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Some Truths Are Unprovable

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Posted by Richard Ruquist on March 14, 2001 23:49:36 UTC

At least in mathematics that is true. And mathematics is the one field where you might expect to find absolute truth.

It has been proven by Godel and expressed as his famous incompleteness theorem. It says that no set of axioms which define a particular system of mathematics can be used to derive or prove everything in that system that is true. There are always theorems in any branch of mathematics that are true but that cannot be proved from the axioms of the math. And this is true for all math but the most simple, which turns out to be plain geometry. It's true for arithmetic.

So I would like you to accept this theorem of mathematics as given, or true if you like, and discuss its consequences, especially in terms of physics and possibly religion.

Being fond of coincidences as shown in my other posts, I have always been fascinated that the guy who cameup with this theorem has a name which means god in both english and hebrew. El is god in Hebrew.

So anyway, one way that you know about unprovable truth in a system is to examine it with another more inclusive system. It is said that the more complete system subsumes the less complete system. And of course, no system is complete all the way to infinity.

In physics, for example, we have the theory of electrocity and the theory of magnetism, both with separate set of equations, which are like the axioms of the respective physics. Then we have the theory of electromagnetism that subsumes both less complete theories. The incompleteness theorem would suggest that there are aspects of electricity and magnetism that are true, but cannot be derived from either of the less complete theories. For instance the propagation of light can be derived from electromagnetic theory, but not either of the other theories.

If we add electrons to the picture, they appear in E&M theory as current, we then need quantum electrodynamics to obtain a complete theory. And as we keep adding particles we get more and more complete theories, until finally with the addition of the graviton we supposedly have a theory of everything. But since such a theory is extremely mathematical, the incompleteness theorem certainly applies. We will never get the whole truth.

Chaitin has more recently extended Godel's work to demonstrate that numbers, or number theory, the basis of math, is much more random than it is regular. I do not understand how he comes to that conclusion. Search on-line for his homepage and read his explanations for yourself. He describes math as a bunch of isolated self-consistent little theories that do not connect to each other, in an ocean of randomness. Perhaps our universe just happened to exploit a particular set of self-consistent mathematics and turned it into physics. Maybe the primordial foam is a sea of random mathematics that springs to life whenever so self-consistency is achieved. If so then god is the island of mathematics we happen to live in. And loop and string theory is the search for god.

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