> I'm inclined to say the corresponding
> natural number is: m = ...951413 --

First, according to your definition and the common convention, the number should have been 314159... which is clearly unbounded and short of being a nonsense it's surely not an integer. Every integer by definition is a finite, hence bounded, number. How can you write anything as ...951413 I can't fathom.

> Does it HAVE to be bounded in
> order to be a natural number?

Yes of course. By definition a cardinality is finite if it equals one of the integers.

> I suspect this is the
> essential idea I didn't take
> into account: the natural
> numbers are generated
> successively (or inductively)
> from zero.

That's right.

> These numbers seem to be a
> "granddaddy" extension of the
> natural numbers -- uncountable
> and having the cardinality of the
> continuum.

You lost me here.

> Interesting to contemplate.
> I think you've helped me understand an
> important distinction.

I could not wish for more.

> Thanks for your help and for
> this wonderful site.

You are welcome.