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What does the downward Löwenheim–Skolem theorem assert fo... | QuestionStock

Representative image for question: What does the downward Löwenheim–Skolem theorem assert for countable first-order languages?

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What does the downward Löwenheim–Skolem theorem assert for countable first-order languages?

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What does the downward Löwenheim–Skolem theorem assert for countable first-order languages?

Options:

If a theory has any model then it has a finite model.
If a theory has an infinite model then it has a countable model.
Every countable theory is complete.
Any model can be expanded to an uncountable elementary extension.

Correct answer: If a theory has an infinite model then it has a countable model.

Explanation: Downward Löwenheim–Skolem (Löwenheim 1915; Skolem 1920s) states that an infinite model in a countable language has a countable elementary submodel. This gives countable models.

Created 2026-03-10T18:50:00.697Z. Updated 2026-03-10T18:50:00.697Z.