A finite algebraic target theorem for Kreisel’s conjecture

Abstract

Friedman’s Problem 34, attributed there to Kreisel, states that for Peano arithmetic formalized precisely as in Kleene, a uniform bound on the proof lengths of all numeral instances \(A(\bar{n})\) entails the provability of the universal closure \(\forall x\,A(x)\). We refer to this statement as Kreisel’s conjecture (KC). Parikh proved the corresponding implication for a relational presentation of Peano arithmetic. Bounded proof analyses have a finite automata-theoretic shadow: suppressing formula instantiations leaves a finite partial automaton whose states record only which schematic continuations remain possible. We characterize exactly when the map sending a word to the idempotent power of its transformation factors through a finite semilattice. The finite criterion separates a failure of commutativity among the idempotents associated with the generators from one that first appears for a composite word. The latter always has a witness whose length is bounded by the size of the transition monoid. In the positive case, the idempotent-power map retracts the transition monoid onto its full set of idempotents. Parikh’s case changes the picture. For the prefix automata of bounded proof trees, already before minimization every nonempty word has the empty partial map as its idempotent power, and the nonidentity transformations form a nilpotent ideal. Yet the universal implication still holds. What Parikh’s proof retains is not continuation behaviour but realization: which analyses prove which numeral instances. The resulting realization profiles determine the coarsest finite quotient of \(\mathbb {N}\) retaining every realization set. In Parikh’s presentation, the resulting equivalence relation and its classes are effectively definable in Presburger arithmetic, and the assignment sending each numeral to its realization profile is effectively ultimately periodic. The comparison separates exact algebraic compression of continuations from the arithmetic information needed for the passage to the universal closure.