The End of an Era - That of The Natural Numbers
in [Math]
|
From: Nam Nguyen on 1 May 2010 11:55 William Hughes wrote: > On May 1, 12:11 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: >>> However, I cannot see why you should think the >>> fact that you have "any intuition about the >>> naturals" means that cGC is undecidable. > >> My memory might be bad, but where did I say that? > > You said > > "if we have any intuition about the naturals > then we'd also have the intuition that > we can't know the arithmetic > truth or falsehood of (1)" > > Now apply the fact that (1) is decidable iff cGC > is decidable So, for example, suppose cGC is provable, how would you demonstrate (1) is decidable?
From: William Hughes on 1 May 2010 11:59 On May 1, 12:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: > > On May 1, 12:11 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > William Hughes wrote: > >>> However, I cannot see why you should think the > >>> fact that you have "any intuition about the > >>> naturals" means that cGC is undecidable. > > >> My memory might be bad, but where did I say that? > > > You said > > > "if we have any intuition about the naturals > > then we'd also have the intuition that > > we can't know the arithmetic > > truth or falsehood of (1)" > > > Now apply the fact that (1) is decidable iff cGC > > is decidable > > So, for example, suppose cGC is provable, how would you demonstrate > (1) is decidable? pGC is provably true. So if cGC is provably true then (1) is provably flase and if cGC is provably false then (1) is provably true.
From: Nam Nguyen on 1 May 2010 12:08 Nam Nguyen wrote: > William Hughes wrote: >> On May 1, 12:11 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> > William Hughes wrote: >>>> However, I cannot see why you should think the >>>> fact that you have "any intuition about the >>>> naturals" means that cGC is undecidable. >> >>> My memory might be bad, but where did I say that? >> >> You said >> >> "if we have any intuition about the naturals >> then we'd also have the intuition that >> we can't know the arithmetic >> truth or falsehood of (1)" >> >> Now apply the fact that (1) is decidable iff cGC >> is decidable > > So, for example, suppose cGC is provable, how would you demonstrate > (1) is decidable? Assuming of course we're talking about (un)decidability in formal systems "as strong as arithmetic", such as Q, PA, PA+pGC, PA+pGC+(1), PA+pGC+~(1), etc...
From: Nam Nguyen on 1 May 2010 12:14 William Hughes wrote: > On May 1, 12:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> William Hughes wrote: >>> On May 1, 12:11 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> William Hughes wrote: >>>>> However, I cannot see why you should think the >>>>> fact that you have "any intuition about the >>>>> naturals" means that cGC is undecidable. >>>> My memory might be bad, but where did I say that? >>> You said >>> "if we have any intuition about the naturals >>> then we'd also have the intuition that >>> we can't know the arithmetic >>> truth or falsehood of (1)" >>> Now apply the fact that (1) is decidable iff cGC >>> is decidable >> So, for example, suppose cGC is provable, how would you demonstrate >> (1) is decidable? > > pGC is provably true. First, what would your definition of a formula being decidable be? (Note: you were using the term "decidable"). Secondly, can you prove pGC is _true_ in the naturals? > So if cGC is provably true then (1) > is provably flase and if cGC is provably false then (1) is > provably true. >
From: William Hughes on 1 May 2010 12:23
On May 1, 1:14 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: > > On May 1, 12:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> William Hughes wrote: > >>> On May 1, 12:11 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> William Hughes wrote: > >>>>> However, I cannot see why you should think the > >>>>> fact that you have "any intuition about the > >>>>> naturals" means that cGC is undecidable. > >>>> My memory might be bad, but where did I say that? > >>> You said > >>> "if we have any intuition about the naturals > >>> then we'd also have the intuition that > >>> we can't know the arithmetic > >>> truth or falsehood of (1)" > >>> Now apply the fact that (1) is decidable iff cGC > >>> is decidable > >> So, for example, suppose cGC is provable, how would you demonstrate > >> (1) is decidable? > > > pGC is provably true. > > First, what would your definition of a formula being decidable be? (Note: you > were using the term "decidable"). A formula P is decidable, iff P is provable or ~P is provable. > Secondly, can you prove pGC is _true_ in the > naturals? Yes, it follows almost trivially from the fact that there are infinitely many primes. > > > So if cGC is provably true then (1) > > is provably flase and if cGC is provably false then (1) is > > provably true. > > |