Weak essentially undecidable theories of hereditarily finite multisets

Abstract

We introduce two first-order theories of hereditarily finite multisets: a schematic theory WF^- and a finitely axiomatized theory F^-, in the language with the empty multiset, singleton formation, multiset union, and a containment relation. We prove that WF^- is mutually interpretable with Robinson&#39;s theory R, and F^- with Robinson arithmetic Q; in particular, F^- is essentially undecidable. Multisets thereby join numbers, strings, trees, sets, and sequences in the mutual-interpretability classes of R and Q. The distinctive obstacle of the multiset case is the simultaneous failure of the standard devices for recovering ordered pairs: positional order, local order on immediate constituents, and idempotence-based Kuratowski pairing. We show that order is recoverable from bare multiplicity: the term pi(x,y) = <x> u <x> u <y> is provably injective in F^-, yielding a direct interpretation of the Kristiansen-Murwanashyaka tree theory T; conversely, F^- is interpreted in Q by arithmetizing a normal-form calculus for multiset terms within bounded arithmetic. Each structural axiom of F^- is shown independent of the others, with finite or Presburger-definable decidable witnesses, and the containment axiom is conservative. As an application, we identify Spencer-Brown&#39;s forms modulo commutative juxtaposition with hereditarily finite multisets and locate the boundary of essential undecidability within the calculus of indications.