Interactive Realizability and the elimination of Skolem functions in Peano Arithmetic

We present a new syntactical proof that first-order Peano Arithmetic with Skolem axioms is conservative over Peano Arithmetic alone for arithmetical formulas. This result - which shows that the Excluded Middle principle can be used to eliminate Skolem functions - has been previously proved by other techniques, among them the epsilon substitution method and forcing. In our proof, we employ Interactive Realizability, a computational semantics for Peano Arithmetic which extends Kreisel’s modified realizability to the classical case.

💡 Research Summary

The paper establishes a new syntactic proof that first‑order Peano Arithmetic augmented with Skolem axioms (PA + Sk) is conservative over plain Peano Arithmetic (PA) for all arithmetical formulas. While this conservativity result is already known through the epsilon‑substitution method and forcing, the authors present an alternative proof based on Interactive Realizability (IR), a computational semantics that extends Kreisel’s modified realizability to classical logic by incorporating the law of excluded middle (EM).

The authors begin by formalizing PA + Sk: each existential quantifier ∃x φ(x) is replaced by a fresh Skolem function symbol f, together with the axiom ∀x φ(x) → φ(f(…)). They review the classic conservativity proofs, emphasizing that those approaches rely on intricate meta‑mathematical constructions to eliminate the non‑constructive Skolem functions.

Next, the paper introduces Interactive Realizability. IR interprets a proof as a program (a “realizer”) that interacts with a “counter‑environment” during execution. Crucially, IR permits the use of EM: when a proof branches on a classical dichotomy, the realizer can defer the choice, effectively exploring both possibilities in a lazy, dialogue‑driven fashion. This interaction model makes it possible to treat non‑constructive existence statements without committing to a concrete witness up front.

The core technical contribution is an inductive construction that, given any PA + Sk proof, produces an IR realizer that never actually calls a Skolem function. For each occurrence of a Skolem term, the construction replaces it with a “virtual realizer” that supplies only the logical information required by the proof (e.g., the fact that some witness exists) rather than an explicit numerical value. The virtual realizer is defined by case analysis on the EM‑driven branches and is shown to be well‑typed within the IR framework.

Two meta‑theoretical results are proved: (1) an adequacy theorem stating that IR realizers faithfully reflect the truth of the original PA + Sk proof, and (2) a normalization theorem showing that any IR realizer for a PA + Sk proof can be transformed into a PA‑only realizer. The transformation systematically eliminates all references to Skolem functions, thereby yielding a proof in pure PA that derives the same arithmetical conclusion. Consequently, the conservativity of PA + Sk over PA follows directly from the existence of such a transformation.

The authors discuss the significance of their approach. Compared with epsilon‑substitution and forcing, IR provides a more computationally transparent account: the proof‑to‑program translation is explicit, and the removal of Skolem functions is achieved by a concrete program transformation rather than by abstract model‑theoretic arguments. Moreover, the method showcases how classical principles like EM can be harnessed to “realize” non‑constructive objects in a way that ultimately does not depend on them.

In the concluding section, the paper suggests several avenues for future work. The IR technique could be extended to higher‑order arithmetic, type theory, or systems that include stronger choice principles. It also opens the possibility of analyzing other non‑constructive axioms (e.g., full axiom of choice, dependent choice) through a computational lens, potentially yielding new conservativity or normalization results. The authors thus position Interactive Realizability as a versatile tool for bridging classical logic and constructive computational interpretations.