Editing Curry's paradox (section)

=== Naive set theory ===

Even if the underlying mathematical logic does not admit any self-referential sentences, certain forms of naive set theory are still vulnerable to Curry's paradox. In set theories that allow [[Axiom schema of specification#Unrestricted comprehension|unrestricted comprehension]], we can prove any logical statement ''Y'' by examining the set
<math display="block">X \ \stackrel{\mathrm{def}}{=}\  \left\{ x \mid (x \in x) \to Y \right\}.</math>One then shows easily that the statement <math>X\in X</math> is equivalent to <math>(X\in X) \to Y</math>. From this, <math>Y</math> may be deduced, similarly to the proofs shown above. ("<math>X\in X</math>" stands for "this sentence".)

Therefore, in a consistent set theory, the set <math>\left\{ x \mid (x \in x) \to Y \right\}</math> does not exist for false ''Y''. This can be seen as a variant on [[Russell's paradox]], but is not identical. Some proposals for set theory have attempted to deal with Russell's paradox not by restricting the rule of comprehension, but by restricting the rules of logic so that it tolerates the contradictory nature of the set of all sets that are not members of themselves. The existence of proofs like the one above shows that such a task is not so simple, because at least one of the deduction rules used in the proof above must be omitted or restricted.