Torkel Franzen on truth
in [Logic]
|
From: george on 26 Nov 2007 17:49 On Nov 24, 8:26 pm, Newberry <newberr...(a)gmail.com> wrote: > I do not understand this. Let's first decide if the argument that > leads to > "No matter how long we generate theorems of PA we will never > derive a contradiction" is true is formalizable. We are not going to be able to decide that. We are especially not going to be able to watch *PA* itself decide it, since PA cannot and therefore DOES not decide this. Stronger theories can decide it, but they don't so much "decide" it as *presume* it. Every theorem is basically ALREADY ASSUMED by the axioms.
From: george on 27 Nov 2007 17:03 On Nov 26, 12:46 pm, "LordBeotian" <pokips...(a)yahoo.it> wrote: > 1) We consider the issue from a sintactic point of view and prove it by > transfinite induction (Gentzen). > 2) We find a model for the axioms. .... > Both 1 & 2 are formalizable in ZFC Right. > *BUT* we DON'T know that PA is consistent > because we formalized it in ZFC (*), Of course we do. > we instead DO know it by mere > consideration of the intuitive not-formalized arguments 1 or 2 above. No, really, you don't know much of ANything, especially not anything infinitary, THAT way. > (*) Otherwise you could ask why should ZFC influence our belief given the > fact that we don't even know if it is consistent. True. But THAT is when you say that it looks intuitively right. The point is, once you say this for ZFC, you never have to say it for anything else.
From: LordBeotian on 28 Nov 2007 07:09 "george" <greeneg(a)cs.unc.edu> ha scritto >> 1) We consider the issue from a sintactic point of view and prove it by >> transfinite induction (Gentzen). >> 2) We find a model for the axioms. > ... > >> Both 1 & 2 are formalizable in ZFC > > Right. > >> *BUT* we DON'T know that PA is consistent >> because we formalized it in ZFC (*), > > Of course we do. > >> we instead DO know it by mere >> consideration of the intuitive not-formalized arguments 1 or 2 above. > > No, really, you don't know much of ANything, especially > not anything infinitary, THAT way. I desagree. The majority of mathematicians usually talk about infinitary things and are convinced about their arguments without even know what ZFC is (and also knowing nothing about first order formal systems).
From: george on 28 Nov 2007 17:22 On Nov 28, 7:09 am, "LordBeotian" <pokips...(a)yahoo.it> wrote: > >> we instead DO know it by mere > >> consideration of the intuitive not-formalized arguments 1 or 2 above. > > > No, really, you don't know much of ANything, especially > > not anything infinitary, THAT way. > > I desagree. The majority of mathematicians usually talk about infinitary > things and are convinced about their arguments without even know what ZFC is But you said "arguments". "Argument" implies a chain of REASONING, of logical INFERENCE. In other words, there are some axioms and rules of inference being invoked to support these beliefs. It doesn't have to be ZFC specifically. Lots of things will work. ZFC is just the currently usual choice for people who choose to care about this aspect of it. > (and also knowing nothing about first order formal systems). It's not like religion. It works whether you believe in it or not.
From: LordBeotian on 29 Nov 2007 02:31
"george" <greeneg(a)cs.unc.edu> ha scritto >> >> we instead DO know it by mere >> >> consideration of the intuitive not-formalized arguments 1 or 2 above. >> >> > No, really, you don't know much of ANything, especially >> > not anything infinitary, THAT way. >> >> I desagree. The majority of mathematicians usually talk about infinitary >> things and are convinced about their arguments without even know what ZFC >> is > > But you said "arguments". "Argument" implies a chain of REASONING, of > logical INFERENCE. In other words, there are some axioms and rules > of inference being invoked to support these beliefs. Axioms and rules of inference are connected only to *formal* reasoning. Informal reasoning don't imply any defined set of axioms or rules, even if it can if course be formalized in many different formal deductive systems. You should also agree that we definitely don't need a reasoning to be framed in a set of axioms and rules to be convinced. Of course it can help, but we are convinced (for example) of infinity of primes or Pythagorean theorem long before we hear about PA, Q, ZFC or Hilbert's axioms for geometry. >> (and also knowing nothing about first order formal systems). > > It's not like religion. It works whether you believe in it or not. I know. The point is that even if you could be able to formalize the reasoning of mathematicians in ZFC, this formalization is not really relevant for their work and their beliefs. |