The End of an Era - That of The Natural Numbers
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From: Nam Nguyen on 21 Apr 2010 23:50 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. 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"): 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). Second observation: as a consequent of the 1st observation, no extension of Q can prove nor disprove (1) unless it's inconsistent. 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. 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.
From: William Elliot on 22 Apr 2010 02:39 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? > 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)? GC? > 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). > > Second observation: as a consequent of the 1st observation, no > extension of Q can prove nor disprove (1) unless it's inconsistent. > > 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. > > 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. >
From: Nam Nguyen on 22 Apr 2010 02:45 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? 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)? GC? L(PA) = L(0,S,+,*,<) GC = Goldbach Conjecture > >> 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). >> >> Second observation: as a consequent of the 1st observation, no >> extension of Q can prove nor disprove (1) unless it's inconsistent. >> >> 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. >> >> 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. >>
From: Aatu Koskensilta on 23 Apr 2010 11:59 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. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: William Elliot on 22 Apr 2010 04:49
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? > 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. 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. >>> 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)? >>> 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. Why are you waisting your efforts disputing with Godel when you could become instantly famous simply by proving (or disproving) GC? |