Godels theorem ends in paradox-simply proof
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From: Nam Nguyen on 8 Jun 2010 22:06 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Aren't these a bit circular: you're explaining the naturals using the >> naturals? > > Why do you think Daryl's definition of a model was an explanation of > anything? My question above doesn't have anything to do with "anything"! It just asked about the circularity of using the naturals to define the naturals. > >> But no one has asked for the the truth of every formula. I just >> requested for only _1_ formula: (1)! > > Why should Daryl know whether the formula you described is true or > false? The why should he, or you, or any of us, care about whether or not this 1 formula, say, Ax[~(Sx=0)] is true or false?
From: Aatu Koskensilta on 8 Jun 2010 22:26 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > It just asked about the circularity of using the naturals to define > the naturals. You didn't ask for an explanation or definition of the naturals. You asked Daryl to spell out the intended model of PA. > The why should he, or you, or any of us, care about whether or not > this 1 formula, say, Ax[~(Sx=0)] is true or false? Everyone will of course have to decide for themselves what they care about. Why is it relevant here whether or not we know the truth of a particular formula related to Goldbach's conjecture? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 8 Jun 2010 22:44 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Meantime, are you then saying the naturals is collectively a model > of PA? No -- I don't understand what is meant by "the naturals is collectively a model of PA". Collectively as opposed to what? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 8 Jun 2010 22:49 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> It just asked about the circularity of using the naturals to define >> the naturals. > > You didn't ask for an explanation or definition of the naturals. You > asked Daryl to spell out the intended model of PA. You must have misinterpreted the conversation. There's is the FOL definition of model of a formal system. Then there was Daryl's saying that the naturals is the "*intended* model" of PA. We explain the FOL definition of model but we _verify_ what's intended as a model be in fact meeting the definition of model. "Spelling out" is a form of verification! > >> The why should he, or you, or any of us, care about whether or not >> this 1 formula, say, Ax[~(Sx=0)] is true or false? > > Everyone will of course have to decide for themselves what they care > about. Why is it relevant here whether or not we know the truth of a > particular formula related to Goldbach's conjecture? So you could, for example, say (1) is true in the naturals while I could _equally_ say ~(1) is false? Not to mention that similarly we each could _equally_ say G(PA) true and false, because out of (1) each could use infinite sets of primes for encoding: sets which could be empty or non empty but which could be impossible to know - as impossible as the truth value of (1)!
From: herbzet on 8 Jun 2010 23:02
Aatu Koskensilta wrote: > > herbzet writes: > > > If you're not disagreeing with anything, I'm not sure what you're > > driving at. Could you elaborate? > > I was just pointing out that while your reasoning was perfectly fine > there's no need to invoke classical logic to conclude from G�del's proof > that for every sufficiently expressive consistent formal theory there's > an arithmetical truth it doesn't prove. Generally, I don't invoke classical logic, it just kicks down the door and muscles its way in without invitation. -- hz |