Name the thesis: "Formal sentences capture informal ones"
in [Theory]
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From: Torkel Franzen on 30 Jan 2005 06:54 Helene.Boucher(a)wanadoo.fr writes: > Not at all. To show that two concepts are not equivalent, it is > sufficient to present an environment in which they are clearly not the > same. The environment that I have proposed does that - thus Con(PA) > and "PA is consistent" cannot be intensionally equivalent; they do not > "say the same thing." Why this raises "basic questions" is for you to > elucidate. Because it is trivially true that 2^n exists for every n, so your problem with the translation Con(PA) has no obvious connection with ordinary mathematics.
From: Helene.Boucher on 30 Jan 2005 08:57 Why is it trivially true?
From: Torkel Franzen on 30 Jan 2005 09:06 Helene.Boucher(a)wanadoo.fr writes: > Why is it trivially true? That's one of the "basic questions".
From: Helene.Boucher on 30 Jan 2005 09:06 " so we have to change the axioms of PA" Since I would suggest changing the axioms of PA - specifically eliminating the successor axiom - you do not need to draw this conclusion. I grant it. "an enormous part of mathematics (which is all connected with arithmetic)" In such a system you cannot prove (x)(y)(x + y = y + x). On the other hand, one can prove (x)(y)(z)(x + y = z => y + x = z) I would suggest there is not an essential loss.
From: LordBeotian on 30 Jan 2005 09:25
<Helene.Boucher(a)wanadoo.fr> ha scritto > " so we have to change the axioms of PA" > > Since I would suggest changing the axioms of PA - specifically > eliminating the successor axiom - you do not need to draw this > conclusion. I grant it. > > "an enormous part of mathematics (which is all connected with > arithmetic)" > > In such a system you cannot prove > (x)(y)(x + y = y + x). > On the other hand, one can prove > (x)(y)(z)(x + y = z => y + x = z) > I would suggest there is not an essential loss. By successor axiom do you mean the axiom (Sx=Sy)->(x=y) ? |