Name the thesis: "Formal sentences capture informal ones"
in [Theory]
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From: Lee Rudolph on 31 Jan 2005 06:27 tchow(a)lsa.umich.edu writes: .... >Let me first refer you to my article `What makes a representing formula >"reasonable"?' that I posted to sci.logic on Dec 12 2002. (You can find >it on Google Groups; I'd give the URL, but Google currently has an annoying >beta version whose URL's are ephemeral.) .... The old behavior can still be evoked (who knows for how much longer? with a URL of the form http://groups.google.co.uk/groups?selm=au69np%24ppl%242%40galois.mit.edu (or one with ".ca" rather than ".co.uk" in place of ".com"). Lee Rudolph
From: Helene.Boucher on 31 Jan 2005 08:21 LordBeotian wrote: > <Helene.Boucher(a)wanadoo.fr> ha scritto > > > No, it's the axiom "(x)(Nx => there exists y such that Sxy)", i.e. > > every natural number has a successor. > > As far as I know it is not an axiom of PA... It is not usually explicitly stated, but it is assumed when one supposes that the sequential operator is total, i.e. that for every natural number n, (n') exists.
From: LordBeotian on 31 Jan 2005 09:06 <Helene.Boucher(a)wanadoo.fr> ha scritto > > > No, it's the axiom "(x)(Nx => there exists y such that Sxy)", i.e. > > > every natural number has a successor. > > > > As far as I know it is not an axiom of PA... > > It is not usually explicitly stated, but it is assumed when one > supposes that the sequential operator is total, i.e. that for every > natural number n, (n') exists. Ok, so we would have to change PA in such a way that S is just a relation symbol... In the case you suggest PA has the model given by {0,1}. Are you suggesting that {0,1} *could be* the whole set of natural numbers? If not we can add the axiom "1 has a successor". Are you suggesting that {0,1,2} *could be* the whole set of natural numbers? If not we can add the axiom "2 has a successor". And so on... unless there is a point when you say "yes, I suggest that the set {0,1,..,890} (for example) *could be* the whole set of natural numbers". If this never happen we have added enough axioms to have a system that believe that any natural number has a successor.
From: tchow on 31 Jan 2005 09:10 In article <vcby8eahxxm.fsf(a)beta19.sm.ltu.se>, Torkel Franzen <torkel(a)sm.luth.se> wrote: <tchow(a)lsa.umich.edu writes: < <> I would say that *something like* (+) is implicitly accepted by most <> people, and forms the basis for concluding that Goedel's 2nd theorem <> effectively kills Hilbert's program as originally conceived. < < Certainly. However, for this we don't need <"intension-preserving". It is sufficient that the equivalence of <Con(PA) and a "direct" formalization of "PA is consistent" is provable <in a weak theory. However, even then you still need the assumption that the "direct" formalization is intension-preserving. -- 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: Torkel Franzen on 31 Jan 2005 09:22
tchow(a)lsa.umich.edu writes: > However, even then you still need the assumption that the "direct" > formalization is intension-preserving. Yes, but this is not really an assumption, any more than when we translate "Every apple is green" into "(x)(x is an apple -> x is green)". |