Meaning, Presuppositions, Truth-relevance, Gödel's Theorem and the Liar Paradox
in [Math]
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From: Nam Nguyen on 9 Aug 2010 11:16 Daryl McCullough wrote: > In article <3f8cee92-6dbb-4f44-beb7-0878d02b9b8d(a)p11g2000prf.googlegroups.com>, > Newberry says... >> On Aug 9, 6:33=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >>> Newberry says... >>> >>>> The question was what is Goedel sentence. The formula I exhibited is >>>> Goedel's formula in many kinds of logic including PA. >>> If it is sufficiently similar to the Godel formula for PA, then it >>> is nonsensical to say that it is neither true nor false. >> Would you care to define "sufficiently similar" and show how your >> conclusion follows? > > The main ideas behind Godel's proof is > 1. Invent a coding for formulas so that every formula is associated > with a natural number (or an element of whatever the domain of the > theory is about) > 2. Define a formula Pr(x) such that Pr(x) holds of a natural number > x if and only if x is the code of a provable formula of whatever theory > we are talking about. So GIT is a meta theorem, NOT a FOL [syntactical] theorem, and would require our intuitive knowledge of the truths about the natural numbers. Right? > 3. Construct a sentence G such that G <-> ~Pr(#G) is a theorem, > where #G means the code for G. > > 1-3 is what I consider the essential features of what it means > for G to be a "Godel sentence". There a few details that can be > tweaked---for instance, 3 presupposes that there are constant > terms (e.g. numerals) for each element of the domain. That's > not essential; instead, we can have a formula Q(x) such > that > > G <-> Ax (Q(x) -> ~Pr(x)) > > and such that Q(x) holds if and only if x is the code for G. > > Anyway, in terms of 1-3, it is nonsensical to say that G is > neither true nor false. G is a specific formula. If that formula > is provable, then Pr(#G) holds (by definition, Pr(#G) holds > if G is provable). But G is the negation of that formula. So > G is the negation of a true sentence, and so is a false sentence. > > So if you say that G is not false, then it follows that G is > not provable, and from that it follows that ~Pr(#G) is true, > and from that, it follows that G is true. > > -- > Daryl McCullough > Ithaca, NY > -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt ----------------------------------------------------------- |