Meaning, Presuppositions, Truth-relevance, Godel's Theorem and
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From: Newberry on 12 Aug 2010 10:16 On Aug 12, 5:05 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > >On Aug 11, 10:03=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> Take an example: Every even number greater than 2 can be written as > >> the sum of two prime numbers. What is "blocking" me from knowing that > >> this is true? Well, there is nothing blocking me from checking each > >> even number, one at a time, to see that it can be written as the sum > >> of two primes. But there will never be a point at which I've checked > >> *all* of them. So if Goldbach's conjecture is *true*, I might never > >> know that it is true (because I can't check an infinite number of cases). > > >First of all this sounds as if mathematics were an experimental rather > >than deductive science, which itself makes this argument suspect. > > For a given axiom system, there are true universal statements that we > cannot prove, but we can "observe" one at a time. As someone else pointed > out, there is no statement that is absolutely unprovable, we may not > be able to prove it in PA, but perhaps in a stronger theory. > > >Secondly there are sentences, which we actually know that are true, > >such as that Goedel's formula is not derivable. > > It's not provable in PA. It's provable in stronger theories such > as ZFC. > > >I know that you will deny this. People either deny or confirm this > >depending on in which phase of the disputation they are. Let me just > >say that many people are unshakeably convinced that ZFC and PA are > >consistent. Where does this certainty come from? > > In the case of PA, we have a clear concept of the natural numbers, > and the axioms of PA are clearly tree of this conception. So you CAN derive that PA is consistent after all. > In the > case of ZFC, the cumulative hierarchy gives a pretty strong conception > of the universe of sets, but it's a little hazier (to me, anyway). > > -- > Daryl McCullough > Ithaca, NY
From: Daryl McCullough on 12 Aug 2010 11:28
Newberry says... > >On Aug 12, 5:05=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> In the case of PA, we have a clear concept of the natural numbers, >> and the axioms of PA are clearly tree of this conception. That's supposed to be "true" not "tree". >So you CAN derive that PA is consistent after all. Sure. It's provable model-theoretically. -- Daryl McCullough Ithaca, NY |