Curry's Paradox

Paradox
Let $P$ be an arbitrary proposition.

Let a proposition $C$ be defined:
 * $C \implies P$

that $\neg C$.

Then $C \implies P$ is vacuously true.

By definition, $C$ is true, a contradiction.

This contradiction shows $C$ to be true.

By definition, $C \implies P$.

By Modus Ponendo Ponens, we conclude $P$, where $P$ is arbitrary.

But this means that any proof system expressing the above is inconsistent.

Also known as
Curry's Paradox can also be seen referred to as Löb's Paradox after.

Also see

 * Löb's Theorem