Skolem's paradox

Lore & Background

Skolem's paradox stems from the Löwenheim–Skolem theorem, which shows that any consistent first-order theory of sets has a countable model. This appears to conflict with Cantor's theorem, which proves the existence of uncountable sets and is itself provable from the axioms. Skolem resolved the paradox in 1922 by explaining that countability is not absolute but relative to the model in which it is measured; a set may be countable in one model and uncountable in another, depending on whether an enumerating function exists within that model.

Reader's Guide

Skolem's paradox is significant because it exposes limitations of first-order logic in capturing set-theoretic concepts. It shows that first-order axioms cannot uniquely determine the cardinality of sets, leading to the notion of non-absoluteness. The paradox was harshly received by Ernst Zermelo, who argued against the limitations of first-order logic, but it was quickly accepted by the mathematical community. Philosophically, it raises questions about whether any first-order sentence truly states 'there are uncountable sets' and whether any set is uncountable in an absolute sense. Scholars like Hilary Putnam have applied the paradox and Skolem's relativity to the philosophy of language. The paradox is not a true antinomy like Russell's paradox, but it remains a key example of the interplay between model theory and set theory.

Did You Know?

• Skolem's paradox is not an actual antinomy like Russell's paradox, but was described as a 'paradoxical state of affairs' by Skolem.
• The paradox arises from the Löwenheim–Skolem theorem, which shows that any consistent first-order model of set theory has an equivalent countable model.
• Skolem resolved the paradox by concluding that countability is relative to the model, and that countable, first-order models are incomplete.
• Ernst Zermelo harshly received Skolem's work, arguing against the limitations of first-order logic and Skolem's notion of relativity.