Theorem of Löwenheim-Skolem

The theorem of Löwenheim-Skolem belongs to the Théorie of the models. Its simplicity and its power make of it a major theorem - with the Théorème of compactness.

Theorem

That is to say T a first order theory.

; Statement

If T admits an infinite model, or arbitrarily large finished models, she admits a model of any cardinal larger than that of T .

In particular, when T is finiment axiomatisable , and admits an infinite model, it admits a countable model .

; Proof

With arbitrarily large finished models, one can add, with the theory, constants

c_i

two to two distinctesc_i \ neq c_j-->. Very finished part of the theory admits a model; by compactness, one obtains an infinite model.

Alternative

If the model is infinite, the theorem of Löwenheim-Skolem makes it possible to increase its cardinal with any higher cardinal.

Corollaries

the theorem of Löwenheim-Skolem makes it possible for example to show that the Logique first order is strictly lower than that of the second order.

If one applies the theorem to the set theory ZFC, or with another axiomatic theory intended to found the theorems of Cantor, one obtains a countable universe of all the units defined in ZFC. But one can prove in ZFC which there exist indénombrables units. In other words ZFC affirms that there exist more units than it cannot about it define: it is the Paradoxe of Skolem.

See too

Biographies:

Leopold Löwenheim .
Thoralf Skolem .

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