Christian_Szegedy comments on Deleting paradoxes with fuzzy logic - Less Wrong

6 [deleted] 11 August 2009 04:27AM

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Comment author: gjm 11 August 2009 04:33:53PM 3 points [-]

With that definition of truth, a Goedel sentence is not "true", because there are models in which it fails to hold; neither is its negation "true", because there are models in which it does. But that's not the only way in which the word "true" is used about mathematical statements (though perhaps it should be); many people are quite sure that (e.g.) a Goedel sentence for their favourite formalization of arithmetic is either true or false (and by the latter they mean not-true). There's plenty of reason to be skeptical about the sort of Platonism that would guarantee that every statement in the language of (say) Principia Mathematica or ZF is "really" true or false, but it hardly seems reasonable to declare it wrong by definition as you're doing here.

Comment author: Christian_Szegedy 11 August 2009 06:01:43PM 0 points [-]

You are right: you may come up with another consistent way of defining truth.

However, my comment was a reaction to silas's comment, in which he seemed to confuse the notion syntactic and semantic truth, taking provability as the primary criterion. I just pointed out that even undergraduate logic courses treat semantic truth as basis and syntactic truth enters the picture as a consequence.