Re: abundance of irrationals!)


Randy Poe wrote:
> W. Mueckenheim wrote:

> > There is no other reliable way to prove
> > anything about properties and existence of n e N than by induction.
>
> That may or may not be true, but at any rate induction
> doesn't run 1, 2, 3, ... to N. It runs 1, 2, 3, ... n
> for any finite n you are interested in.

But how do you find Card(N) = aleph_0 > n e N from the Peano-axioms?
>
> > > Again, why must there be a "greatest difference"? Do you
> > > think all limits exist?
> >
> > All differences between two finite natural numbers exist by
> > definition, because the sum of two n does exist and is a natural
> > number.
>
> Why does that imply there is a largest one? Why does
> "exist" imply "there's a maximum"?

No. But EVERY difference that can be defined by two natural numbers
does exist. Therefore, there are only finite differences (= distances)
in N.

Regards, WM

.

Loading