nLab Curry's paradox

Idea

In mathematical logic and mathematical foundations Curry’s paradox is a paradox which is a version of Russell's paradox that does not involve the use of negation.

Russell’s paradox is a proof of falsehood, rendering a certain class of formal systems (those including some kind of unlimited axiom of comprehension) “inconsistent” in the sense that they prove false, which by the law of ex falso quodlibet entails that they prove anything at all. Curry’s paradox is a direct proof of “anything at all” without the detour through falsehood. Thus, it applies more generally than Russell’s paradox, e.g. to theories lacking a notion of “false”, or to paraconsistent theories whose notion of “false” does not satisfy ex falso quodlibet.

Curry’s paradox does, however, still depend on the structural rule of contraction. Thus, it can be avoided by systems based on linear logic.

Argument

Let $P$ be any statement at all, and consider the set

$$C = \{ x \mid (x\in x) \Rightarrow P \}.$$

Then if $C\in C$, by definition we have $(C\in C) \Rightarrow P$, and hence by modus ponens we have $P$. Therefore, $(C\in C) \Rightarrow P$. But by definition this means that $C\in C$, and therefore (as we just proved) $P$.

Note that, if negation is defined as $\neg P = (P \Rightarrow \bot)$ for some notion of falsehood $\bot$, then Russell’s paradox is the special case of Curry’s paradox for $P=\bot$.

Related concepts

• axiom of full comprehension

References

In relation to the axiom of full comprehension:

• Frederic Fitch, A method for avoiding the Curry paradox, in Essays in Honor of Carl G. Hempel, Reidel, Dordrecht, Holland 1969, pp. 255–265 (doi:10.1007/978-94-017-1466-2)