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The minimum size at which a (downward) Lowenheim-Skolem-type theorem applies in a logic is known as the Lowenheim number, and can be used to characterize that logic's strength.
Moreover, if we go beyond first-order logic, we must give up one of three things: countable compactness, the Downward Lowenheim-Skolem Theorem, or the properties of an abstract logic.[5]:134

https://en.wikipedia.org/wiki/L%C3%B6wenheim_number
Lowenheim number
In mathematical logic the Lowenheim number of an abstract logic is the smallest cardinal number for which a weak downward Lowenheim-Skolem theorem holds.[1] They are named after Leopold Lowenheim, who proved that these exist for a very broad class of logics.
Examples
・The Lowenheim-Skolem theorem shows that the Lowenheim-Skolem-Tarski number of first-order logic is ?0. This means, in particular, that if a sentence of first-order logic is satisfiable, then the sentence is satisfiable in a countable model.
・It is known that the Lowenheim-Skolem number of second-order logic is larger than the first measurable cardinal, if there is a measurable cardinal.[3] (And the same holds for its Hanf number.) The Lowenheim number of the universal (fragment of) second-order logic however is less than the first supercompact cardinal (assuming it exists).

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