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 statement 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.
Note that unlike Russell's paradox, this paradox does not depend on what model of negation is used, as it is completely negation-free. Thus paraconsistent logics still need to take care.
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:
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