The End of an Era - That of The Natural Numbers
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From: William Elliot on 26 Apr 2010 05:00 On Sun, 25 Apr 2010, Nam Nguyen wrote: >>>>>>>>> It's widely believed our intuition of the natural numbers has >>>>>>>>> led to foundational understandings of mathematical reasoning and >>>>>>>>> not the least of which is the validity of GIT proof. In this >>>>>>>>> post, however, we'll demonstrate that if there's an intuition of >>>>>>>>> knowing the natural numbers, there's also an intuition about not >>>>>>>>> knowing them that would invalidate GIT proof. >>>>>>> >>>>>>> GIT = Godel Incompleteness Theorem (The 1st Theorem) >>>>>>> >>>>>>>>> Let F and F' be 2 formulas, we'll define the logical operator >>>>>>>>> xor as: F xor F' <-> (F \/ F') /\ ~(F /\ F') >>>>>>>>> >>>>>>>>> Let's now consider thew 2 formulas in L(PA): pGC ("pro GC") and >>>>>>>>> cGC("counter GC"): >>>>>>>>> >>>>>>> L(PA) = L(0,S,+,*,<) >>>>>> A first order language with equality, constant symbol 0, binary >>>>>> function constants +,* and binary relation constant < with Peano's >>>>>> axioms? >>>>>> >>>>>>> GC = Goldbach Conjecture >>>>>> Every even number greater than two is the sum of two primes? >>>>> >>>>>>>>> pGC <-> "There are infinitely many examples of GC" >>>>>>>>> cGC <-> "There are infinitely many counter examples of GC" >>>>>>>>> >>>>>>>>> Now, let's consider the following formula: >>>>>>>>> >>>>>>>>> (1) pGC xor cGC >>>>>>>>> >>>>>>>>> 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). >> >>> First of all, in arguing foundational issues we should be very precise >>> in what we say (and I think you'd agree with this). >> >> Indeed. What's a GC example? What's a GC counter example? > > GC(x) <-> (x is even >=4) /\ Ep1p2[(p1, p2 are primes) /\ (x=p1+p2)] > cGC(x) <-> ~GC(x) x is restriced to positive intergers. Let GC = { x | GC(x) }. (1) is GC is infinite xor N\GC is infinite. > and given a unary property P, the statement "There are infinitely many > examples of P", denoted say by *P, is defined as: > > *(P) <-> AxEy[P(x) -> ((x<y) /\ P(y))] > > then we'd now have: > > pGC <-> *(GC) > cGC <-> *(cGC) > >>> In summary, if you _precisely_ let me know >>> what I should substantiate then I'll try. >> >> I agree, we currently don't know the truth or falsehood of (1). > I'd agree. > >> From that known observation, show that we can't ever know >> the truth or falsehood of (1). > > But again, William, you weren't precise: was that what I really > had said? If you read my (first) observation carefully, you'd see > what I said basically is: if we only intuit the natural numbers > the way we currently do, then such intuition would never let us > to know the truth value of (1) be. And that's not the same as > saying something like "since we don't know something we can't > ever know that something". > I'm asking you to prove, in the meta language in which your observation is stated, that we don't know if (1) is true or false implies we can't know if (1) is true or false. -- >> In the year 1900, the truth or falsehood of FLT wasn't known. >> By using the method you use to substantiate that we'll never >> know the truth or falsehood of (1), people in 1900 would >> conclude that they can never know. Explain how your method >> and it's conclusion contradicts the now known proof of FLT. > > But this is a reasonable concern and I agree if not-knowing is part > of reasoning framework, care should be taken to make a distinction > between the 2 different cases: that you don't know something but > such something is possible to know, and that it's impossible to know > something because intrinsically it's impossible so within the > reasoning framework. > The difference is temporal. Simply prove one of these two. If time t = now and P(t), then for all time t, P(t). If it is now, it always was and forever shall be.
From: William Elliot on 26 Apr 2010 05:55 On Mon, 26 Apr 2010, Phil Carmody wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> William Elliot wrote: >>> Indeed. What's a GC example? What's a GC counter example? >> >> I hope you don't mind me saying that there's a distinction between >> assuming the trivial knowledge such as formalizing an example (or a >> counter example) of GC and not being precise in mis-recognizing, mis- >> interpreting what people stated or claimed. (No big deal really, >> but just so you know). Any rate, here are some of the relevant >> definitions: >> >> GC(x) <-> (x is even >=4) /\ Ep1p2[(p1, p2 are primes) /\ (x=p1+p2)] >> cGC(x) <-> ~GC(x) > > So clearly there are an infinitude of "counterexamples" to the > Goldbach conjecture. 1 is one. 3 is one. 5 is one too... > Oh my, I missed that. OP needs to carefully clarify the range of x. Since there are an infinite number of counter examples, if (1), then there can't be infinitely many examples, ie GC is false. (1) -> ~GC. >> and given a unary property P, the statement "There are infinitely many >> examples of P", denoted say by *P, is defined as: >> >> *(P) <-> AxEy[P(x) -> ((x<y) /\ P(y))] > > Let P(x) <-> (x is a positive integer) /\ (x is minus pi) > > Clearly *P holds, as P(x) never holds, and your implication is > always vacuously holds. I.e., using your logic, because there are > no examples of P, there are infinitely many examples P. > Such sloppiness by OP, not explicating the range of x. > I always try skip past your long drawn-out threads, and have never > spend enough time to ascertain whether you are a loon or not. I think > finally I have the evidence that you really don't have a clue what > you're talking about. > Yes, he's prone to long winded non-sequitures, known better as erudite distractions. > Which means, with no guilt or fear of missing anything useful, I never > have to see another post of yours again - *PLONK*. > No no, stick around for awhile until, as I've asked, he proves if now, forever shall be.
From: Tim Golden BandTech.com on 26 Apr 2010 09:37 On Apr 24, 1:12 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Aatu Koskensilta wrote: > > Nam Nguyen <namducngu...(a)shaw.ca> writes: > > >> 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). > > > This isn't an observation. It's a bald assertion that is, on the face of > > it, vague, implausible, and entirely arbitrary. > > For what it's worth, I wish Torkel Franzen were still with us so that > arguments about foundational issues of reasoning don't deteriorate > into Inquisition-like decrees, blasting the opponent' arguments without > due analysis of what he has repeatedly said with some technical details, > post after posts. > > The kind of decrees you have done to my arguments - posts after posts. > > I don't mind _if_ you point out precisely what's wrong with my arguments, > with the basis of my arguments. Many times in many threads you've only > made vague, subjective, decrees like what you've made above, without > due respect to the points, counter points I've made or raised, and then > kept silent. And then come back later to just make another similar "blasting" > ones on the same subjects, without any analysis at all. How frustrating > it is arguing with you! > > As much as I respect you knowledge on mathematical formalism, I wish that > when you don't substantially have anything to argue with me, you'd kind > of keep in mind Wittgenstein's wisdom: > > "Wovon man nicht sprechan kann, darüber muss man schweigen"! > > *** > In any rate, what are you clear, plausible, specific technical reasons > for your belief that the knowledge of the naturals aren't of intuitive > nature, or that you'd know the arithmetic truth or falsehood of (1)? > > Iow, what are your _technical, non-bias, objective_ grounds for attacking > my observation above? I'm not honestly able to follow your OP, but do take interest in the idea. I do not accept that the continuum is built out of a discrete basis, as the traditional number theory develops it. Instead, granting a continuum, it is much easier to yield the discrete. On the observed continuum we see no standard unity value; it is instead arbitrarily chosen. Near to the natural numbers are numbers with modulo behaviors, which are in some ways simplifications of the natural number, where the natural number arguably is within this class of the modulo family as modulo infinity. So we could start out considering