation must be verified, and the Tarski–Vaught test is the only general way to do it. Elementary equivalence does not imply isomorphism, and the one general converse, Keisler–Shelah, recovers it only through ultrapowers whose index sets may be far larger than the structures. Upward Löwenheim–Skolem guarantees elementary extensions in every larger cardinality, so no infinite structure is pinned down and nonstandard models are unavoidable. In set theory the method has a ceiling: Kunen (1971) proved that in ZFC there is no nontrivial elementary j: V → V, so embedding-defined large cardinals stop below the Reinhardt level.