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