# Paradoxes of set and property

DOI
10.4324/9780415249126-Y024-1
DOI: 10.4324/9780415249126-Y024-1
Version: v1,  Published online: 1998
Retrieved September 21, 2020, from https://www.rep.routledge.com/articles/thematic/paradoxes-of-set-and-property/v-1

## 10. Zermelo, von Neumann and Gödel: turning paradoxes into theorems

From a mathematical perspective, what is most striking is how paradoxes have repeatedly been turned into theorems. We consider four examples.

For Zermelo, Russell’s paradox became the theorem of his axiomatic set theory that there is no set of all sets (1908).

For von Neumann, the paradox of the largest ordinal became the theorem of his axiomatic set theory (1925) that every set can be well-ordered. The proof was based on the axiom that every proper class (and no set) can be mapped one-to-one onto the class of all sets. Thus proper classes, which were ‘too big’ to be sets, were rehabilitated by no longer being considered to be contradictory, as they had been for Cantor and Zermelo. In effect, von Neumann made precise Cantor’s distinction between consistent and inconsistent collections.

Gödel’s incompleteness theorem, which states that any sufficiently rich formal system contains propositions that cannot be proved or refuted in the system (see Gödel’s theorems), was obtained, in part, by applying Richard’s paradox to the notion of provability (1931).

Finally, Tarski’s theorem on the indefinability of truth (1933) had a close connection with the liar paradox. Tarski saw the universality of natural languages as the source of the semantic paradoxes, since this universality allows truth to be defined in such languages (see Tarski’s definition of truth).