Re: Successor Axiom: on what grounds TF?

From: |-|erc (h_at_r.c)
Date: 02/07/05 

Date: Mon, 7 Feb 2005 21:50:44 +1000

"David C. Ullrich" <ullrich@math.okstate.edu> wrote in
> >> >The idea is not to derive interesting math given a replacement axiom
> >> >for the Successor Axiom. It is to see what interesting math can be
> >> >derived without assuming the Successor Axiom or its alternative.
> >>
> >> Strikes most of us as an extremely silly idea. It's like you're
> >> saying you want to see what you can derive about the natural
> >> numbers without using the fact that they're the natural numbers.
> >>
> >>
> >
> >I don't know why you think this is "like". I think Q is more like
> >what you're suggesting, because you don't have access to induction.
> >And a lot of people like to look at Q.
>
> Not at all. Q is an actual structure, not a list of axioms.
> And it's a structure that people are interested in for
> natural reasons - what is the intended model for PA minus
> the successor axiom, and what does that model have to do
> with anything else in mathematics?
>
> >Here are things you can prove in such a system:
> >(1) (x)(y)(x + y = z => y + x = z)
>
> Seems a little implausible that one can _define_
> x + y without successors. What's the definition
> of x + 1?

you could make a strict theory of computers that disallowed TMs because of the
infinite tape. then you are stuck with F.S.Ms and indeed not every number has
a successor for any specified model.

<-----finite memory---->
suc(suc(suc(suc(suc(0)))))

Herc

Relevant Pages

  • Re: Successor Axiom: on what grounds TF?
    ... Joe Kearney wrote: ... Clearly you have to have the successor relationship in order to talk ... but you don't need the Successor Axiom to do ... What one supposes is induction, ...
    (sci.logic)
  • Re: Group on arbitrary ordinal
    ... Arturo Magidin wrote: ... The successor issue was for the following reason: ... Constructing N, a monoid, on omega, we used finite induction: ...
    (sci.math)
  • Re: doubting successor
    ... > way and take the belief part in earnest. ... > You are calling PA3 the Successor axiom. ... > successor. ... I say this not to convince you - because I ...
    (sci.logic)
  • Re: doubting successor
    ... You are calling PA3 the Successor axiom. ... So the formalization is to help explain things that I want to be true ... Mitch Harris ...
    (sci.logic)
  • Re: Group on arbitrary ordinal
    ... Arturo Magidin wrote: ... The successor issue was for the following reason: ... Constructing N, a monoid, on omega, we used finite induction: ...
    (sci.math)
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