The End of an Era - That of The Natural Numbers
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From: Nam Nguyen on 2 May 2010 03:29 William Hughes wrote: > On May 1, 5:12 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> William Hughes wrote: >>> On May 1, 3:15 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> William Hughes wrote: >>>>> On May 1, 2:21 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>> <snip> >>>>>> The definition of pGC was given before, in a high level as: >>>>>> pGC <-> "There are infinitely many examples of GC" >>>>>> From "there are infinitely many primes", how do you conclude >>>>>> pGC is true? >>>>> Clearly. I do not understand what is meant by >>>>> "There are infinitely many examples of GC" >>>>> I take it to mean there are infinitely many even >>>>> integers greater than 4 that are the sum of two primes. >>>>> This follows immediately from the fact that there are >>>>> an infinite number of primes (just add 3 to every prime, >>>>> if the prime is not 2 you get an x for which GC(x) is true). >>>> Ah, yes. It's trivial indeed as you said. (As long long as we don't >>>> in meta level conclude from "there are infinitely many primes" as >>>> a 1st order theorem to "pGC is true" as a [meta] true statement. >>>> We were talking about "decidable" which is of syntactical notion. >>>> I forgot that I had asked you about being "_true_ in the naturals". >>>> My bad!). >>>> OK then, let's get back to my question before this issue: >>>> NN asked: >>>> > So, for example, suppose cGC is provable, how would you demonstrate >>>> > (1) is decidable? >>>> to which WH responded: >>>> > pGC is provably true. >>>> My question is based entirely on syntactical notion, free of notion >>>> of truth of the naturals. Why does your answer have the word "true"? >>>> Iow, why does your answer have something to do with my question? >>> O.k, we lose true entirely, and take P is provable iff there >>> is a derivation whose last line is P. And P is decidable >>> iff P is provable or ~P is provable >> Ok. Let's loose "true", here. (My line of questions - here - was a bit >> sidetracked with your cross-path truth and provability). I think though >> you'd agreed with me that "provable" always means provable in a formal >> system T and that in this context T is supposed to be "as strong as >> arithmetic", which is an intuitive notion. (If you don't please let me >> know). >> >> >> >>> We start with "there are an infinite number of primes" >>> is provable, (This is certainly a reasonable assumption) from which it >>> follows that pGC is provable. >>> If cGC is provable then ~(1) is provable and (1) is decidable. >> Now then going back further to our original argument: >> >> NN [originally] stated: >> >> > 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) >> >> WH later responed: >> > However, I cannot see why you should think the fact that you >> > have "any intuition about the naturals" means that cGC is >> > undecidable. >> >> which NN asked: >> > but where did I say that? > > a rather important snip here > > 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)" > >> which WH answer: >> > Now apply the fact that (1) is decidable iff cGC >> > is decidable >> >> So, what does the fact that in T "If cGC is provable then ... (1) >> is decidable" have anything to do with what I originally stated as >> the first observation [above]? Can you elaborate on how that fact >> would relate to my observation, > > We now have > > (1) is decidable iff cGC is decidable. > pGC is true > pGC is provable > > We will assume that T is sound (another reasonable > and common assumption). Why do you think it's a reasonable assumption, when no one has been able to prove T is consistent? > Consider your statement > > "we can't know the arithmetic > truth or falsehood of (1)" > > However, if (1) is decidable we can know the arithmetic truth > or falsehood of (1). So if cGC is decidable then > (1) is decidable and we can know the arithmetic truth > or falsehood of (1). [If cGC not decidable, then (1) is not > decidable. It may still be possible to know the arithmetic truth > or falsehood of (1) but this seems unlikely.] Now, "cGC is > decidable" is a perfectly reasonable intuition, so if we > assume T is sound, observation 1 is false. > > >> especially, e.g., when being provable >> or decidable doesn't necessarily mean being true? > > If we do not assume T is sound, then there is no connection > between provable and true. Now we have to phrase things: > (1) is true iff cGC is false. However, "cGC is false" is a > perfectly reasonable intuition, so even if we don't assume > T is sound, observation 1 is false. > > - William Hughes
From: Nam Nguyen on 2 May 2010 03:39 William Hughes wrote: > > If we do not assume T is sound, then there is no connection > between provable and true. Now we have to phrase things: > (1) is true iff cGC is false. > However, "cGC is false" is a > perfectly reasonable intuition, so even if we don't assume > T is sound, observation 1 is false. I'm sorry: that simply doesn't cut it. Anyone else could equally say "cGC is true" is a perfectly reasonable intuition! So who would be correct: you or they? And why would that _intuition_ be correct at the expense of others?
