Skolem's paradox

In mathematical logic, specifically model theory, Skolem's paradox is a direct result of the (downward) Löwenheim-Skolem theorem, which states that every model of a sentence of a first-order language has an elementarily equivalent countable submodel.

Related Topics:
Mathematical logic - Model theory - Löwenheim-Skolem theorem - Model - Elementarily equivalent - Countable

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The paradox is seen in Zermelo-Fraenkel set theory. One of the earliest results, published by Georg Cantor in 1874, was the existence of uncountable sets, such as the powerset of the natural numbers, the set of real numbers, and the well-known Cantor set. These sets exist in any Zermelo-Fraenkel universe, since they follow directly from the axioms. Using the Löwenheim-Skolem Theorem, we can get a model of set theory which only contains a countable number of objects. However, it must contain the fore-mentioned uncountable sets, which appears to be a contradiction. However, the sets in question are only uncountable in the sense that there does not exist within the model a bijection from the natural numbers onto the sets. It is entirely possible that there is a bijection outside the model.

Related Topics:
Paradox - Zermelo-Fraenkel set theory - Georg Cantor - 1874 - Uncountable - Powerset - Natural numbers - Real numbers - Cantor set - Axiom - Contradiction - Bijection

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