Posted: Sep 15, 2015 1:18 am
by scott1328
I have listened to the video and read the paper you quoted. And in a video and paper aimed at popularizing deeply technical ideas, I would expect such language and metaphors to make his ideas accessible. I do not believe he is claiming what you think he is.

And if he is claiming that reals have infinite precision, then he is playing fast and loose with the mathematics.

Chaitin does make a particularly egregious error in the paper you cited

So in fact there are more uncomputable reals than computable reals. From Cantor's theory of infinite sets, we see that the set of uncomputable reals is just as big as the set of all reals, while the set of computable reals is only as big as the set of whole numbers. The set of uncomputable reals is much bigger than the set of computable reals.

#{uncomputable reals} = #{all reals} = ℵ1
#{computable reals} = #{computer programs} = #{whole numbers} = ℵ0
ℵ1 > ℵ0

In the text I've quoted Chaitin asserted that the cardinality of the reals, typically referred to as C, is equivalent to Aleph-1. This assertion, called the Continuum Hypothesis is formally undecidable in standard formulations of mathematical set theory, that he would assert it as true in his popular paper, leads me to doubt the seriousness of this entire paper.