[FOM] Re: Indispensability of the natural numbers

[Timothy Y. Chow]

On Wed, 19 May 2004, Vladimir Sazonov wrote:
> Timothy Y. Chow wrote:
> > Do you agree that our mental concepts of symbols and rules are also vague
> > illusions of something solid?  
> *General* mental concepts - of course! See also below.
[...]
> First, the (highly informal, vague and floating) entity (N)
> does not vanish. Only our understanding and intuition on it
> may be changed in some way.

Thanks for your long and detailed message.  With this clarification of 
your point of view, and with the new (to me) insight that I described in 
my last post to FOM, I find that I disagree with you much less than I 
originally thought.  A few small points remain.

For example, I now see no reason why the inconsistency of PA should 
threaten platonism.  Platonism has successfully survived set-theoretic
paradoxes, Goedel's theorems, and the independence of the continuum
hypothesis.  The notion that N is a determinate, independently existing
object will not be refuted by a discovery that we were wrong about one
of its properties.

> No standard N can be explicated. Only inside strong theories like ZFC
> we can do that, say via the least infinite ordinal omega. But from
> the "outside" point of view this omega still remains vague.

In fact, I would say that ZFC does not eliminate the vagueness of which 
you speak, since it doesn't settle every first-order question about N.
ZFC proves only the existence and uniqueness of N.

Questions (for anyone on FOM):

1. Is it provable in a weak system that "If PA is inconsistent, then
   such-and-such a fast-growing function is not total"?

2. PA doesn't prove that "the length of the nth Goodstein sequence" is
   a total function.  Is there an analogous arithmetical statement
   that is unprovable in ZFC?

Tim