Re: Successor Axiom: on what grounds TF?

From: ken quirici (kquirici_at_yahoo.com)
Date: 02/23/05 

Date: 23 Feb 2005 09:15:22 -0800

Helene.Boucher@wanadoo.fr wrote:
> Torkel Franzen wrote:
> > Helene.Boucher@wanadoo.fr writes:
> >
> > This is indeed a traditional approach, for example in
> > Yesenin-Volpin's thinking. But there isn't anything vague about
> > "natural number" in standard mathematics. For any natural number n,
> > n+1 is also a natural number.
>
> I think here you're conflating two separate things, the property of
> being a natural number and the existence of n+1. Being a natural
> number is, I'd agree, clear. If n is a natural number and (n+1)
> exists, then (n+1) obviously is a natural number. But I don't think
> one can claim that the Successor Axiom is inherent in the meaning of
> the (clear) concept "natural natural." An argument against this is
> that the existence of a thing cannot be inherent in the existence of
a
> property (Hartry Fields make this argument) - or at least this is the
> standard reasoning why one cannot prove the existence of God by the
> ontological argument otherwise, for instance.
>
> Unlike Yesenin-Volpin (what little I understand), I do not subscribe
to
> either the Successor Axiom or the alternative. I am agnostic about
> either possibility, since I don't see grounds for accepting either
one.
> That is one reason why I want to know what your grounds are - maybe
> you have good grounds and you'll convince me!
>
> By the way I'm worried about your appendage of "in standard
> mathematics." Obviously, if there is anything standard in
mathematics,
> it is the Peano Axioms, which include the Successor Axiom. My
question
> is about your grounds for believing in the Successor Axiom. What is
or
> is not standard mathematics is irrelevant, or at least should be
> irrelevant, unless your reply is (the very disappointing!), "I
believe
> in everything that is standard mathematics!"

Does anybody take the approach that suc(n) always exists, but in some
cases suc(n) = n?

Thanks.

Ken

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