80-річчя члена-кореспондента НАН України О.С. Ємельянова

In this paper we study local induction w.r.t. Σ1-formulas over the weak arithmetic PA- The local induction scheme, which was introduced in, says roughly this: for any virtual class X that is progressive, i.e., is closed under zero and successor, and for any non-empty virtual class Y that is definable by a Σ1-formula without parameters, the intersection of X and Y is non-empty. In other words, we have, for all Σ1-sentences S, that S implies SX, whenever X is progressive. Since, in the weak context, we have (at least) two definitions of Σ1, we obtain two minimal theories of local induction w.r.t. Σ1-formulas, which we call Peano Corto and Peano Basso. In the paper we give careful definitions of Peano Corto and Peano Basso. We establish their naturalness both by giving a model theoretic characterization and by providing an equivalent formulation in terms of a sentential reflection scheme. The theories Peano Corto and Peano Basso occupy a salient place among the sequential theories on the boundary between weak and strong theories. They bring together a powerful collection of principles that is locally interpretable in PA- Moreover, they have an important role as examples of various phenomena in the metamathematics of arithmetical (and, more generally, sequential) theories. We illustrate this by studying their behavior w.r.t. interpretability, model interpretability and local interpretability. In many ways the theories are more like Peano arithmetic or Zermelo Fraenkel set theory, than like finitely axiomatized theories as Elementary Arithmetic, |Σ1 and ACA0. On the one hand, Peano Corto and Peano Basso are very weak: they are locally cut-interpretable in PA- On the other hand, they behave as if they were strong: they are not contained in any consistent finitely axiomatized arithmetical theory, however strong. Moreover, they extend IΠ1-, the theory of parameter-free Π1-induction

Arithmetic, Set Theory, Reduction and Explanation

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences