Re: Infinities

Hal Finney (hal@rain.org)
Fri, 7 Nov 1997 14:06:15 -0800

John K Clark, <johnkc@well.com>, writes:
> In my last post I may have been unclear about that I *I = I bit. My point was
> that a decimal with a countable number numbers of numbers in it can be
> converted into a fraction, it's known that there is only a countable number
> of such fractions, so there is only a countable number of such decimals.

I don't believe this is true. Consider the following number, whose name
I forget:

.123456789101112131415161718192021222324...979899100101102103104105...

This is created by writing the series 1,2,3,4,5,6,7,8,9,10,11,12,13,...
and concatenating all the digits. It is known to be an irrational number.
Clearly the number of digits in it is countably infinite, as with any
decimal. Yet you will not be able to convert it into a fraction.

Hal