| The only notion of Big Omega I've ever seen that's worth mentioning is the one discovered by Chaitin and applied by Becher. Chaitin is an expert on topics of mathematical logic and researches uncertainty. One of his results is that nearly all mathematical facts have no relation to any other mathematical facts. The theorems and axiomatic systems of what we usually call mathematics are only a miniscule part of all mathematics.
Chaitin's Big Omega is a real number between 0 and 1, which represents the probability that a Turing program picked at random will halt. One property of this Big Omega is that no digit in the number has any relation to any of the other digits. It's completely random. Further, even if we could find a relation between the digits of Omega, we could construct an Omega prime whose digits we couldn't interrelate. Omega prime and Omega double prime, et cetera are the super-Omegas. The surprising thing is that the super Omegas are represented in reality by the probability that an open-ended computer program (like an OS) will halt after finite input.
There is an excellent article on the subject in the March 10 issue of New Scientist, 2001. Just disregard the unfathomably dumb conclusion that a Theory Of Everything is impossible. Since physics uses only a part of mathematics and not the whole of it, (and physics is an experimental science, not a branch of applied mathematics) even if we cannot know all math we might know all physics. -- Richard Kulisz
Omega as constructed above is far from a real number. The term "Big Omega" is dreadfully overloaded, and it's a shame there isn't a better word for its meaning qua big fella. Since we're bending the math notion all out of shape here anyway, I'd say no one could blame us if we picked a name for ourselves. Let's make a contest out of it: Name Big Omega! -- Peter Merel
The above is darn interesting, though. What does it mean that nearly all mathematical facts have no relation to the others - that a randomly picked set of sentences, whatever that entails, will be consistent with probability one? -- Josh Grosse
Chaitin claims that it means that mathematics is an empirical science. -- Keith Braithwaite
If he thinks it means that, he's wrong - that different axioms give different mathematical systems has been well-known for a very long time. But I'm asking about the formal version of the statement, not its implications. -- Josh Grosse
Then I recommend that you read his books. As far as I understand what he's claiming, it goes far beyond getting different systems by choosing different axioms. -- Keith Braithwaite
Maximally uncomputable
Big Omega as constructed above is far from a real number
How so? It's, by definition a probability, so lies in the interval [0,1]. It happens to be one of the uncomputable real numbers (maximally so, again by definition), but what of that? The first few digits of its binary expansion are well known (and are all 1, as it happens).
That's no co-incidence. It can be shown, apparently, that our current mathematics is only capable of calculating as many digits of Chaitin's number as form the initial block of 1s. However many that turns out to be. There's no way of calculating how many that is, and working out each successive digit requires a much worse that exponential increase in computational resources, if I understand correctly. Chaitin's number is known to have a uniform distribution of digits in any base, so there must be some 0s in there somewhere.
Chaitin's number, although maximally uncomputable, is not the hardest to deal with of its class. It is possible to concoct numbers with similar definitions for which we can't calculate any digits. -- Keith Braithwaite
Simultaneous
Thanks for the reference and summary. Did the Big Omega learn anything during this process I wonder? -- Richard Drake
How could you tell? Big Omega both already knew, and never will know everything here. Big Omega learned all this while simultaneously forgetting it all. -- Peter Merel
Connections
God Constrained By Mathematics
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