Why a Norwegian mathematician proved that our concept of infinity might be flawed

Ideas & Philosophy · Published 3 October 2026 Swipe more like this →

In short: Thoralf Skolem was a quiet, modest researcher who fundamentally challenged how we understand set theory. By demonstrating that first-order formalizations of set theory must have countable models, he exposed a paradox suggesting that the very notion of uncountability might be more fragile than mathematicians previously assumed.

Born in 1887, Thoralf Skolem was a prolific Norwegian mathematician who published roughly 180 papers across fields like logic, set theory, and number theory. Despite his immense output, he often worked in relative isolation, publishing much of his research in Norwegian journals. This habit meant that many of his groundbreaking discoveries, such as the Skolem-Noether theorem, were later rediscovered by international peers who were unaware of his earlier work.

Skolem is perhaps best known for his contributions to model theory and his refinement of Zermelo's axiomatic set theory. In 1920, he expanded upon Leopold Löwenheim’s earlier research to formulate the Löwenheim-Skolem theorem. This result established that if a theory within first-order predicate calculus possesses a model, it must also possess a countable model. Skolem famously applied this to create what is now known as Skolem's paradox, which suggests that if set theory is consistent, it can be satisfied within a countable domain. This challenged the prevailing belief that the existence of uncountable sets was definitively proven by axioms.

Beyond his work on sets, Skolem was a pioneer in metalogic. In 1933 and 1934, he constructed a nonstandard model of arithmetic. This system included elements that behaved like infinite quantities and infinitesimals, effectively extending the traditional domain of nonnegative integers. By utilizing these methods, he sought to avoid the paradoxes of the infinite that arise when applying universal quantifiers to infinite classes. His approach was deeply skeptical of the idea that axioms alone could fully determine the nature of mathematical objects, arguing that the evidence for uncountable sets remained inconclusive.

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