Complexity, Randomness and Impossible Tasks

Examples such as "number of hairs on my head," " number of different states of a Rubik cube" and "speed of light in centimeters per decade," each specify, using fewer than the 20 words in the given sentence, some particular whole number. The paradoxical nature of the task becomes clear when we realize that the Berry sentence specifies a particular whole number that, by its very definition, the sentence contains too few words to specify. (Not quite identical, but suggestive is this directive: You're a short person on an elevator in a very tall building with a bank of buttons before you, and you must press the first floor that you can't reach.)

What yields a paradox about numbers can be modified to yield mathematical statements about sequences that can be neither proved nor disproved. Consider a formal mathematical system of axioms, rules of inference and so on. Like almost everything else these axioms and rules can be systematically translated into a sequence of 0's and 1's, and if we do so, we get a computer program P.

We can then conceive of a computer running this program and over time generating from it the theorems of the mathematical system (also encoded into 0's and 1's). Stated a little differently, the program P generates sequences of 0's and 1's that we interpret as the translations of mathematical statements, statements that the formal system has proved.

Now we ask whether the system is complete. Is it always the case that for a statement S, the system either proves S or it proves its negation, ~S?

To see that the answer to this question is "No," Chaitin adapts the Berry sentence to deal with sequences and their complexity. Specifically, he shows that the following task is also impossible (though not paradox-inducing) for our program P: "Generate a sequence of bits that can be proved to be of complexity greater than the number of bits in this program."

The program P cannot generate such a sequence, since any sequence that it generates must, by definition of complexity, be of complexity less than P itself is. Stated alternatively, there is a limit to the complexity of the sequences of 0's and 1's (translations of mathematical statements) generated (proved) by P. That limit is the complexity of P, considered as a sequence of 0's and 1's, and so statements of complexity greater than P's can be neither proved nor disproved by P

This is a sketch of a sketch of Godel's incompleteness theorem. It can be interpreted to be a consequence of the limited complexity of any formal mathematical system, a limitation affecting human minds as well as computers.

The above leaves out almost all the details and is, I realize, almost impenetrably dense if you have never seen this kind of argument. It may, however, give you a taste of Chaitin's profound work.

Professor of mathematics at Temple University, John Allen Paulos is the author of best-selling books, including Innumeracy and A Mathematician Plays the Stock Market. His Who's Counting? column on appears the first weekend of every month.

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