Re: abundance of irrationals!)


W. Mueckenheim wrote:
> "Randy Poe" <poespam-trap@xxxxxxxxx> wrote
> Tony Orlow wrote:
> >
> > That does not follow. There's no induction that will
> > ever get you from 1, 2, 3, ... to N.
>
> Then you never get to N.

I never get to the "last" element of N, since there is
no last element.

But I will get to every natural number n in a finite number
of inductive steps (n of them). This holds for every n.

> 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.

> > 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"?

- Randy

.

Relevant Pages

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