Every Computably Enumerable Random Real Is Provably Computably Enumerable Random

TL;DR

This paper proves that all computably enumerable (c.e.) random reals are provably c.e. random in Peano Arithmetic, and demonstrates limitations in proving universality and randomness of certain universal Turing machines within PA.

Contribution

It establishes that every c.e. random real can be proven in PA to be c.e. random and provides a simpler proof of the key theorem linking c.e. randomness and halting probabilities.

Findings

All c.e. random reals are provably c.e. random in PA.

Existence of universal machines whose universality cannot be proved in PA.

Existence of universal machines with halting probabilities whose randomness PA cannot prove.

Abstract

We prove that every computably enumerable (c.e.) random real is provable in Peano Arithmetic (PA) to be c.e. random. A major step in the proof is to show that the theorem stating that "a real is c.e. and random iff it is the halting probability of a universal prefix-free Turing machine" can be proven in PA. Our proof, which is simpler than the standard one, can also be used for the original theorem. Our positive result can be contrasted with the case of computable functions, where not every computable function is provably computable in PA, or even more interestingly, with the fact that almost all random finite strings are not provably random in PA. We also prove two negative results: a) there exists a universal machine whose universality cannot be proved in PA, b) there exists a universal machine $U$ such that, based on $U$, PA cannot prove the randomness of its halting probability. The paper also includes a sharper form of the Kraft-Chaitin Theorem, as well as a formal proof of this theorem written with the proof assistant Isabelle.