Re: abundance of irrationals

From: David Kastrup (dak_at_gnu.org)
Date: 01/12/05 

Date: Wed, 12 Jan 2005 13:15:40 +0100

David Kastrup <dak@gnu.org> writes:

> mueckenh@rz.fh-augsburg.de writes:
>
>> The only criterion for the existence of a number is in my eyes, that
>> it can be identified so that it can be distinguished from any other
>> number. If I gave you two strings, one representing pi, the other
>> one differeing by 1 in the digit number 10^1000, you could not (any
>> nobody could ever) determine, which of them was pi.
>
> Yes, I could. None of them is pi, since pi's decimal representation
> does not terminate. Not after 10 digits, not after 10^1000 digits.
>
>> Therefore, this number does not exist. Pi is a name and a programme
>> for approximation.
>
> By rational sequences? Yes, that's what irrational numbers are. The
> limits of rational-bounded intervals. Congratulations, you are slowly
> catching up with Dedekind.
>
>> By the way, Cantor said Sqrt(3) is but a name. Here he was correct.
>
> Sure, "sqrt(3)" is a name for a number. So is 2 sin(pi/3). But both
> are names for the same number. Likewise, "13" and "10+3" and "8+5"
> are names for the same number. You can't write down the number, but
> you can write down names for it.

Actually, sqrt(3) and 2 sin(pi/3) are not completely equivalent
according to my somewhat oversexed newsreader. He made a clickable
link from sqrt(3), but not from 2 sin(pi/3). I was unable to resist
clicking this link, and got

SQRT(3) Linux Programmer's Manual SQRT(3)

NAME
       sqrt, sqrtf, sqrtl - square root function

SYNOPSIS
       #include <math.h>

[...]

I bet Cantor did not think of _that_ one. 

-- 
David Kastrup, Kriemhildstr. 15, 44793 Bochum