Incompleteness theorems for random reals

Abstract

We obtain some dramatic results using statistical mechanics-thermodynamics kinds of arguments concerning randomness, chaos, unpredictability, and uncertainty in mathematics. We construct an equation involving only whole numbers and addition, multiplication, and exponentiation, with the property that if one varies a parameter and asks whether the number of solutions is finite or infinite, the answer to this question is indistinguishable from the result of independent tosses of a fair coin. This yields a number of powerful Gödel incompleteness-type results concerning the limitations of the axiomatic method, in which entropy-information measures are used. © 1987.