Torkel Franzen on truth
in [Logic]
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From: Aatu Koskensilta on 19 Dec 2007 07:23 On 2007-12-18, in sci.logic, george wrote: > You are beginning by presuming the existence of an intended model. > > That is just not the way it's generally done any more. Mathematicians don't think, or do not begin by thinking, the natural numbers -- or the reals, or sets of sets of sets of reals, or what have you -- exist and they can state and proves stuff about them? A very curious idea. > The completeness theorem is fundamentally a proof that first-order > semantics SIMPLY DOESN'T EXIST. Right. Just as the fundamental theorem of analysis establishes no-one knows how to ride a bicycle. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 19 Dec 2007 07:25 On 2007-12-18, in sci.logic, tchow(a)lsa.umich.edu wrote: > George, you've *got* to be kidding here. Do you not have any clue how > idiosyncratic your viewpoint is? George knows very well by now that his views on many issues are quite bizarre and idiosyncratic. He's just convinced every one else is wrong. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 19 Dec 2007 08:33 On 2007-12-15, in sci.logic, Daryl McCullough wrote: > Whether I "know" that or not depends on what you mean by "know". > For practical purposes, you know something if you believe it and > there are goods reasons for believing it and believing otherwise > seems difficult. That's the case with ZFC. I don't have any absolute > argument for why it is consistent, it's just that it seems very unlikely > to me that it could be inconsistent and to have that inconsistency > undiscovered before now. Why? We don't have any good data on which to judge the likelihood of that. Happily, as you later note, we have much better grounds to think ZFC is consistent, sound for arithmetical statements, and so on. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 19 Dec 2007 08:40 On 2007-12-15, in sci.logic, abo wrote: > Secondly, are any proofs of PA's consistency "interesting or useful"? Sure. From Gentzen's consistency proof we learn, for example, that if PA proves a statement of the form "AxEyP(x,y)" with P decidable, there is a function F, below the epsilon-0'th level of a hierarchy of fast growing recursive functions, such that AxP(x,F(x)). > Or is it only when a theory proves its own consistency that it is not > interesting or useful? Given that inconsistent theories prove their own consistency, if we merely know of some theory T that it proves its own consistency nothing interesting at all can be concluded. If, inspecting the proof, we find that only principles we accept as correct are in fact used in the proof, its then of course useful in convincing us of the consistency of T. For most theories we're interested we know this can't happen, since the conditions of the second incompleteness theorems apply. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: tchow on 17 Dec 2007 11:33
In article <xLw9j.7491$hQ3.4060(a)pd7urf3no>, Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >Unfortunately mathematical reasoning isn't religion where "beliefs" >would be much relevant. You don't think beliefs are relevant in mathematical reasoning? Then how do you become convinced that *anything* is true? Are you convinced, for example, that sqrt(2) is irrational? On what basis? On the basis of the proof? But the proof starts with some axioms. On what basis do you become convinced of the correctness of the axioms? Or are you *not* convinced of the axioms? But if you're not convinced of the axioms, then what good is a proof of "sqrt(2) is irrational" from those axioms? -- 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 |