You might appreciate the additional lens of Chaitin's constant omega then. The probability that a particular universal machine will halt when fed a random progr

You might appreciate the additional lens of Chaitin's constant omega then. The probability that a particular universal machine will halt when fed a random program is uncomputable but also algorithmically random -- meaning the (finite) binary expansions of omega always form a maximally complex sequence, in the Kolmogorov sense. Why? We can't calculate omega... I mean, that figures. But on top of that the mysterious number, which could be 1/2 or 1/e for all we might expect at first, is instead just a completely uncorrelated sequence of bits. At a certain point it just becomes a mess. No process can keep up. In general most logical systems (all of them become relevant as n grows) are completely random, with no characteristic structure.