Solomon Feferman

Toward Useful Type-Free Theories. I

There is a distinction between semantical paradoxes on the one hand and logical or mathematical paradoxes on the other, going back to Ramsey [1925]. Those falling under the first heading have to do with such notions as truth, assertion (or proposition), definition, etc., while those falling under the second have to do with membership, class, relation, function (and derivative notions such as cardinal and ordinal number), etc. There are a number of compelling reasons for maintaining this separation but, as we shall see, there are also many close parallels from the logical point of view.The initial solutions to the paradoxes on each side—namely Russells theory of types for mathematics and Tarskis hierarchy of language levels for semantics— were early recognized to be excessively restrictive. The first really workable solution to the mathematical paradoxes was provided by Zermelos theory of sets, subsequently improved by Fraenkel. The informal argument that the paradoxes are blocked in ZF is that its axioms are true in the cumulative hierarchy of sets where (i) unlike the theory of types, a set may have members of various (ordinal) levels, but (ii) as in the theory of types, the level of a set is greater than that of each of its members. Thus in ZF there is no set of all sets, nor any Russell set {x∣x∉x} (which would be universal since ∀x(x∉x) holds in ZF). Nor is there a set of all ordinal numbers (and so the Burali-Forti paradox is blocked).

Iterated inductive definitions and subsystems of analysis : recent proof-theoretical studies

Inductive definitions and subsystems of analysis.- Proof theoretic equivalences between classical and constructive theories for analysis.- Inductive definitions, constructive ordinals, and normal derivations.- The ??+1-Rule.- Ordinal analysis of ID?.- Proof-theoretical analysis of ID? by the method of local predicativity.

Theories of Finite Type Related to Mathematical Practice

Publisher Summary This chapter deals with the systems of second order, including a number that was selected for study only on syntactic grounds. The present exposition differs by concentrating on finite-type theories that directly reflect logical features of practice and in which everyday mathematics is readily developed. The chapter also presents closure conditions on universes of sets and functions. The elementary closure conditions lead to finite-type structures. Further, closure conditions include quantification functionals for each domain that is regarded as fixed (or definite) and recursion functionals for natural numbers. A number of finite-type theories based on these closure conditions are also presented in the chapter followed by recursion-theoretic models for finite-type theories.

Categorical Foundations and Foundations of Category Theory

This paper is divided into two parts. Part I deals briefly with the thesis that category theory (or something like it) should provide the proper foundations of mathematics, in preference to current foundational schemes. An opposite view is argued here on the grounds that the notions of operation and collection are prior to all structural notions. However, no position is taken as to whether such are to be conceived extensionally or intensionally.

Hilbert's Program Relativized: Proof-Theoretical and Foundational Reductions

In the Symposium on Hilberts Program to which the following was contributed, I was asked to talk about the metamathematical aspects of Hilberts Program (H.P.), while the other two speakers (Simpson and Prawitz) were to deal with the mathematical and philosophical aspects respectively. However, more so than for other foundational schemes, these three aspects of H.P., both as originally conceived and in its subsequent developments, are intimately linked. Here I shall survey a body of proof-theoretical results stemming from H.P., but organized in a way that is closely tied to various reductive foundational aims, albeit going beyond those advanced by Hilbert. I believe this view of reductive proof-theory (not original with me) helps one to better understand what has been achieved thereby than other, more familiar accounts.

Systems of explicit mathematics with non-constructive μ-operator. Part I

Feferman, S. and G. Jager, Systems of explicit mathematics with non-constructive μ-operator. Part I, Annals of Pure and Applied Logic 65 (1993) 243-263. This paper is mainly concerned with the proof-theoretic analysis of systems of explicit mathematics with a non-constructive minimum operator. We start off from a basic theory BON of operators and numbers and add some principles of set and formula induction on the natural numbers as well as axioms for μ. The principal results then state: (i) BON(μ) plus set induction is proof-theoretically equivalent to Peano arithmetic PA; (ii) BON(μ) plus formula induction is proof-theoretically equivalent to the system (Π0∞-CA)

Finitary inductively presented logics

Abstract A notion of finitary inductively presented (f-i.p.) logic is proposed here, which includes all syntactically described logics (formal systems) met in practice. A f.i.p. theory FS0 is set up which is universal for all f.i.p. logics; though formulated as a theory of functions and classes of expressions, FS0 is a conservative extension of PR A. The aims of this work are (i) conceptual, (ii) pedagogical and (iii) practical. The system FSQ serves under (i) and (ii) as a theoretical framework for the formalization of metamathematics. The general approach may be used under (iii) for the computer implementation of logics. In all cases, the work aims to make the details manageable in a natural and direct way.

Formal Theories for Transfinite Iterations of Generalized Inductive Definitions and Some Subsystems of Analysis

Publisher Summary This chapter discusses the formal theories for transfinite iterations of generalized inductive definitions and some subsystems of analysis. The first order systems express a principle for defining specific sets of natural numbers can be iterated υ times. The systems in the language of classical analysis (containing variables for sets of natural numbers) express roughly that there are hierarchies obtained by iterating the hyperjump operation any number less than υ times. An arithmetic formula is one without set quantifiers or set constants; it may contain set parameters. An arithmetical formula in which all quantifiers are bounded is said to be elementary. The usual notations of recursion theory are used formally and informally. These are special cases of Kreisels results concerning intuitionistic systems. The chapter suggests definite technical advantages of the reductions and some limitations of their foundational significance.

Why a Little Bit Goes a Long Way: Logical Foundations of Scientifically Applicable Mathematics

Does science justify any part of mathematics and, if so, what part? These questions are related to the so-called indispensability arguments propounded, among others, by Quine and Putnam; moreover, both were led to accept significant portions of set theory on that basis. However, set theory rests on a strong form of Platonic realism which has been variously criticized as a foundation of mathematics and is at odds with scientific realism. Recent logical results show that it is possible to directly formalize almost all, if not all, scientifically applicable mathematics in a formal system that is justified simply by Peano Arithmetic (via a proof-theoretical reduction). It is argued that this substantially vitiates the indispensability arguments.

Logics for Termination and Correctness of Functional Programs

In the literature of logic and theoretical computer science, types are considered both semantically as some kinds of mathematical objects and syntactically as certain kinds of formal terms. They are dealt with in one way or the other, or both, in all programming languages, though most attention has been given to their role in functional programming languages. One difference in treatment lies in the extent to which types are explicitly represented in programs themselves and, further, can be passed as values. Familiar examples, more or less at the extremes, are provided by LISP—an untyped language— and ML—a polymorphic typed language. The aim of this paper (and the more detailed ones to follow) is to provide a logical foundation for the use of type systems in functional programming languages and to set up logics for the termination and correctness of programs relative to such systems. The foundation provided here includes LISP and ML as special cases. I believe this work should be adaptable to other kinds of programming languages, e.g. those of imperative style.