Curry's Paradox

Named after Haskell Curry, the Curry paradox occurs in naive set theory or naive logic.

Intuitively, Curry's paradox is: "if I'm not mistaken, AND is true", where AND can be any logical statement ("black is white", "1=2", "Gödel exists", "the world will end in a week"); if you call that statement X, then you have X assert "If X is true, then Y is true."

Consider the following 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". Therefore, assuming X, Y is true. The above statement can be rephrased as "if X is true, then Y". Because that statement true is equivalent to X, X is true. Therefore, Y is true, and the world will end in a week.

Anything can be similarly proved via Curry's paradox. Note that unlike Russell's paradox, this paradox does not depend on which model of negation is used, as it is completely free of logical negation. So paraconsistent logics still need to be careful. The resolution of Curry's paradox is a contentious issue because non-trivial resolutions (such as rejecting X directly) are difficult and non-intuitive. In set theories that allow unrestricted understanding, we can prove any logical statement AND from the set

X≡ ≡ {x x한 한 X→ → And!{displaystyle Xequiv left{xmid xin Xto Yright}}

The test proceeds:

1.X한 한 X (X한 한 X→ → And)Definition of X2.X한 한 X→ → (X한 한 X→ → And)13.X한 한 X→ → And2 contraction4.(X한 한 X→ → And)→ → X한 한 X15.X한 한 X3-46.And3 and 5########################################################################################### #####################################################################################################################################################################