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**On interpretations of bounded arithmetic and bounded set theory.**
*(English)*
Zbl 1183.03029

In [“On interpretations of arithmetic and set theory”, Notre Dame J. Formal Logic 48, No. 4, 497–510 (2007; Zbl 1137.03019)], R. Kaye and 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 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 \(V_n\).

Reviewer: Roman Kossak (New York)

### MSC:

03C62 | Models of arithmetic and set theory |