
My Favorite Strange Number: Ω - llambda
http://scienceblogs.com/goodmath/2008/12/my_favorite_strange_number_cla.php
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Symmetry
And oddly enough, one of my favorite numbers is the Ω of S=k*ln(Ω) fame.

<http://en.wikipedia.org/wiki/Microcanonical_ensemble>

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sehugg
Different but related -- The busy beaver game and the Σ function:

<http://en.wikipedia.org/wiki/Busy_beaver>

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JonnieCache
This is discussed in the following google Tech Talk, "Incompleteness: a
personal perspective" from Cristian Claude

<http://www.youtube.com/watch?v=tYjmiT422yQ>

He discusses, amongst other things, the work done on computing this
uncomputable number, and why that isn't a contradiction. I think one of the
people involved is in the audience.

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ced
I don't understand how omega can be be incompressible. Compression, from the
information theory perspective, allows me to send the string

    
    
      Digits #10000 to #100000 of the decimal representation of omega.
    

which any competent human could "decompress" into the corresponding digit
sequence (inasmuch as it's computable)

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andrewcooke
you're right that the isolated assertion is not at all sensible, but the wider
context is that you are considering all possible algorithms and the sequences
that they generate, and looking for some way of compressing as many of those
as possible. you should read one of chaitin's books for a better idea (they're
generally "accessible", but at some point you just give up trying to make
complete sense of it all...)

in other words, you are choosing your compression for one particular algorithm
(the one that generates omega - you could go further and say it's 1 for omega
and 0 for everything else). this is a common problem in arguments about
compressibility (if we're only trying to compress beethoven's ninth then i can
do that in one bit too) - you have to be talking in an "average" or
statistical sense for things to be useful.

having said that, the exact nature of what they're saying is still not clear
to me (ie it's not clear exactly how the details of the argument work out, or
how this eventually ties back to
<http://en.wikipedia.org/wiki/Shannons_source_coding_theorem>).

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andrewcooke
i think another replier here may also be correct that chaitin, in particular,
includes the length of the decompression code. unfortunately i can't delete my
answer for some reason (too old i guess).

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ced
No, I think that your answer made the most sense. The length of the
decompression code is a constant, whereas Omega has an infinity of digits, so
it doesn't matter.

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kzrdude
The link to Calude's paper in the article is broken, it is this:
<http://www.cs.auckland.ac.nz/~cristian/Calude361_370.pdf>

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doodyhead
The Halting Problem theoretically defines the limits of computation, but has
anyone in a pragmatic, everyday coding situation actually reached that limit?
In the practical sense I've never seen it have any bearing. Perhaps the
problems I'm working on are too mundane.

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khafra
I had just remarked, the other day, that people obsessed with Chaitin's Omega
are much more interesting than people obsessed with Pi.

