Name the thesis: "Formal sentences capture informal ones"
in [Theory]
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From: Helene.Boucher on 31 Jan 2005 14:49 LordBeotian wrote: > <Helene.Boucher(a)wanadoo.fr> ha scritto > > > > 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. > > > > If there is a problem with the successor axiom itself, there is a > > problem with you using "and so on..." as you have done. > > The only "problem" I see is if 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", is this the case? I would say the problem in both cases is that you are supposing that you are always able to continue an operation ad infinitum. In one case it's the successor operation; in your argument it's the step which is being repeated "and so on". You may think (and you may well be right) that there is nothing incorrect about this assumption, but still it is an assumption - and really a basic one, which you're not going to get around except by assuming it in some other guise. I'm going to leave it with two comments, and then finish my part of this discussion. (Maybe some other time?) (1) In Frege Arithmetic, in some definite sense, the Successor Axiom requires impredicativity, unlike other arithmetic axioms (See Predicative Fragments of Frege Arithmetic, by Oystein Linnebo). (2) Systems without the Successor Axiom can have some interesting properties, which other, more ontologically ambitious, systems cannot, e.g. the ability to prove oneself consistent. So I think the Successor Axiom sticks out in more than one way.
From: Torkel Franzen on 31 Jan 2005 15:05 Andrew Boucher says: > I would say the problem in both cases is that you are supposing that > you are always able to continue an operation ad infinitum. In one case > it's the successor operation; in your argument it's the step which is > being repeated "and so on". You may think (and you may well be right) > that there is nothing incorrect about this assumption, but still it is > an assumption - and really a basic one, which you're not going to get > around except by assuming it in some other guise. So you're raising a basic question?
From: Jamie Andrews; real address @ bottom of message on 31 Jan 2005 16:55 In comp.theory tchow(a)lsa.umich.edu wrote: > (*) Formal sentences (in PA or ZFC for example) adequately express > their informal counterparts. > Any candidates for a catchy name for (*)? Is this not the central tenet of "logicism"? (It could well not be, because I may have misunderstood the precise meaning of the word in philosophical discourse.) See for example http://www.eecs.umich.edu/~rthomaso/documents/logicism/ --Jamie. (a Dover edition designed for years of use!) andrews .uwo } Merge these two lines to obtain my e-mail address. @csd .ca } (Unsolicited "bulk" e-mail costs everyone.)
From: Torkel Franzen on 31 Jan 2005 17:03 me(a)privacy.net (Jamie Andrews; real address @ bottom of message) writes: > Is this not the central tenet of "logicism"? I don't think it has anything in particular to do with logicism.
From: Stephen Harris on 31 Jan 2005 17:53
"Jamie Andrews; real address @ bottom of message" <me(a)privacy.net> wrote in message news:367nq8F4qcoivU1(a)individual.net... > In comp.theory tchow(a)lsa.umich.edu wrote: >> (*) Formal sentences (in PA or ZFC for example) adequately express >> their informal counterparts. >> Any candidates for a catchy name for (*)? > > Is this not the central tenet of "logicism"? > > (It could well not be, because I may have misunderstood the > precise meaning of the word in philosophical discourse.) > > See for example > http://www.eecs.umich.edu/~rthomaso/documents/logicism/ > Logicism is the theory that mathematics is an extension of logic and therefore all mathematics is reducible to logic. Kurt Gýdel's incompleteness theorem ultimately undermined the purpose of the project. The attempted resurrection of this theory is styled neo-logicism. When considering translation from natural/informal language, and I think natural languages can have be formally constructed, there are always some phrases which do not have an identical tranlsation into a second language given that both languages do not have identical symbols and quantifiers. I think that is a theorem. So the exception proves the rule, Which does not seem to qualify as thesis potential, Stephen |