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

From: |-|erc (H_at_r.c)
Date: 01/31/05 

Date: Mon, 31 Jan 2005 11:09:15 +1000

"The Ghost In The Machine" <ewill@sirius.athghost7038suus.net> wrote
> >> then omega_U = infinity.
> >>
> >> This is not in error although one might question the assumptions.
> >>
> >
> > why is it not an error?
>
> Because it is not. Given your assumptions and the computation
> given of bitsize, it's a foregone conclusion.

Are probabilities allowed to reach oo?

>
> > why are u adding exponential sums?
>
> The issue is similar to the harmonic series, and diverges for much
> the same reason. If bitsize(x) = n then x must be between
> 2^n (inclusive) and 2^(n+1) (exclusive), which gives 2^n - 1
> numbers to play with, each of weight 2^(-n). The factor
> is therefore (1 - 1/2^n), and the total sum diverges without
> bound if one includes every program, not merely the halting
> ones (one might call that omegamax_U).
>
> Even if only a third of the programs halt, one still sees
> divergence without bound.
>
> If you're not familiar with the harmonic series:
>
> Let S = 1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ...

Your sequence is 1 + 2 + 4 + 8 + 16...

For what halt values would it ever converge?

Herc

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