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The Birth of Model Theory by Calixto Badesa

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Preface

The name “Löwenheim-Skolem theorem” is commonly given to a variety of results to the effect that if a set of formulas has a model of some (infinite) cardinality, it also has models of some other infinite cardinality. The first result of this type, proved by Löwenheim in “Über Möglichkeiten im Relativkalkül” [1915], asserts (though he put it rather differently) that if a first-order sentence has a model, then it has a countable model as well.

Because of the significance of this theorem, “Über Möglichkeiten im Relativkalkül” is mentioned in every history of logic; but the extraordinary historical interest of the paper does not reside in the theoretical importance of any of the results it contains (in the same paper Löwenheim also proved ...

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