Torkel Franzen on truth
in [Logic]
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From: Newberry on 6 Dec 2007 23:55 On Dec 6, 2:54 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > On 6 Dec, 18:04, george <gree...(a)cs.unc.edu> wrote: > > > In his book on GIT, the late TF (to his great credit) challenges > > > > "... the mistaken idea that "Gödel's theorem states that > > > in any consistent system which is strong enough to > > > produce simple arithmetic there are formulas which > > > cannot be proved in the system, but which we can see to be true." > > > The theorem states no such thing. > > > This forces us to ask, "Does Prof.Peter Smith know Prof.Daniel > > Isaacson?" > > Oh? Why does it force us to ask that?? Actually I do know Dan, and of > course we'd both agree with Torkel Franzen's point. (It is one anyone > teaching this stuff stresses: sure, we need to assume that the system > we are dealing with is indeed consistent if we are get to see that the > canonical Gödel sentence is true on the standard interpretation.) > > > Because Isaacson, TEACHING THIS STUFF RIGHT NOW ("Michaelmas > > term, 2007), flaunts his ignorance of this Torkelian perspective, > > lecturing, > > >>>>> The Godel sentence is demonstrably true, > > >>>>> though not demonstrable in the system for which it is constructed.. > > Hold on, hold on. You've rather ripped this out of context from Dan's > lecture notes. He two paragraphs earlier introduces the assumption > that we are dealing with systems of arithmetic S which are *sound* > (and hence consistent). And on *that* assumption, which I take to be > still in force when we comments on the proof he sketches, the > canonical Gödel sentence will indeed be true on the standard > interpretation. It does not matter that the system is "sound" i.e. it does not help to introduce yet another concept. If we can formally prove S to be consistent we can only do it in another system, whose consistency we not know. The "proof" is an empty game with symbols. It does NOT tell that no matter how long we keep generating theorems in S we will never derive a contradiction. It has zero cogency. If we are 100% sure that a system is consistent then it cannot be because of a formal proof. There are three possibilities: 1) We do not have the foggiest idea if PA is consistent and we will never know. Hence we do not know if Gödel sentence is true. 2) The human mind surpasses any computer 3) There exists a formalization of arithmetic that can prove its own consistency. But it is not possible to reject 1 and 3 and at the same time claim that the human mind does NOT surpass any machine.
From: herbzet on 7 Dec 2007 02:24 Alan Smaill wrote: > kleptomaniac666_(a)hotmail.com writes: > > On Dec 5, 4:29 pm, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > >> george <gree...(a)cs.unc.edu> writes: > >> > On Dec 4, 8:32 pm, kleptomaniac6...(a)hotmail.com wrote: > >> >> Also, would it not be the case that independence from PA would imply that > >> >> FLT is true? > >> >> As FLT can be expressed as a pi-1 sentence. > >> > >> > I do not see how to do that (express FLT as a Pi-1 sentence in the > >> > language of PA), since the language of PA does not include an > >> > exponentiation operator. > >> > >> the expressibility of exponentiation in PA > >> was done by Goedel in his proof of incompleteness (and it's serious work). > >> Not sure if this would allow a pi-1 formulation, though. > >> > >> -- > >> Alan Smaill > > > > I thought (though I could be wrong) that there was a general principle > > that any statement in the language of first order arithmetic which > > took the form: every positive integer has the property p, where p is a > > property that can be algorithmically checked can be expressed as a > > pi-1 sentence. > > I'm not convinced; > according to Smullyan, the relation x^y = z is sigma-1 wrt PA. In "Godel, Escher, Bach" Hofsteder mentions the difficulty of defining "z is a power of x" in PA and challenges the reader to define the simpler "z is a power of two". I came up with "z has no odd factor greater than one". I don't know if that's pi-1 or sigma-1. :-) -- hz
From: Herman Jurjus on 7 Dec 2007 04:29 Newberry wrote: > > There are three possibilities: > 1) We do not have the foggiest idea if PA is consistent and we will > never know. Hence we do not know if G�del sentence is true. > 2) The human mind surpasses any computer > 3) There exists a formalization of arithmetic that can prove its own > consistency. > > But it is not possible to reject 1 and 3 and at the same time claim > that the human mind does NOT surpass any machine. In the strictest sense, the correct answer is 1). But: if PA should ever turn out to be inconsistent, then the whole subject of foundations of mathematics as we now know it will have to be redone: we assume everywhere in mathematics that the natural number sequence makes sense, and that it has the PA properties. We even presume it in definitions of the formal language of FOL. In short: 1) is correct: we only -do as if- we know PA is consistent, and we do so for pragmatic reasons. -- Cheers, Herman Jurjus
From: tchow on 7 Dec 2007 10:15 In article <8f772a36-9d58-4183-859a-860e968e51ce(a)a35g2000prf.googlegroups.com>, Newberry <newberryxy(a)gmail.com> wrote: >> > >>>>> The Godel sentence is demonstrably true, >> > >>>>> though not demonstrable in the system for which it is constructed. [...] >It does not matter that the system is "sound" i.e. it does not help to >introduce yet another concept. If we can formally prove S to be >consistent we can only do it in another system, whose consistency we >not know. The "proof" is an empty game with symbols. It does NOT tell >that no matter how long we keep generating theorems in S we will never >derive a contradiction. It has zero cogency. If we are 100% sure that >a system is consistent then it cannot be because of a formal proof. You've changed the subject by introducing the concept of "being 100% sure." The original statement under discussion said only that the Goedel sentence is demonstrably true, which it is: There does indeed exist a demonstration of its truth. What attitude we take towards that demonstration is a separate question. You can choose to doubt it, but the "it" that you're doubting still exists (otherwise, what is it that you're doubting?). -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
From: LordBeotian on 7 Dec 2007 10:32
"Newberry" <newberryxy(a)gmail.com> ha scritto >There are three possibilities: >1) We do not have the foggiest idea if PA is consistent and we will >never know. Hence we do not know if G�del sentence is true. >2) The human mind surpasses any computer >3) There exists a formalization of arithmetic that can prove its own >consistency. You could think that the human mind surpasses PA (in the sense that it can convince itself of thing not provable in PA) but (for example) does not surpass other unknown extensions of PA. For example there could be a *true* arithmetical statement U about which human mind cannot ever be convinced. So the human mind would never surpass PA+U, and will also never know which statement is U. |