Re: abundance of irrationals

mueckenh_at_rz.fh-augsburg.de
Date: 01/21/05 

Date: 21 Jan 2005 08:53:34 -0800

David Kastrup wrote:

> That is complete hogwash. The Peano axioms don't talk about
> computation at all. They talk about existence. And the existence
> does not depend on an ability to compute. They don't say "ok, I
think
> if you dig among the atoms of the universe, you might be able to find
> a number". The universe is not concerned. The ability of the
> universe or a calculator or a brain to count is not concerned. To
> each natural number, a successor exists. Period. By assumption.
> That is what an axiom is.
>
> If you want to use different axioms, feel free to do so. But then
you
> are not talking about the same thing as mathematicians, and you are a
> liar and fraud if you claim that your different choice of axioms
makes
> the mathematics built on the standard axioms in any way invalid.

Why did Peano formulate his axioms as he did? Don't you believe that
there is some remote relationship to the basic principle of counting?
Counting needs natural numbers, and the fundamental property of natural
numbers is this axiom: if n is a number, then n + 1 is a number. That
is even the fundament of mathematics (except geometry). Mathematics
could exist and, in fact, has existed without Peano and without set
theory, but not without this axiom.

But you are right, mathematics has been perverted and abused to such a
degree by set theoretical notions, that its basic principles sound mad
and useless. People smile about counting as a branch of mathematics
whereas: "An active research program strives to gain insight into very
finite situations by invoking transfinite numbers that dwarf even
Cantor's remoter alephs. There is the first inaccessible Cardinal and
after it, Hyper-Inaccessibles; then the First Mahlo Cardinal. There are
Cardinals Indescribable, Huge, Supercompact, Rowbottom and (it may be)
Ramsey Cardinals, and then the Extendible and perhaps the Ineffable
Cardinals, not to mention those that are Inexpressible. Devising them
isn't only a game of one-upmanship on a gigantic scale, but a serious
attempt to prove important theorems which are unprovable without their
condescenting help: fetching from afar carried to its logical extreme."
(Kaplan)

Regards, WM

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