Meaning, Presuppositions, Truth-relevance, Godel's Theorem and
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From: Newberry on 11 Aug 2010 09:23 On Aug 11, 5:46 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > >On Aug 10, 8:48=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> If I axiomatize > >> computer programs and naturals, then there will be statements of > >> the form "Program P does not halt on input 0" that are not provable > >> and not disprovable. > > >Yet you are convinced that some of these unprovable sentences are > >true. I still do not understand how you arrive at the conclusion that > >something is true without deriving that it is true. > > Suppose I flip a coin and don't show you the result. Then you know > that *one* of the following statements is true: > > 1. The coin landed heads up. > 2. The coin landed tails up. > > You have no way of knowing which one is true, but you know > one of them is. > > The things that are provable are just different from the things > that are true. There is no reason to think that they should be > the same. It would be nice if they were, but why should you expect > it? If you do not show a coin you have blocked the flow of information. If something is true in a formal system how did the information get blocked? > > -- > Daryl McCullough > Ithaca, NY
From: Daryl McCullough on 11 Aug 2010 13:03 Newberry says... >If you do not show a coin you have blocked the flow of information. If >something is true in a formal system how did the information get >blocked? 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). -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 11 Aug 2010 14:17 Daryl McCullough wrote: > > Suppose I flip a coin and don't show you the result. Then you know > that *one* of the following statements is true: > > 1. The coin landed heads up. > 2. The coin landed tails up. > > You have no way of knowing which one is true, but you know > one of them is. Exactly! This, btw, firmly reveals the logic, the rationale, behind the relativity of mathematical reasoning! -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Newberry on 12 Aug 2010 00:40 On Aug 11, 10:03 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > >If you do not show a coin you have blocked the flow of information. If > >something is true in a formal system how did the information get > >blocked? > > 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. Secondly there are sentences, which we actually know that are true, such as that Goedel's formula is not derivable. 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? > > -- > Daryl McCullough > Ithaca, NY
From: Daryl McCullough on 12 Aug 2010 08:05
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. 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 |