Torkel Franzen on truth
in [Logic]
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From: Aatu Koskensilta on 12 Nov 2007 06:39 On 2007-11-09, in sci.logic, Herman Jurjus wrote: > aatu.koskensilta(a)xortec.fi wrote: >> Yes, PA is obviously consistent. > > So what? Nothing much. Why? -- 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 12 Nov 2007 06:43 On 2007-11-09, in sci.logic, george wrote: > Come on. > The Gentzen proof is not obviously understandable. > N doesn't obviously exist at all. Gentzen's proof and the existence of the set of natural numbers don't really have much to do with obviousness of consistency of PA. -- 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 12 Nov 2007 06:49 On 2007-11-11, in sci.logic, Newberry wrote: > On Nov 9, 4:17 am, aatu.koskensi...(a)xortec.fi wrote: >> Yes, PA is obviously consistent. > > OK, how do we reconcile it with this? Reconcile in what sense? There is no apparent contradiction between Torkel's explanation concerning... ... the mistaken idea that "G�del'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. As has been emphasized, in general we simply have no idea whether or not the G�del sentence of a system is true, even in those cases when it is in fact true. What we know is that the G�del sentence is true if and only if the system is consistent, and that much is provable in the system itself. and the observation that PA is obviously consistent. -- 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 12 Nov 2007 06:56 On 2007-11-11, in sci.logic, abo wrote: > What is "good reason" to you may not be "good reason" to someone > else. For instance, a hard-core theist would hold that there is not > "good reason" to discuss the existence of God. An indifferent atheist might also well find discussing the existence of God somewhat pointless. Regardless of the question of whether "good reason" is or is not subjective, it remains a rather trivial platitude that people will in fact be interested in subjecting this or that to scrutiny, reflection, doubt, only if presented some incentive to, a "good reason" in a more mundane sense. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Daryl McCullough on 12 Nov 2007 06:57
Aatu Koskensilta says... > >On 2007-11-09, in sci.logic, abo wrote: >> Conclusive! Obvious! Who could doubt what one learned as a young boy >> in Sunday school? > >I don't know. What does one learn about Peano arithmetic as a young boy in >Sunday school? I'm not sure. But an acquaintance of mine explained how the natural numbers can be represented using lambda calculus. He told me he learned it in Church. -- Daryl McCullough Ithaca, NY |