Name the thesis: "Formal sentences capture informal ones"
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From: Helene.Boucher on 30 Jan 2005 10:03 (1) Why this raises "basic questions" is for you to elucidate. (2) Because it is trivially true that 2^n exists for every n. (3) Why is it triviallly true? (4) That's one of the 'basic questions'. Hardy har har ! So your elucidation why something raises basic questions (such as "Why is it trivially true that 2^n exists for every n?") is the statement that "Because it is trivially true that 2^n exists for every n." Well done ! You've constructed your own vicious circle !
From: Helene.Boucher on 30 Jan 2005 10:05 No, it's the axiom "(x)(Nx => there exists y such that Sxy)", i.e. every natural number has a successor.
From: Torkel Franzen on 30 Jan 2005 10:15 Helene.Boucher(a)wanadoo.fr writes: > So your elucidation why something raises basic > questions (such as "Why is it trivially true that 2^n exists for every > n?") is the statement that "Because it is trivially true that 2^n > exists for every n." Right. Perhaps this will be clearer if I say "trivially true in ordinary mathematics". Hence, when you question whether 2^n exists for every n, you raise a basic question, and the question of the faithfulness of the translation Con(PA) becomes a side issue.
From: Helene.Boucher on 30 Jan 2005 10:38 Torkel Franzen wrote: > Helene.Boucher(a)wanadoo.fr writes: > > > So your elucidation why something raises basic > > questions (such as "Why is it trivially true that 2^n exists for every > > n?") is the statement that "Because it is trivially true that 2^n > > exists for every n." > > Right. ! > Perhaps this will be clearer if I say "trivially true in > ordinary mathematics". I don't think so. Presumably you would mean by 'ordinary mathematics' something which includes the truth of the successor axiom, so your additional phrase answers the question "Why is it trivially true...?" in a trivial way (the answer being, "because it's true by the definition of 'ordinary' mathematics") or turns the question into one of causality instead of grounds ("why has ordinary mathematics come to include the successor axiom?"). > Hence, when you question whether 2^n exists > for every n, you raise a basic question, and the question of the > faithfulness of the translation Con(PA) becomes a side issue. Except (again!) the faithfulness of the translation was the issue of the thread. And the intensional equivalence of two sentences should not turn on whether something else is true or not. And ... well I won't repeat myself !
From: Torkel Franzen on 30 Jan 2005 10:51
Helene.Boucher(a)wanadoo.fr writes: > Presumably you would mean by 'ordinary mathematics' > something which includes the truth of the successor axiom, so your > additional phrase answers the question "Why is it trivially true...?" > in a trivial way (the answer being, "because it's true by the > definition of 'ordinary' mathematics") or turns the question into one > of causality instead of grounds ("why has ordinary mathematics come to > include the successor axiom?"). It's not an answer at all to the question why it is trivially true. It is merely the observation that since you put in question trivial theorems of ordinary mathematics, your regarding Con(PA) as not being a faithful translation of "PA is consistent" becomes a side issue. > Except (again!) the faithfulness of the translation was the issue of > the thread. And the intensional equivalence of two sentences should > not turn on whether something else is true or not. Naturally it turns on whether we take other things to be true. You yourself referred to an "environment". |