New essay on Goedel's Incompleteness Theorem
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From: Scott H on 27 Sep 2009 17:06 On Sep 27, 4:53 pm, Scott H <zinites_p...(a)yahoo.com> wrote: > [(x)P(t,x)] <=> [(x)P([(x)P(t,x)],x)], Correction: that should be [(x)P([(x)P(y,x)])]. Either way, the idea is that G isn't speaking directly about itself, but about a 'reflection' of itself in a model. It's an open question whether the G in the model is 'the same G.' Proponents of G seem to think that it is, while a proponent of ~G might not.
From: Scott H on 27 Sep 2009 17:09 On Sep 27, 4:58 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > What is G'? If we have a nest of theories T > T' > T'' > T''' ... all with the same axioms, then G is a statement belonging to the theory T, while G' is the equivalent statement formulated within T'. We can also formulate G'', G''', and so on.
From: Aatu Koskensilta on 27 Sep 2009 17:09 Scott H <zinites_page(a)yahoo.com> writes: > Either way, the idea is that G isn't speaking directly about itself, > but about a 'reflection' of itself in a model. It's an open question > whether the G in the model is 'the same G.' Proponents of G seem to > think that it is, while a proponent of ~G might not. There are no "proponents" of G or ~G. This business about reflections, models and what-not is pure waffle. Some theories, such as ZFC + ~G, are simply mistaken about the provability of their G�del sentence. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 27 Sep 2009 17:11 Scott H <zinites_page(a)yahoo.com> writes: > If we have a nest of theories > > T > T' > T'' > T''' ... > > all with the same axioms, then G is a statement belonging to the > theory T, while G' is the equivalent statement formulated within T'. > We can also formulate G'', G''', and so on. I see. So T = T' = T'' ... are all the same theory, and G = G' = G'' .... the same sentence. Is there some point to this notational gymnastics? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 27 Sep 2009 23:09
Marshall wrote: > On Sep 26, 11:57 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> If I want to know what 2+2 is, do >>> you say 4, or do you say "it depends"? >> The truth of 2+2=4 require finite knowledge. What would happen if you >> want to know the _arithmetic_ truth/falsehood of GC, cGC, or !GC? > > These are all cute little rhetorical techniques you are using, > but the fact remains that we can add, subtract, multiply and > divide natural numbers with no trouble, without knowing the > truth of, or even having heard of, Goldbach's Conjecture. > We could do so just as well had Goldbach never been born. You're right. If all there's to it is just a child math of addition, subtraction, multiplication, division, then we don't need to hear anything about Goldbach. But nor need we to hear anything about complex numbers, transcendental numbers, Hilbert, GIT, Skolem paradoxes, cardinalities, Compactness, etc... In fact, you don't even need to know the so called the naturals numbers (since this concept would have unknown-abilities such as hinted by GoldBach Conjecture), because you could do such a child math with 10 fingers and 10 toes! That is, according to your perception of what mathematics or arithmetic be! |