Naive Set Theory
Even if the underlying mathematical logic does not admit any self-referential sentence, in set theories which allow unrestricted comprehension, we can nevertheless prove any logical statement Y by examining the set
The proof proceeds as follows:
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- Definition of X
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- from 1
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- from 2, contraction
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- from 1
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- from 3 and 4, modus ponens
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- from 3 and 5, modus ponens
Therefore, in a consistent set theory, the set 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.
Read more about this topic: Curry's Paradox
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