Predicative Justification and Development of a Second Order Theory of Finite Sets

Abstract

We will predicatively justify the induction axioms for arithmetical sets, which are the induction axioms of (the first-order) Peano Arithmetic, in the predicative second order theory of hereditarily finite sets FSS. The comprehension axioms of FSS permit the predicative formation of infinite classes of hereditarily finite sets. The predicative formation means that the infinite classes are comprehended without quantification over infinite classes. We interprete into FSS the theory FSI which includes also the axiom of induction usable with the classes of FSS. This will show that the induction is consistent with FSS. We also show FSI equivalent to ACA0. The second order theory FSI was developed for its use in computer programming. For the motivation and more details see [8]. The predicative justification of FSI within FSS was inspired by S. Feferman and G. Hellman's paper Challenges to Predicative Foundations of Arithmetic [2]