On interpretations of bounded arithmetic and bounded set theory (Q1038653)

In [``On interpretations of arithmetic and set theory'', Notre Dame J. Formal Logic 48, No. 4, 497--510 (2007; Zbl 1137.03019)], \textit{R. Kaye} and \textit{T. L. Wong} proved that Peano Arithmetic is bi-interpretable with the suitably formulated ZF set theory with the axiom of infinity negated. Pettigrew proves an analogous result for the theory \(\text{I}\Delta_0+\exp\). The corresponding set theory is a variant of the axiom system EA introduced by \textit{J. P. Mayberry} [The foundations of mathematics in the theory of sets. Encyclopedia of Mathematics and Its Applications. 82. Cambridge: Cambridge University Press (2000; Zbl 0972.03001)]. EA is a first-order theory in the language including the constant \(\emptyset\), function symbols for power set, sum set, pair set, and rank operations, as well as unary function symbols \(\{x\in y: \Phi(x)\}\) for each formula \(\Phi(x)\) in which each quantifier is bounded by a term. The system, called EA\(^*\), includes the axioms of Extensionality, Pair Set, Sum Set, Power Set, Foundation, Axiom Schema of Subset Separation for Bounded Formulas, Dedekind Finiteness, and a special axiom, called Weak Hierarchy Principle, which guarantees that for every set there exists the first level of the cumulative hierarchy at which that set occurs. The interpretation of EA\(^*\) in \(\text{I}\Delta_0+\exp\) is the standard Ackermann interpretation. The inverse interpretation uses arithmetic operations defined using a function that takes each level of the cumulative hierarchy \(V_n\) to a linear ordering of \(V_n\).