Re: ******** CAN ANYONE HERE DEFINE CHAITIN'S OMEGA ? ***********

From: r.e.s. (r.s_at_ZZmindspring.com)
Date: 01/31/05 

Date: Mon, 31 Jan 2005 01:57:46 GMT

"The Ghost In The Machine" <ewill@sirius.athghost7038suus.net> wrote ...
> <H@r.c> wrote ...

>> So you CANNOT define Omega then?
>>
>> NOBODY IN SCI.MATH CAN DEFINE OMEGA!!!

> You win this round. Congratulations!

A correct definition has been given many times in
multiple threads -- it's a mystery to me why Herc won't
admit it. And it's equally mysterious why you encourage
his delusions.

Aside from my clarification of the definition at MathWorld,
it's easy to find the definition in
C. S. Calude, et al, "Computing a Glimpse of Randomness"
http://www.expmath.org/expmath/volumes/11/11.3/Calude361_370.pdf
-- which is link that has been posted many times.

> Not that it matters all that much. I will have
> to do some research (or perhaps some kindly soul
> can do it for me :-) ) as to, not Chaitin's Omega,
> but some other constant which might be described
> as
>
> K_U = sum(X = Enc(all halting (P,I) pairs)) ( (2^(-X)))
>
> where:
>
> K_U is a binary constant, and
> Enc(P,I) encodes a program/machine P with its input tape I
> into a unique integer. (There are corresponding
> DecProg() and DecInput() forms, complementary to Enc().)
>
> And just to beat this to death, assume that we have a
> program H that can take as input Enc(P,I) and return true
> if it halts, false if it does not.

In the thread
"Herc defines the HOLY GRAIL OF MATHEMATICS"
I'd already explained to Herc about the problems with the
MathWorld definition of Omega, *and* how to fix it, *and*
provided a reference to the paper by Calude, et al, *and*
explained differences between Omega and the quantity tau,
which is special case of your K_U.

--r.e.s.

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