The Torah and Chaitin's Omega Number

It was a key belief of the ancient Greeks that our universe is rationally comprehensible, that it is possible to explain all of the diversity, richness and apparent complexity of the natural world using only a few basic principles. To this day, many scientists are still searching for such a theory, with the expectation that such a theory exists.

In the early twentieth century, the famous mathematician David Hilbert followed this ancient Greek tradition, proposing the idea that all mathematical facts can be derived from only a handful of axioms. In the 1930's, the mathematician Kurt Goedel proved that such a scenario is impossible by showing that for any proposed finite axiom system for arithmetic, there must always be true statements which are unprovable within the axiom system, if we are to assume that the axiom system has no inconsistencies. The mathematician, Alan Turing, extended this result to show that it is impossible to design a computer program which can determine whether any other computer program will eventually halt. In the latter half of the 20th century, the mathematician Gregory Chaitin defined a real number between zero and one, which he calls "Omega", to be the probability that a computer program halts. And Chaitin showed that:

1) For any mathematical problem, the bits of Omega, when Omega is expressed in binary, completely determine whether that problem is solvable or not.

2) The bits of Omega are random, and only a finite number of them are even possible to know. (Therefore, an infinite number of them are impossible to know.)

3) Hence, most mathematics problems are impossible to solve, and most mathematical facts are impossible to prove.

From this, we can conclude that most of mathematics is not rationally comprehensible. In other words, most mathematical facts are, as Chaitin says, "true for no reason". Chaitin's work completely destroys the Greek paradigm that our universe is rationally comprehensible, since he logically proves that the majority of the laws of mathematics (which are part of our universe) are not even rationally comprehensible.

So what does all of this have to do with Torah? Well, the Torah has a different way of looking at the world than the ancient Greeks had. While the ancient Greeks believed that the universe is rationally comprehensible, the Torah view is that as a whole, the universe is incomprehensible to a finite human being. For instance, when the Torah says, "Thou shalt not murder", it does not give a rational explanation as to why murdering is wrong; the Torah viewpoint is that it is wrong to murder because G-d said so. In fact, Jews are commanded to uphold all of the 613 commandments of the Torah not because they make sense to us, but for no other reason than because G-d commanded us to do so. So Chaitin's Omega number shows us that mathematics is more in line with the Torah religious view of the universe than the ancient Greek philosophical view of the universe. And this is the first theorem in the history of mathematics to do such.

In my opinion, Chaitin's theorem should be called the "Fundamental Theorem of Mathematics", because it completely describes the nature of all of mathematics using just one number, the halting probability. I have used some of his ideas in an article that I wrote here: http://arxiv.org/abs/cs.CC/0507008

Acknowledgements: I would like to thank Gregory Chaitin for his helpful comments. His great work can be found here: http://www.cs.auckland.ac.nz/CDMTCS/chaitin/