# Infinities

John K Clark (johnkc@well.com)
Fri, 7 Nov 1997 09:59:14 -0800 (PST)

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On Thu, 6 Nov 1997 Hal Finney <hal@rain.org> Wrote:

>>Me
>>But can you identify ALL the points or just some of them ?

>This seems to be the main point. You can identify all of the points.
>Every real number in [0,1] corresponds to an infinite binary fraction.

Infinite? OK, but the big question is what order infinity, just how many
decimal places do you need to represent all the points on that line? We know
it can't be the smallest infinity, that of the integers (call it I) because
we already know that I *I = I so the line would only have I points on it and
Cantor proved that's not true.

>The slight complication is that some numbers have two representations,
>such as 1.0 and 0.1111..., the analog of 0.9999... in decimal. But
>this can be dealt with since there are only countably many such
>numbers.)

The complication is not so slight BECAUSE there are only countably many such
numbers. It's very easy to convert a decimal number of any base into a
fraction and that means decimals can only deal with the rational numbers.
Decimals can't even represent a simple irrational number like the square root
of 2, much less PI, and don't even think about using them for Chaitin numbers.

Using decimals will never work because if you give me 2 rational numbers,
I don't care how close together they are, I can find an aleph-one (and maybe
more) number of irrational numbers between them.

John K Clark johnkc@well.com

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