Logic MT24, Löwenheim-Skolem theorem

Created: January 01, 2025 Read markdown About these notes View in context Study these flashcards

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@Define what it means for a structure to be countable.

countable-structure

It has a countable domain.

@State the weak downwards Löwenheim-Skolem theorem.

weak-downward-lowenheim-skolem

Every consistent set of sentences has a countable model.

Builds on ∆consistent-theory, ∆countable-structure, ∆satisfiable-theory

@Justify the weak downwards Löwenheim-Skolem theorem (∆weak-downward-lowenheim-skolem) using the proof of Gödel’s Completeness Theorem, i.e. that every consistent set of sentences has a countable model.

weak-downward-lowenheim-skolem-proof

The model used to show a consistent set of sentences is satisfiable (∆model-existence-theorem-proof) was the Henkin model, which was countable (its domain is made of equivalence classes of closed $\mathcal L'$-terms, and there are only countably many of those).

Builds on ∆weak-downward-lowenheim-skolem, ∆model-existence-theorem-proof, ∆henkin-construction



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