Curry's paradoxNamed after Haskell Curry, Curry's paradox occurs in naive set theory or naive logics.
Intuitively, Curry's paradox is: "If I'm not mistaken, Y is true", where Y can be any logical statement ("black is white", "1=2", "Gödel exists", "the world will end in a week")
If we call that statement X, then we have that X asserts "If X is true, then Y is true."
Consider the statement X "If this statement is true, the world will end in a week," which will be abbreviated as "If X is true, then Y" (For a more rigorous phrasing of self-reference, see Quine.) Therefore, assuming X, Y is true.
The previous statment can be rephrased to "If X is true, then Y". Because that true statement is equivalent to X, X is true. Therefore, Y is true, and the world will end in a week. Anything else can similarly be "proven" via Curry's paradox.
The resolution of Curry's paradox is a contentous issue because nontrivial resolutions (such as disallowing X directly) are difficult and not intuitive.
In set theories which allow unrestricted comprehension, we can prove any logical statement Y from the set
The proof proceeds: