The End of an Era - That of The Natural Numbers
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From: Nam Nguyen on 26 Apr 2010 23:32 William Elliot wrote: > On Sun, 25 Apr 2010, Nam Nguyen wrote: >> >> 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. I've made corrections in my response to Phil Carmody. 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))] >> 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. I already explained that to you but you're not listening. That's NOT how/what my observation was stated. Read it carefully. What I said which I'm repeating here is: Nam said: >>> 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). That observation is of the form: If H then C. Does my H _really_ state "we don't know if (1) is true or false"? If I asked you to explain something you didn't argue, wouldn't that be ridiculous of me? Why have you kept doing that then? I already suggest to you what your request should be, but since you seem to have ignored, there's no need for me to respond further until you're precise on what I did or didn't state.
From: William Elliot on 27 Apr 2010 02:23 On Mon, 26 Apr 2010, Nam Nguyen wrote: >>> >The correct versions are: > > GC(x) <-> (x is even >=4) -> Ep1p2[(p1, p2 are primes) /\ (x=p1+p2)] By that definition, GC(pi) is true. The correct definition is: GC(x) <-> (x is even integer) & (x >= 4 -> Ep1p2[(p1, p2 are primes) /\ (x=p1+p2)]) > (*)P <-> Ex[P(x)] /\ AxEy[P(x) -> (P(y) /\ (x < y))] > >>>> 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). > Based upon that intuition, I'll concur that we don't know the truth or falsehood of (1). Based upon that intuition, you claim we'll never know the truth or falsehood of (1). That's an unsubstatiated claim. Give some insights why I should accept it.
From: Aatu Koskensilta on 27 Apr 2010 05:20 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > My observation is of the form: If H then C. What is it that you're > objecting? In your original post you described an arithmetical statement (1), related to the Goldbach conjecture, and went on to write: 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). My objection consists simply in the observation that this is not an observation at all, that is, it is not something we can find out by inspection, reflecting on (1), recalling some salient mathematical facts, but rather a somewhat baffling assertion for which you didn't offer any argument whatever; and further, that what we should take your purported observation to amount to is not clear, since in particular the notion of intuition you have in mind has not been spelled out. > To see why such knowledge is just an intuition, we can (in meta level) > equate the truth of any mathematical formula F in L(PA) to a syntactical > notion as: > > F is true <-> (PA isn't inconsistent) and (PA |- F) What does it mean for a piece of knowledge to be or fail to be "of intuitive nature"? How does the above stipulation -- taking here the charitable view that you intend the above equivalence to introduce non-standard technical terminology, not as a bizarre claim about truth of arithmetical statements in any usual sense -- help us to see why our "knowledge of the natural numbers is of intuitive nature"? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 27 Apr 2010 06:03 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > 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))] According to this definition "There are infinitely many examples of P" holds in case there is no x such that P(x). The standard formalization in the language of arithmetic of "There are infinitely many Ps", for an arithmetical property P, is: (x)(Ey)(x < y & P(y)) (And, as Phil notes, your formalization of "x is a counter-example to the Goldbach conjecture" wasn't quite right either...) > However there's reason why TF, e.g., coined the term "Goldbach-like" > statement and not "FLT-like" statement. The point being certain class > of formulas might have certain relevant foundational implication and > certain other classes might not. Sure, but the relevant class of statements -- Pi-0-1 -- includes both Fermat's last theorem and the Goldbach conjecture. In _G�del's Theorem_ Torkel presumably chose the Goldbach conjecture as the prototypical Pi-0-1 sentence because it can be directly and straightforwardly expressed in the language of arithmetic, while Fermat's last theorem and other famous arithmetical conjectures that are Pi-0-1 require a bit of coding, e.g. to express exponentiation. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 27 Apr 2010 23:44
Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> My observation is of the form: If H then C. What is it that you're >> objecting? > > In your original post you described an arithmetical statement (1), > related to the Goldbach conjecture, and went on to write: > > 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). > > My objection consists simply in the observation that this is not an > observation at all, that is, it is not something we can find out by > inspection, reflecting on (1), recalling some salient mathematical > facts, but rather a somewhat baffling assertion for which you didn't > offer any argument whatever; and further, that what we should take your > purported observation to amount to is not clear, since in particular the > notion of intuition you have in mind has not been spelled out. > >> To see why such knowledge is just an intuition, we can (in meta level) >> equate the truth of any mathematical formula F in L(PA) to a syntactical >> notion as: >> >> F is true <-> (PA isn't inconsistent) and (PA |- F) > > What does it mean for a piece of knowledge to be or fail to be "of > intuitive nature"? It's actually not that difficult as one might suspect. First, we'd define knowledge that are of non-intuitive nature, and then an intuitive knowledge is one that is not non-intuitive. A reasoning knowledge is of *non-intuitive* nature iff it's a knowledge of a syntactical proof. For instance, given the T = {Ax[x>=0]}, then the knowledge of the proof of (Ax[x>=0] \/ ~Ax[x>=0]) in T is a non-intuitive knowledge. An intuitive reasoning knowledge is one that is not non-intuitive. For example, the knowledge of the T above being consistent is an intuitive knowledge. > How does the above stipulation -- taking here the > charitable view that you intend the above equivalence to introduce > non-standard technical terminology, not as a bizarre claim about truth > of arithmetical statements in any usual sense -- help us to see why our > "knowledge of the natural numbers is of intuitive nature"? By the definition above of a formula F being true and by the fact that PA's perceived consistency is an intuitive knowledge, the knowledge of the naturals is an intuitive knowledge. Before I would continue about (1) and the observation in question, would you agree with me so far on what is or isn't of intuitive nature or on "the natural numbers is of intuitive nature"? If you don't agree, would you please kindly by technical reasons explain why. |