Chaitin's Number - It's Not As Uncomputable As You Thought

Chaitin's number is the probability that a randomly chosen programme for a Turing computer will halt. Strictly the numerical value of Chaitin's number is not defined until some standard computing code is agreed. But this is an essentially trivial complication. The received wisdom (well, theorem) is that Chaitin's number is uncomputable. The claim goes that, were it computable knowledge of its numerical value would enable a wide range of intractable mathematical theorems to be proved, such as Goldbach's Conjecture and the Riemann Hypothesis. I have never found the argument for this convincing, but that's probably just my ignorance.

It comes as something as a shock, then, to find that the first 64 binary digits of Chaitin's number have already been computed. They are,


(see Cristian Calude) Since 64 binary digits is roughly 19 decimal places, this means that Chaitins' number is known with far greater precision than any quantity in the whole of physical science. Even the much vaunted anomalous magnetic moments of the electron and muon have been measured (and calculated) only to about 10 decimal places.

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"Blue Lagoon", M8 (the Lagoon Nebula) spans about 30 light-years at an estimated distance of 5,000 light-years toward the constellation Sagittarius: Russell Croman, 2006