Incompleteness theorems for random reals
Using approaches from thermodynamics and information theory, the author shows several results in diophantine theory with stochastic proofs. Indeed, these ideas can be used to construct an exponential diophantine equation whose solutions are, in a very definite sense, random; e.g., in the sense of Martin-Löf. Many limitative theorems of logic follow from such results; e.g., modified incompleteness theorems of Gödel type. In the process, the author shows the equivalence of several different senses of random: (1) Martin-Löf; (2) Chaitin and (3) Solovay. The author seems to believe that God plays dice even with (pure) mathematics.
modified incompleteness theorems of Gödel type
scientific article; zbMATH DE number 4059395