Math 2.0

Mathematical logic and foundations First-order logic Theorem

Löwenheim–Skolem theorem

Builds on SignatureStructureElementary substructureCardinal number

Statement

Let σ\sigma be a signature, and let κ=max⁡(∣σ∣,ℵ0)\kappa = \max(\abs{\sigma}, \aleph_0). Let MM be an infinite σ\sigma-structure, and let λ\lambda be an infinite cardinal.

  1. Downward: if κ≤λ≤∣M∣\kappa \le \lambda \le \abs{M} and A⊆MA \subseteq M has ∣A∣≤λ\abs{A} \le \lambda, then MM has an elementary substructure NN with A⊆NA \subseteq N and ∣N∣=λ\abs{N} = \lambda.
  2. Upward: if λ≥max⁡(∣M∣,κ)\lambda \ge \max(\abs{M}, \kappa), then MM has an elementary extension of cardinality λ\lambda.

Remarks

The downward part closes AA under Skolem functions, which pick witnesses for existential formulas, and uses the Tarski–Vaught test. The upward part adds λ\lambda new constants that must be pairwise distinct, uses the compactness theorem, and then applies the downward part. For example, the field ℝ\R has a countable elementary substructure: the real algebraic numbers form one, by quantifier elimination for real closed fields, the Tarski–Seidenberg theorem. The hypothesis that MM is infinite is needed: a finite structure has no elementary extension of a different size. As a consequence, a countable first-order theory with an infinite model has models of every infinite cardinality. So no first-order theory with an infinite model determines its models up to isomorphism; Skolem's paradox is one effect of this. In Mathlib the downward part is FirstOrder.Language.exists_elementarySubstructure_card_eq.

Going deeper

For a student

The theorem is a basic tool of model theory. It shows that categoricity in all infinite cardinalities is impossible, so model theory studies categoricity in one cardinality, as in Morley's categoricity theorem. The Łoś–Vaught test combines it with categoricity in one cardinality to prove that a theory is complete. Second-order logic has no such theorem, since the second-order Peano axioms determine ℕ\N up to isomorphism.

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Cardinal number, concept, no evidence levelCardinal numberconceptElementary substructure, concept, no evidence levelElementary substructu…conceptSignature, concept, no evidence levelSignatureconceptStructure, concept, no evidence levelStructureconceptLöwenheim–Skolem theorem, statement, no evidence levelLöwenheim–Skolem theo…statement
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  1. Cardinal number concept
  2. Elementary substructure concept
  3. Signature concept
  4. Structure concept
  5. Löwenheim–Skolem theorem statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Löwenheim–Skolem_theorem
About this page
Tag484M0TZ1S7
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Kindstatement, theorem
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UsesSignature, Structure, Elementary substructure, Cardinal number
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Changed2 November 2026

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