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XOR’s Hammer

Some things in mathematical logic that I find interesting

Category: Provability Logic

Written by mkoconnorAugust 19, 2008August 23, 2008

Rosser’s Clever Improvement to Gödel’s Original First Incompleteness Theorem

Sometimes it’s the case that in a first-order system T in which you can do number theory there is a property P(n) of natural numbers such that the following two seemingly contradictory statements hold: For every n, T proves P(n) and T does not prove that for all n, P(n) To see that these statements […]

Written by mkoconnorAugust 13, 2008August 25, 2008

Löb’s Theorem: Santa Claus and Provability

Consider the following argument for the existence of Santa Claus (which is called Curry’s paradox): Let S be the sentence If S is true, then Santa Claus exists. Lemma: S is true. Proof. S is of the form “If P, then Q.” so to show S we just have to assume P and show Q. […]

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