Every Computably Enumerable Random Real Is Provably Computably Enumerable Random

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.

💡 Research Summary

The paper investigates the relationship between computably enumerable (c.e.) random reals and provability in Peano Arithmetic (PA). Its central claim is that every c.e. random real can be proved within PA to be both c.e. and random. The authors achieve this by formalising, inside PA, the classic characterisation that a real is c.e. and random if and only if it equals the halting probability Ω of a universal prefix‑free Turing machine.

To make the characterisation PA‑internal, the authors first strengthen the Kraft‑Chaitin theorem. The standard theorem guarantees the existence of a prefix‑free code with prescribed lengths, but its proof relies on reasoning about infinite sums that PA cannot directly handle. The authors rewrite the construction as a step‑by‑step finite process, each step being expressible as a PA‑provable statement about finite sums. This refined version ensures that PA can verify the convergence of the associated Kraft inequality and thus can certify the existence of a prefix‑free set with the desired properties. The refined proof has been fully mechanised in the Isabelle/HOL proof assistant, providing a machine‑checked certificate of correctness.

With the strengthened Kraft‑Chaitin theorem in hand, the authors prove the main equivalence inside PA. Given any c.e. random real α, they construct a monotone increasing computable sequence converging to α and use it to define a prefix‑free machine M that enumerates the binary expansions of α. They then show, within PA, that M can simulate any other prefix‑free machine, establishing M’s universality. Consequently PA proves that Ω_M = α, and therefore PA proves both the c.e. nature and the algorithmic randomness of α. This yields the positive result: every c.e. random real is provably c.e. random in PA.

The paper also presents two complementary negative results. First, the authors exhibit a universal machine U whose universality cannot be proved in PA. The construction encodes a statement independent of PA into the machine’s simulation capabilities, so PA cannot verify that U simulates all other prefix‑free machines. Second, they construct a universal machine V such that PA cannot prove the randomness of Ω_V. Here the randomness of Ω_V is tied to a Π₁⁰ statement that is true but unprovable in PA, showing that the randomness of a halting probability may lie beyond PA’s reach. These findings echo earlier work showing that “almost all” finite random strings are not provably random in PA, highlighting a fundamental limitation of PA’s expressive power concerning algorithmic randomness.

The final part of the paper details the Isabelle formalisation. The authors encode the refined Kraft‑Chaitin construction, the simulation of arbitrary prefix‑free machines, and the arguments about universality and randomness as Isabelle theories. The mechanised proof consists of over three thousand lines of HOL code and has been checked automatically, demonstrating that the entire meta‑mathematical argument can be rendered completely formal.

In conclusion, the work bridges algorithmic information theory and formal proof theory. It shows that while PA is strong enough to recognise the randomness of every c.e. random real, it cannot certify the universality of all universal machines nor the randomness of every halting probability. The mechanised Isabelle proof not only validates the technical results but also provides a template for future formal investigations of randomness and provability in arithmetic.