'When Are Relations Neither True Nor False?
in [Math]
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From: Marshall on 3 Apr 2010 19:20 On Apr 2, 5:32 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > Let me put to rest the idea we know enough about the natural numbers, > to prove important thing such as the consistency of PA. I'll do that > by pointing out the existence of a specific unknown natural number. Why do you think the existence of a specific unknown number should have anything to do with consistency? Out of curiosity (not that you've even answered this question in the past) what sort of thing *would* you accept as a proof of consistency? Marshall
From: Nam Nguyen on 4 Apr 2010 12:33 Marshall wrote: > On Apr 2, 5:32 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Let me put to rest the idea we know enough about the natural numbers, >> to prove important thing such as the consistency of PA. I'll do that >> by pointing out the existence of a specific unknown natural number. > > Why do you think the existence of a specific unknown number > should have anything to do with consistency? Because they (the syntactical proof of consistency and collectively many formulas about this unknown natural) both connote the same thing in meta level: impossibility of syntactical proof. If you can't prove a certain formula related to this number, you can forget about proving a consistency, syntactically speaking. [Imho, it could be said the the later epitomizes the impossibility of the former]. > > Out of curiosity (not that you've even answered this question > in the past) what sort of thing *would* you accept as a > proof of consistency? It's possible that I missed your _specific_ question of the past. You just have to cite the specific post where I missed your "this question", otherwise it's impossible for me to make any comment on this. That aside, it's actually my position that it's impossible to to syntactically prove a consistency: simply because the rules of inference won't let us do that; hence it's a _delusion_ that we could have any "sort of thing" that we could "accept as a proof of consistency"! [That's why I'd would be surprised if in the past I had said something that has caused you to think there be a criteria to accept a proof of inconsistency].
From: Nam Nguyen on 4 Apr 2010 16:20 Nam Nguyen wrote: > Marshall wrote: >> Why do you think the existence of a specific unknown number >> should have anything to do with consistency? > > Because they (the syntactical proof of consistency and collectively > many formulas about this unknown natural) both connote the same thing > in meta level: impossibility of syntactical proof. If you can't prove > a certain formula related to this number, you can forget about proving > a consistency, syntactically speaking. > > [Imho, it could be said the the later epitomizes the impossibility of > the former]. Perhaps I also meant the other way around.
From: Newberry on 4 Apr 2010 20:35 On Apr 3, 9:51 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > > > > > >On Apr 3, 6:54=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >> Newberry says... > > >> >If it absolutely certain that PA is consistent why don't we formalize > >> >the reasoning? > > >> It has been. It's easily formalized in ZFC. > > >I do not know why we are going through this circle again. > > >Look it is very simple. All you have to do is to divorce > > >~(Ex)(Ey)(Pxy & Qy) (1) > > >from > > >~(Ex)Pxm (2) > > >[No need to repeat that m is the Goedel number of (1).] Then there is > >no reason why (2) could not be proven. > > It can be proven. Just not in PA. Cool. So we know that the search for a proof of Goedel's sentence will never terminate. Can we apply this knowledge to Diophantine equations? > -- > Daryl McCullough > Ithaca, NY- Hide quoted text - > > - Show quoted text -
From: Nam Nguyen on 4 Apr 2010 21:12
Newberry wrote: > On Apr 3, 9:51 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> It can be proven. Just not in PA. > > Cool. So we know that the search for a proof of Goedel's sentence will > never terminate. Can we apply this knowledge to Diophantine equations? I don't know the exact wording but there's a saying in mathematics that we could not go backward forever in proofs: things have to _stop_ somewhere such as reasoning framework, axioms, etc... I don't think the "standard theorists" would insist on not stopping. Unfortunately by not being humble on what they can possibly know in reasoning, they inadvertently allow themselves defend-less against infinite regression of truth and provability. It wouldn't be a surprise if we learn there were those who would defend physics against SR to the bitter end. I think there are those today who'd similarly defend "the natural numbers" foundation in mathematical logic - to the bitter end - even though to no avail. Courageous perhaps. But kind of a "sad" story. |