From: William Hughes on 2 May 2010 07:42 On May 2, 4:39 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: > > > If we do not assume T is sound, then there is no connection > > between provable and true. Now we have to phrase things: > > (1) is true iff cGC is false. > > However, "cGC is false" is a > > perfectly reasonable intuition, so even if we don't assume > > T is sound, observation 1 is false. > > I'm sorry: that simply doesn't cut it. Anyone else could equally > say "cGC is true" is a perfectly reasonable intuition! So who > would be correct: you or they? And why would that _intuition_ > be correct at the expense of others? Your first observation is First observation: 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). It turns out that by "any intuition" you do not mean "there are a finite number of natural numbers" or "cGC is false" There are lots of other things that make your First observation false, e.g "cGC is true" or "we can know the truth or falsity of cGC" or "GC is true" So what do you mean by "any intuition"? -William Hughes
From: Daryl McCullough on 2 May 2010 08:17 William Hughes says... >Clearly. I do not understand what is meant by > > "There are infinitely many examples of GC" > >I take it to mean there are infinitely many even >integers greater than 4 that are the sum of two primes. >This follows immediately from the fact that there are >an infinite number of primes (just add 3 to every prime, >if the prime is not 2 you get an x for which GC(x) is true). >What do you mean by the statement? You're completely missing the point by bringing up actual mathematics. This is logic! -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 2 May 2010 11:35
William Hughes wrote: > On May 2, 4:39 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> William Hughes wrote: >> >>> If we do not assume T is sound, then there is no connection >>> between provable and true. Now we have to phrase things: >>> (1) is true iff cGC is false. >>> However, "cGC is false" is a >>> perfectly reasonable intuition, so even if we don't assume >>> T is sound, observation 1 is false. >> I'm sorry: that simply doesn't cut it. Anyone else could equally >> say "cGC is true" is a perfectly reasonable intuition! So who >> would be correct: you or they? And why would that _intuition_ >> be correct at the expense of others? > > > Your first observation is > > First observation: 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). > > It turns out that by "any intuition" you do not mean > > "there are a finite number of natural numbers" > or > "cGC is false" > > There are lots of other things that make your First observation > false, e.g > > "cGC is true" > or > "we can know the truth or falsity of cGC" > or > "GC is true" > > So what do you mean by "any intuition"? It's not the same situation: because a) I'm supposed to give some _explanation_ for my observation which I said I would and which I'm in the process of doing so (see my latest post in the conversation with WE); and so _you_ would have to _defend_ your counterpoint (at least making plan to defend it), and b) what you and I have to defend are different in nature (knowing vs. impossible to know). In addition I already gave a caveat: >>> It's not true such imprecise knowledge of the naturals would >>> mean we know nothing about them: we do know know some formula >>> being true. It's just that we wouldn't be able know the truth >>> status of all formulas! which means by standard acceptance of mathematicians and logicians the truth of SOME formulas should be in such knowledge, such as the non-induction axioms of PA, which means even by the vague notion of "intuition" in this context we don't have a carte-blanche to claim anything whatsoever. [Schoenfield's book implicitly suggests some formulas that it'd be questionable whether or not they'd be true in the (purported) natural numbers]. In summary, you and I each should give, as William Elliot suggested, "insights" to _defend_ our different positions on this issue about the knowledge of the naturals. And though "insight" is not a precise word, that doesn't at all mean unreasonable insights such as there are only finite number of the naturals could be used to further the arguments. |