Curry's Paradox - Naive Set Theory

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:

  1. Definition of X
  2. from 1
  3. from 2, contraction
  4. from 1
  5. from 3 and 4, modus ponens
  6. 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

Famous quotes containing the words naive, set and/or theory:

    “The naive notion that a mother naturally acquires the complex skills of childrearing simply because she has given birth now seems as absurd to me as enrolling in a nine-month class in composition and imagining that at the end of the course you are now prepared to begin writing War and Peace.”
    —Mary Kay Blakely (20th century)

    “A wise man should order his interests, and set them all in their proper places. This order is often troubled by greed, which puts us upon pursuing so many things at once that, in eagerness for matters of less consideration, we grasp at trifles, and let go things of greater value.”
    —François, Duc De La Rochefoucauld (1613–1680)

    “The theory of rights enables us to rise and overthrow obstacles, but not to found a strong and lasting accord between all the elements which compose the nation.”
    —Giuseppe Mazzini (1805–1872)