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# (reference request) Chaitin's constant is incompressible

I've been looking for a full, detailed proof that Chaitin's constant is incompressible, i.e. there is a universal constant $c$ such that every program writing first $n$ digits of $\Omega$ has length at least $n-c$. I wasn't able to find such detailed proof, so I'm asking this question.

This is in Downey and Hirschfeldt: Algorithmic randomness and complexity, Theorem 6.1.3, which cites

Chaitin, G. Information-theoretical characterizations of recursive infinite strings, Theoretical Computer Science, vol. 2 (1976), 45-48.

See Chaitin's very illuminating book http://arxiv.org/abs/math/0404335 (Meta Math! The Quest for Omega).

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# (reference request) Chaitin's constant is incompressible

I've been looking for a full, detailed proof that Chaitin's constant is incompressible, i.e. there is a universal constant $c$ such that every program writing first $n$ digits of $\Omega$ has length at least $n-c$. I wasn't able to find such detailed proof, so I'm asking this question.

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This is in Downey and Hirschfeldt: Algorithmic randomness and complexity, Theorem 6.1.3, which cites

Chaitin, G. Information-theoretical characterizations of recursive infinite strings, Theoretical Computer Science, vol. 2 (1976), 45-48.

I have two questions: what is $U_s$? I couldn't find the definition anywhere in paper. Second, why does $\Omega_s\uparrow n\neq\Omega\uparrow n$ lead to contradiction? - Wojowu Aug 1 at 17:26

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