Re: abundance of irrationals!)


aeo6 Tony Orlow wrote:
> *** T. Winter said:
> > In article <MPG.1ce4531e5377a768989bd0@xxxxxxxxxxxxxxxxxxxxxxxxx>
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:
> > ...
> > > Perhaps an axiom says that a finite set MUST have a greatest
member. Perhaps
> > > that is why we INSIST that N is infinite, even though we don't
want to deal
> > > with infinite values, because we cannot find an overall largest
finite
> > > number. Sure, that's a problem, but do we need the axiom for
largest
> > > member? What does that really do for us that we can't do another
way that's
> > > better?
> >
> > Give me a finite set of numbers and I will supply you with the
largest.
> > It is *not* an axiom that finite sets of numbers have a largest
number.
> >
> If there is no such axiom, then why do people keep harping on it?

Because it's a theorem.

> It doesn't
> prove anything that there is no definite largest element. The set can
still be
> finite.

A finite set of numbers, with the usual order relationships,
has a definite largest elements. A set without a largest
element can not be finite.

- Randy

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