1. Introduction: The Internal Logic of Arithmetic

The idea of an internal logic of arithmetic or arithmetical logic is inspired by a variety of motives in the foundations of mathematics. The development of mathematical logic in the twentieth century, from Hilbert to the contemporary scene, could be interpreted as a continuous tread leading to arithmetical logic. Arithmetization of analysis with Cauchy, Weierstrass and Dedekind and arithmetization of algebra with Kronecker have lead to the foundational inquiries initiated by Hilbert. Frege’s logical foundations of mathematics, mainly arithmetic, have contributed to clarify philosophical motives and although Frege’s logicism has not achieved its goals, it has given birth to Russell’s theory of types and to some extent to Zermelo’s set-theoretic cumulative hierarchy while launching philosophical logic and philosophy of language. But the “arithmetism” I have in mind here is mostly anti-Fregean in that in turns logicism upside down and asks the question: “how far can we go into logic with arithmetic alone” rather than the Fregean question: “how far can we go into arithmetic with deductive logic alone?” It is Kronecker’s polynomial arithmetic that guides here and the purpose of this book is to see how far a Kroneckian constructivist program can go in the arithmetization (and algebraization) of logic in the twenty-first century. The present work has been conceived has a sequel to my 2002 book Internal Logic, Foundations of Mathematics from Kronecker to Hilbert (Kluwer) and as a continuation of my efforts towards an arithmetical logic.