New essay on Goedel's Incompleteness Theorem
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From: Marshall on 30 Sep 2009 18:05 On Sep 30, 2:47 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Marshall <marshall.spi...(a)gmail.com> writes: > > On Sep 30, 3:40 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> Marshall <marshall.spi...(a)gmail.com> writes: > >> > On Sep 29, 9:26 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > >> >> Nobody in their right mind cares about Google Groups ratings. > > >> > Only insane people care what others think of them? That doesn't > >> > seem right. > > >> You have some sort of delusion that Google's rating system accurately > >> reflects general opinion about your character? > > > I didn't say "accurately reflects general opinion"; nonetheless it is > > certainly a reflection of others' opinions. > > You interpreted "Nobody in their right mind cares about Google Groups > ratings," as "Only insane people care what others think of them." > > Surely, you see how ridiculous this misinterpretation is. I acknowledge that one's Groups rating is only a small subset of the overall set of others' opinions about one. Other than that, no. Although I suppose one could argue that "insane people" is hyperbolic with regards to "nobody in their right mind." We've discussed this in the past, and with respect I think it would be even less useful for us to discuss this than it would be for us to try to convince Nam that a formal system specifies a language first and axioms second. Marshall
From: Scott H on 30 Sep 2009 21:26 On Sep 30, 5:23 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Scott H says... > > >1. Every statement has a Goedel number. > >2. If a statement is 'about' a number, then its Goedel number is also > >'about' that number. > > That doesn't make any sense. What is the number 4 about? 4 isn't the Goedel number of a statement that is about anything. > >3. Some statements are about Goedel numbers. > >4. Therefore, there are Goedel numbers that are about Goedel numbers. > > I don't think it makes any sense to say that a Godel number is "about" > anything. In any case, I don't see where you are getting an infinite > sequence of G, G', G'', etc. I say that a Goedel number is about something when its statement is about something. The infinite sequence arises when the substitution or 'arithmoquine' function is applied repeatedly. > G could be said to be "about" its Godel number, since it says > that its Godel number is not the Godel number of a provable sentence. > But then G' is the Godel number of G, so G' is "about" whatever G is > about. So there is no G'', G''', etc. I'll be more accurate: G' is about G'', the Goedel number of G', which G is about. As I told Aatu, I'll leave it open whether G, G', G'', ... are all basically the same statement.
From: Nam Nguyen on 30 Sep 2009 23:05 Jesse F. Hughes wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Aatu Koskensilta wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> >>>> There might have been more than one points that I might have >>>> "adamantly insisted" in the past. So unless you spell it out I >>>> couldn't know what we're really talking about here. >>> David's statement, that if T and T* have the same axioms they're the >>> same theory, implies that the language of a theory is determined by >>> its axioms. >> By Shoenfield, a theory is determined by axioms. So, L(T) is determined >> by T's axioms. > > > Oh, good Lord, not that again! That really applies to you! > > Shoenfield does *not* claim L(T) is determined by its axioms. Here's > the quote, once again: > > "We consider a language to be completely specified when its symbols > and formulas are specified. This makes a language a purely > syntactical object. Of course, most of our languages will have a > meaning (or several meanings); but the meaning is not considered to > be part of the language. We shall designate the language of a formal > system F by L(F)." > > "The next part of a formal system consists of its /axioms/. Our > only requirement on these is that each axiom shall be a formula of > the language of the formal system." [p. 4 in my copy of Shoenfield] > > The first part of a formal system is the language. The *next* part is > the axioms. Clearly, then, the axioms do not determine the language. You sounded like a kindergarten kid reading technical book for the first time, reading _too literally_ what's in the book! Where in Shoenfield's book did he say that if one doesn't consider language as the first part of formal system, one's reasoning would fall apart? You (and others) were wrong in many different levels. In the first place, then, I already gave a details of how a language of a T can be legitimately defined in term of axioms! (This was in conversation with CM in which the Shoenfield's definition of a "designator" was used as an example). Why don't you go and review that conversation instead of hand-waving "Oh, good Lord, not that again!". To date, you, Aatu, and others kept silent on this example that would show one can define the language last! Why is that? Secondly, as I mentioned above, did Shoenfield absolutely have to say "The first part of a formal system ..." in order for his book to make sense? Couldn't he have said "Part of a formal system is the language ..."? Thirdly, just to illustrate how you've obsessed with Shoenfield's English-but-non-technical-way-of-speaking "The first part...", let me give a simple example. Suppose I was contemplating a language for my theory and I was thinking of using the symbol "add" as a binary function symbol. But a friend of mine just stopped by and told me that his program would _randomly_ output Ascii strings and one which looks like an equation. Without looking first at the _equation_, I told him I'd take it as the axiom of my theory. And when I looked at it, that _random_ string turned out to be: x+y=0 Where did I, you, Aatu, or anybody, contemplate on the language of T = {x+y=0} "first", in this case? Do you understand now the issue that in formulating a formal system, you don't have to define a language "first"? As long as the strings called axioms follow syntactical formation of a formula, then those are the axioms; and the language of the axiom set would be derived/defined from there!
From: Tim Little on 30 Sep 2009 23:12 On 2009-09-30, Scott H <zinites_page(a)yahoo.com> wrote: > At any rate, I have proposed that G refers to its 'reflection' or > Goedel code, which I have called G' instead of t. Yes, statement G refers to the number G'. G' does not literally refer to anything, as it is not a statement. If it is interpreted as a statement via the decoding, that statement is G and refers to G'. There is no G'', and no endless reference. - Tim
From: Nam Nguyen on 30 Sep 2009 23:14
Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> The beauty a of mathematical truth doesn't require a mass of people to >> appreciate it. If - but only if - one sees a mathematics/reasoning >> truth, one doesn't really need anybody else to appreciate it. > > One sits in a dark corner, in total solitude, marveling at the private > mathematical truths one's discovered? Yours may be a very romantic ideal > but to me it sounds quite dreadful. > You seem to have a tendency to interpret things _out of context_, for the worse! Scott seemed so _desperate_ to the point he would "attack" people pointlessly. I just tried to console him that should it be the case only he and nobody else would see the (correct) truth, others don't have to see it and so he wouldn't need to be frustrated/desperate. That doesn't mean _in general_ I'd encourage people to be "in a dark corner, in total solitude...", as you've imagined! |