The End of an Era - That of The Natural Numbers
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From: Nam Nguyen on 4 May 2010 02:39 Nam Nguyen wrote: > Alan Smaill wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Also, I suppose many decades ago truth-equals-provability mathematics >>> was the "actual mathematics" of the time. A kind of "soup-du-jour >>> mathematics" isn't it? [Actually Shoenfield gave a hint this was (and >>> is) the case!] >> >> Given his acceptance of G�del's incompleteness theorem, >> your claim is implausible. >> >> Which "hint" of his do you have in mind? > > First, he said of his version of Robinson Arithmetic system (he > denoted by N) as "[a formal] system for the natural numbers." > (pg. 22). > > But then in a subsequent chapter he had: "The theory N is not a > satisfactory axiom system for studying natural numbers" (pg. 204). [Btw, my use of Shoenfield's "hint" was referring to "soup-du-jour mathematics", not to GIT in particular.]
From: Alan Smaill on 4 May 2010 04:58 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Alan Smaill wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Also, I suppose many decades ago truth-equals-provability mathematics >>> was the "actual mathematics" of the time. A kind of "soup-du-jour >>> mathematics" isn't it? [Actually Shoenfield gave a hint this was >>> (and is) the case!] >> >> Given his acceptance of G�del's incompleteness theorem, >> your claim is implausible. >> >> Which "hint" of his do you have in mind? > > First, he said of his version of Robinson Arithmetic system (he > denoted by N) as "[a formal] system for the natural numbers." > (pg. 22). > > But then in a subsequent chapter he had: "The theory N is not a > satisfactory axiom system for studying natural numbers" (pg. 204). But what on earth has this got to do with your claim that truth-equals-provability is the case today. Sure looks like he's saying that this proof system is too weak, in that there are obvious accepted mathematical statements that cannot be proved from this system. This is distinguishing between truth and provability, *not* identifying them. -- Alan Smaill
From: H.Y. ADDANDSTUFF on 4 May 2010 06:33 On May 4, 1:58 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > Nam Nguyen <namducngu...(a)shaw.ca> writes: > > Alan Smaill wrote: > >> Nam Nguyen <namducngu...(a)shaw.ca> writes: > > >>> Also, I suppose many decades ago truth-equals-provability mathematics > >>> was the "actual mathematics" of the time. A kind of "soup-du-jour > >>> mathematics" isn't it? [Actually Shoenfield gave a hint this was > >>> (and is) the case!] > > >> Given his acceptance of Gödel's incompleteness theorem, > >> your claim is implausible. > > >> Which "hint" of his do you have in mind? > > > First, he said of his version of Robinson Arithmetic system (he > > denoted by N) as "[a formal] system for the natural numbers." > > (pg. 22). > > > But then in a subsequent chapter he had: "The theory N is not a > > satisfactory axiom system for studying natural numbers" (pg. 204). > > But what on earth has this got to do with your claim that > truth-equals-provability is the case today. > > Sure looks like he's saying that this proof system is too weak, > in that there are obvious accepted mathematical statements that > cannot be proved from this system. This is distinguishing > between truth and provability, *not* identifying them. > > -- > Alan Smaill- Hide quoted text - > > - Show quoted text - If - this statement is used to open an exact proof p = np, if this is true or false, do this. = is equals, != equals not { is open function, } is close function //comments here and more there <!---// starts the script, //---> closes it. http://meami.org/
From: William Hughes on 4 May 2010 07:45 On May 4, 2:21 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: <snip> > > It is interesting to note that while you have been presented with > > many putative "intuitions" given which the truth or falsehood > > of (1) is knowable (you have accepted none and explicitly rejected > > one), you have not presented a single "intuition" under which the > > truth or falsehood of (1) is not knowable. > > That's correct: I have not - yet. That doesn't mean I'm not going to. I'm not holding my breath. - William Hughes
From: Nam Nguyen on 4 May 2010 08:30
William Hughes wrote: > On May 4, 2:21 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > > <snip> > > >>> It is interesting to note that while you have been presented with >>> many putative "intuitions" given which the truth or falsehood >>> of (1) is knowable (you have accepted none and explicitly rejected >>> one), you have not presented a single "intuition" under which the >>> truth or falsehood of (1) is not knowable. >> That's correct: I have not - yet. That doesn't mean I'm not going to. > > I'm not holding my breath. If you don't have a good faith on that then that's your issue and isn't my concern. [Btw, the post about imprecision in reasoning and the recent T post are part of the explanation. So in effect I've been doing the explanation, whether or not you're listening to.] |