The End of an Era - That of The Natural Numbers
in [Math]
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From: Nam Nguyen on 24 Apr 2010 01:12 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(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, dar�ber 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?
From: William Elliot on 24 Apr 2010 01:59 On Fri, 23 Apr 2010, Nam Nguyen wrote: > Aatu Koskensilta wrote: >> Nam Nguyen <namducnguyen(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. > I congratulate you for your excellently prolonged ad homenim rampage. > The kind of decrees you have done to my arguments - posts after posts. > I notice you weren't able to refute my rebuttal of your esteemed fantasy. > 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, dar�ber 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? >
From: Nam Nguyen on 24 Apr 2010 02:08 William Elliot wrote: > On Thu, 22 Apr 2010, Nam Nguyen wrote: > >> William Elliot wrote: >>> On Wed, 21 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? Right. That's why it has a name "L(PA)"! > >> GC = Goldbach Conjecture >> > Every even number greater than two is the sum of two primes? Your guess is correct! > >>>> 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). >>>> > Currently we do not know, nor do we know if GC is independent of PA. > In addition there's no proof that we can't know. > Indeed, your > fallacious argument could have been made a hundred years ago about > FCT or FLT which we couldn't know then, but do now. Perhaps you'd want to reflect a bit on what TF said below, before making the judgment above: "So for any formal system S which incorporates a bit of arithmetic - the basic rules needed to carry out computations - a Goldbach-like statement is disprovable in S if false. On the other hand, we can make no similar observation about how a Goldbach-like statement can be proved if it is true." > > If (1) is false, then either there's finite many examples of GC and > finite many counter examples of GC or there's infinitely many examples > of GC and infinitely many counter examples of GC. First of all what you said here isn't correct: given that we're supposed to have infinitely many even numbers, it's impossible that "there's finite many examples of GC and finite many counter examples of GC". Secondly, _what kind of natural numbers_ should we have for (1) to be true, or false? > >>>> Second observation: as a consequent of the 1st observation, no >>>> extension of Q can prove nor disprove (1) unless it's inconsistent. >>>> > Indeed, falsehood implies any and every thing. > >>>> Third observation, as a consequence of the 2nd observation, _if_ a >>>> formal system T is consistent and capable of expressing arithmetic >>>> then (1) is undecidable in T. Hence G(T) is no longer necessary in >>>> T's incompleteness, again, _if_ it is indeed incompleteness. >>>> > What's G(T)? G(T) = Godel's sentence for T. > >>>> Finally, as a consequence of all the above, if T is capable of >>>> expressing arithmetic of the naturals then it's impossible to >>>> demonstrate (prove) T's incompleteness. Hence GIT is invalid. >> > _IF_ GC can't be proved or disproved in T, then you've another proof > of the incompleteness theorem. It does not show Godel's proof is > wrong. Given T is general (it's just "as strong as arithmetic of the naturals"), then GC also "can't be proved or disproved" in T+GC in this case, right? How is that not wrong? > > Why are you waisting your efforts disputing with Godel when you could > become instantly famous simply by proving (or disproving) GC? Why do you think TF coined the term "Goldbach-like statement" in his book about Godel Theorem? Don't you think the GC-related (1) would epitomize that there's something wrong with your intuition about the natural numbers?
From: Nam Nguyen on 24 Apr 2010 02:13 William Elliot wrote: >> > I notice you weren't able to refute my rebuttal of your esteemed fantasy. > Sometimes being too fast isn't a good thing! I've just done my refuting your rebuttal. (It just happens I did that when you wrote this reply post.)
From: William Elliot on 24 Apr 2010 03:06
On Sat, 24 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). >> Currently we do not know, nor do we know if GC is independent of PA. >> In addition there's no proof that we can't know. > >> Indeed, your fallacious argument could have been made a hundred years >> ago about FCT or FLT which we couldn't know then, but do now. > > Perhaps you'd want to reflect a bit on what TF said below, before > making the judgment above: > > "So for any formal system S which incorporates a bit of arithmetic > - the basic rules needed to carry out computations - a Goldbach-like > statement is disprovable in S if false. On the other hand, we can > make no similar observation about how a Goldbach-like statement can > be proved if it is true." > You have jumped from the observation that we don't know a proof or a disproof of GC, to the claim we cannot ever know a proof or a disproof of GC. You have yet to substantiate your claim. In addition TF's statement by itself without the context of his system of logic is vague. I don't see how it fits into the discussion or how it fills the gap in your thinking. For example, what's a Goldback like statement and is the quote other than a distraction? |