the mod-1, then the mod-2, mod-3, ..., and wind up at the natural number eventually, its predecessors being simpler and so more fundamental. Rather, the components of the construction being more fundamental and yielding the family which contains simpler predecessors. Even a modern computer has an upper limit to how high it can count. Yes, its getting huge, but it will never be infinite. These lower mod systems when coupled with continuous magnitude generate raw support for spacetime (emergent spacetime) with unidirectional time: http://bandtechnology.com/PolySigned/index.html As far as I know you have the freedom to construct what you wish in mathematics. If others don't like then so it is. Getting them to understand what you are saying is more the point of the communication. I think you should take the constructional freedom, and if somebody isn't convinced of a step then it should shake out so that at least both people have expressed themselves as cleanly as possible, and perhaps so that others come to understand the two sides better. When I think of my 'intuition of the naturals' I see two things: 3 = 1 1 1 and 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 will get so large that I won't be sure how large it is, but I feel certain that it is large. The next step in technology is actually the usage of modulo symbolism within the number's representation, and so my argument on the modulo family may bear heavily to some formal mathematician. We are taught first to count and then to use the variable. Can it be any other way? The variable whose qualities are unknown does not seem valid. The variable whose qualities are limited does seem more valid. - Tim
From: William Elliot on 26 Apr 2010 23:02 On Mon, 26 Apr 2010, Tim Golden BandTech.com wrote: > ... within this class of the modulo family as modulo infinity. So we > could start out considering the mod-1, Which infinity? Cardinal infinity, ordinal infinity, countable infinity, uncountable infinity, the analytic infinity? What are the integers modulus infinity? Describe the integers modulus one.
From: Nam Nguyen on 26 Apr 2010 23:08
Phil Carmody wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> William Elliot wrote: >>> Indeed. What's a GC example? What's a GC counter example? >> I hope you don't mind me saying that there's a distinction between >> assuming the trivial knowledge such as formalizing an example (or a >> counter example) of GC and not being precise in mis-recognizing, mis- >> interpreting what people stated or claimed. (No big deal really, >> but just so you know). Any rate, here are some of the relevant >> definitions: >> >> GC(x) <-> (x is even >=4) /\ Ep1p2[(p1, p2 are primes) /\ (x=p1+p2)] >> cGC(x) <-> ~GC(x) > > So clearly there are an infinitude of "counterexamples" to the > Goldbach conjecture. 1 is one. 3 is one. 5 is one too... > >> and given a unary property P, the statement "There are infinitely many >> examples of P", denoted say by *P, is defined as: >> >> *(P) <-> AxEy[P(x) -> ((x<y) /\ P(y))] > > Let P(x) <-> (x is a positive integer) /\ (x is minus pi) > > Clearly *P holds, as P(x) never holds, and your implication is > always vacuously holds. I.e., using your logic, because there are > no examples of P, there are infinitely many examples P. So, there were a couple of technical errors [one on GC(x) and the other on *(P)]. The correct versions are: GC(x) <-> (x is even >=4) -> Ep1p2[(p1, p2 are primes) /\ (x=p1+p2)] (*)P <-> Ex[P(x)] /\ AxEy[P(x) -> (P(y) /\ (x < y))] Of course it's not a substitute for my acknowledgment of the errors; it was past 10:30 pm and it had been a long weekend for me. In any rate my whole argument stands unchanged after the correction. > > I always try skip past your long drawn-out threads, and have never > spend enough time to ascertain whether you are a loon or not. I think > finally I have the evidence that you really don't have a clue what > you're talking about. Geez. Are you calling Quinne a "loon" since he had a technical error in initially believing his ML theory were consistent? (Btw, have you ever had any minor technical error in your whole life?) > > Which means, with no guilt or fear of missing anything useful, I never > have to see another post of yours again - *PLONK*. Sure, if that's the excuse you had. If I were you, I'd move forward in argument after Nam's corrections, and not stop at that kind of "victory"? Of course you might have a different opinion. |