New essay on Goedel's Incompleteness Theorem
in [Logic]
|
Prev: Solutions manual to Mechanical Behavior of Materials, 3E Norman E Dowling
Next: Editor of Physical Review A, Dr Gordon W.F. Drake does WRONG subtraction of 8th Class mathematics.
From: Nam Nguyen on 27 Sep 2009 23:17 Nam Nguyen wrote: > 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! By the way, when Goldbach and Euler investigated about the arithmetic truth/falsehood of GC, they were probably, in your assessment, just being "cute little rhetorical" and not doing serious math!
From: George Greene on 27 Sep 2009 23:57 On Sep 27, 5:06 pm, Scott H <zinites_p...(a)yahoo.com> wrote: > Either way, the idea is that G isn't speaking directly about itself, > but about a 'reflection' of itself in a model. This is simply not the case. There are MANY DIFFERENT models of the axioms in question and if G is NOT A THEOREM, then it cannot legitimately be said to be speaking ABOUT ANYthing!!! Absolutely EVERY "referent" of EVERY [referring] part of G is going to be "about" ONE thing in ONE model and ANOTHER thing in ANOTHER model! G's NON-provability is an ESSENTIAL defining characteristic of G, and this means that G is TRUE IN SOME models of the axioms AND FALSE IN OTHERS. The entities "in" DIFFERENT models of the axioms simply cannot be said to be "the same" IN ANY sense. > It's an open question whether the G in the model is 'the same G.' No, it isn't. It's not any kind of question at all. It's KNOWN AND ANSWERED, to the extent that "the model" is known. > Proponents of G seem to think that it is, while a proponent of ~G might not. THERE ARE NO SUCH THINGS as "proponents of G" and "proponents" of ~G, unless you are going to allege that MODELS are proponents. Normally, PEOPLE are proponents, and in this case, THERE ARE NO SUCH people. G is true in SOME models and false in OTHERS, and NO personal "proponent" is relevant IN EITHER case! The facts of the matters simply are what they are!
From: George Greene on 27 Sep 2009 23:59 On Sep 27, 5:09 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > There are no "proponents" of G or ~G. Well, obviously, unless you are going to say that models can "propose". > This business about reflections, > models and what-not is pure waffle. Not exactly. It is purely completely over Scott's head, is what it is. It might have some merit or relevance in the context of some statement by someone who was actually familiar with the concepts. > Some theories, such as ZFC + ~G, are > simply mistaken about the provability of their Gödel sentence. It is NOT possible for a THEORY to be "mistaken" about one of ITS OWN AXIOMS. Rather, it would make more sense to say that the humans are mistaken in CALLING the relevant predicate "a provability predicate".
From: Daryl McCullough on 28 Sep 2009 09:05 Scott H says... > >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. If T and T' have the same axioms, then what is the difference between T and T'? Anyway, your idea that there is an infinite sequence of statements, G, G', G'', etc. is just wrong. G does not say that any other statement besides itself is unprovable. -- Daryl McCullough Ithaca, NY
From: David C. Ullrich on 28 Sep 2009 09:24
On Sun, 27 Sep 2009 14:09:53 -0700 (PDT), Scott H <zinites_page(a)yahoo.com> wrote: >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, For heaven's sake. If T and T* have the same axioms then they're the same theory. > 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. David C. Ullrich "Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to." (John Jones, "My talk about Godel to the post-grads." in sci.logic.) |