From: Kent Crispin <kent@songbird.com>
Date: Mon, 20 Jan 1997 08:29:50 -0800 (PST)
To: Jim Choate <ravage@EINSTEIN.ssz.com>
Subject: Re: [Math Noise] (fwd)
Message-ID: <199701201629.IAA12902@toad.com>
MIME-Version: 1.0
Content-Type: text/plain


Jim Choate allegedly said:
> 
> 
> Forwarded message:
> 
> > Only countably many real numbers, or members of any uncountable
> > set, are denumerable. It is the property of being uncountable,
> > rather than of being real or complex, which is important here.
> 
> In short you are saying there are Reals which can not be expressed in the
> format:
> 
>      AmEm + Am-1Em-1 + ... + A0E0 . B0E-1 + B1E-2 + ... + BnE-n+1

No, that's not what he is saying.  What you have written does not represent 
a *specific* number.  He is saying that IF you have a particular scheme for 
representing *specific* numbers, you can only represent countably many 
-- for any given scheme, there are numbers you can't represent. 

To put it another way a scheme that says "you can represent numbers 
as half infinite strings of digits with a single period somewhere" 
doesn't actually *specify* any numbers.  A scheme that says "start 
with the number 1 and increment it 400 times" actually specifies a 
number. 

> And I contend that ANY number which is Real can be expressed by the decimal
> expansion above. Which clearly qualifies as a formal system.

To be a formal system of the type required, you would also have to 
specify deterministic rules that could generate the "Ai" values.  The 
key distinction is between "expressed by" and "generated by".


-- 
Kent Crispin				"No reason to get excited",
kent@songbird.com,kc@llnl.gov		the thief he kindly spoke...
PGP fingerprint:   5A 16 DA 04 31 33 40 1E  87 DA 29 02 97 A3 46 2F