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Chapter Six

Löwenheim's Theorem

6.1 THE PROBLEM

6.1.1 Today, we use the name Löwenheim-Skolem theorem to refer to all the theorems that guarantee that if a set of formulas has a model of a particular cardinality, it also has a model of some other cardinality. The first theorem in the series (leaving open the question of whether it is also the weakest) is the second of the theorems that Löwenheim proves in “Über Möglichkeiten;” it states:

If the domain is at least denumerably infinite, it is no longer the case that a first-order fleeing equation (Fluchtz ählgleichung) is satisfied for arbitrary values of the relative coefficients. (“Über Möglichkeiten,” p. 450 (235))

If we make explicit the definition of fleeing equation in terms of validity ...

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