The End of an Era - That of The Natural Numbers
in [Math]
|
From: William Hughes on 5 May 2010 00:18 On May 5, 12:56 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: > > On May 4, 11:36 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > >> *IF* GC is genuinely true > >> then intuitively it's impossible to know so. > > > I see this one of "all intuitions" but > > > If GC is genuinely true then intuitively > > this can be shown by induction > > > is not. > > Not sure I understand what you've said here. Are you saying > we can show GC true if it's true? No, this is not yet known, and may never be known. I am saying that my *intuition* is that if GC is true we can show that it is true. It is certainly true that for some P we can show that P(x) holds for every x [1] while for other P we cannot. Why do you rule that the intuition that GC belongs to the first group is not "any intuition"? - William Hughes [1] Formally, we do this by induction, but in most informal proofs (e.g. the vast, vast, majority of mathematical proofs) induction is not explicitly used.
From: Nam Nguyen on 5 May 2010 01:38 William Hughes wrote: > On May 5, 12:56 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> William Hughes wrote: >>> On May 4, 11:36 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> *IF* GC is genuinely true >>>> then intuitively it's impossible to know so. >>> I see this one of "all intuitions" but >>> If GC is genuinely true then intuitively >>> this can be shown by induction >>> is not. >> Not sure I understand what you've said here. Are you saying >> we can show GC true if it's true? > > No, this is not yet known, and may never be known. > > I am saying that my *intuition* is that if GC is > true we can show that it is true. Are you saying in the context of FOL reasoning, truth can be equated to provability? If not, what do you mean by "show" here? Beside, what's the difference between your "if GC is true we can show that it is true" and my "can show GC true if it's true" in my question to you? > > It is certainly true that for some P we > can show that P(x) holds for every x [1] > while for other P we cannot. > > Why do you rule that the intuition that > GC belongs to the first group is not "any intuition"? > > - William Hughes > > > [1] Formally, we do this by induction, but in most informal > proofs (e.g. the vast, vast, majority > of mathematical proofs) induction is not explicitly used.
From: William Hughes on 5 May 2010 07:36 On May 5, 2:38 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > what's the difference between > your "if GC is true we can show that it is true" and > my "can show GC true if it's true" in my question to you? Nothing. However note, I am not claiming that A: we can show GC true if it's true A is not yet known and may never be known. I am claiming that A is my *guess*. (In detail my guess is that T is sound and therefore something provable in T is true (although something true may not be provable in T) and that GC is provable in T) The question is not whether my guess is right or wrong, the question is whether my guess qualifies as an intuition. - William Hughes
From: Alan Smaill on 5 May 2010 07:54 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > William Hughes wrote: >> On May 4, 9:30 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>> 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.] >> >> Can give an intuition without using multiple >> long posts? Forget about Observation 1. >> Just give an example of an intuition >> Any intuition will do. > > OK. If you just want a short description intuition about (1) then here > it is. > > Intuitively, to see _either_ cGC or ~cGC as true or false, you have to > do the same impossible thing: transverse the entire set of natural numbers > to figure it out, hence (again intuitively) it's impossible to know the > truth value of cGC, hence of (1). > > [In contrast, intuitively it's not impossible to see ~GC as true since > a counter example is still a distinct possibility. So in principle, > we can't say it's impossible to know the truth value of GC, though > *IF* GC is genuinely true then intuitively it's impossible to know so.] > > And that is as short as I could put it. Do you also have the intuition that it is impossible to see that the associativity of addition holds for natural numbers, where addition is defined as usual for the recursive definition (0 case and successor case)? -- Alan Smaill
From: Nam Nguyen on 5 May 2010 22:38
Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> William Hughes wrote: >>> On May 4, 9:30 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> 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.] >>> Can give an intuition without using multiple >>> long posts? Forget about Observation 1. >>> Just give an example of an intuition >>> Any intuition will do. >> OK. If you just want a short description intuition about (1) then here >> it is. >> >> Intuitively, to see _either_ cGC or ~cGC as true or false, you have to >> do the same impossible thing: transverse the entire set of natural numbers >> to figure it out, hence (again intuitively) it's impossible to know the >> truth value of cGC, hence of (1). >> >> [In contrast, intuitively it's not impossible to see ~GC as true since >> a counter example is still a distinct possibility. So in principle, >> we can't say it's impossible to know the truth value of GC, though >> *IF* GC is genuinely true then intuitively it's impossible to know so.] >> >> And that is as short as I could put it. > > Do you also have the intuition that it is impossible to see that > the associativity of addition holds for natural numbers, > where addition is defined as usual for the recursive definition > (0 case and successor case)? I don't happen to have that intuition. But what would you mean by the "natural numbers"? Those in which cGC is true? Or ~cGC is true? |