Re: The first great falacy of Chaitin's Theory

Aatu Koskensilta wrote:
"T.H. Ray" <thray123@xxxxxxx> writes:

Not true. Algorithmic compressibility as pursued in
algorithmic information theory is very useful, particularly for experimental mathematics. E.g., Chaitin's (and Kolomogorov's) definition of the complexity of a sequence supports Chaitin's result of a
"maximally unknowable" number, Omega.

In what sense is Chaitin's Omega "maximally unknowable"?

Physical implications may follow, considering that randomness of
the kind that Chaitin's number exhibits is a property of some
chaotic natural systems.

This suggestion is on the face it completely obscure. Perhaps you can
elaborate on what you have in mind.


If he did that would be rather impressive.

Finding natural applications of novel mathematics is generally the (eventual) goal of mathematical research, besides that of knowledge for its own sake.

It's like the other day when there was some news that an experiment on a quantum system was supposed to show some consequence of there being an undecideable theory. They encoded their theory in a set of quantum states, then claimed that if the result was "random" then their theory was incomplete. Yet, measurement error would as well have quite an "effect", where as well there is the observer/detector situation in the quantum where checking the value of an inner variable disturbs it. So, they illustrated either their claim, or that their theory of quantum variables is incomplete (or inconsistent).

That leads into a similar notion about the standard framework of set theory, in terms of any theory ("metatheory") in which to frame it.

Chaitin's constant purports to be the probability that a "random" algorithm halts. Is it almost zero, is it almost one, is it one half, these questions follow from the definitions of "algorithm", "halting", and of course, "random", over the space of algorithms in their interpretation in their encoding, in the structures that contain them (which don't exist in regular set theories because there's no universe in ZF(C), in which these constructions are framed.)

http://en.wikipedia.org/wiki/Chaitin's_constant

It's funny that it is said that they only need N+1 bits of Chaitin's Omega to make use of it, after noting they can never construct that because its uncomputable, generally indicating that computable processes are not computable.

Well-order the reals.

Regards,

Ross F.
.

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