William Tait

• PhD
• Professor Emeritus at University of Chicago

“Finitism” (Tait 1981) presents an argument that finitist number theory is primitive recursive arithmetic (PRA). The argument is based on taking seriously the “finite” in “finitism”. But the question remained: what did Hilbert (and Bernays) mean in the early 1920’s through the early 1930’s by “finitism” and in particular, did they restrict finitist...

A strict version of compositional semantics would have all composite meaningful expressions be of the form \(XY\), where \(X\) and \(Y\) are meaningful and the concatenation expresses application of a function (\(X\)) to an argument (\(Y\)). In proof theory, compositionality is violated because of bound variables both in formulas (quantification) a...

Gentzen's original proof was constructively invalid because it applied induction to trees of which it is only assumed that they are well-founded. The Bar Theorem fixes this by stating that every well-founded tree is built up by an inductive procedure and therefore supports proof by induction. The paper suggests eliminating the middle man: Work with...

A defense against a certain kind of skeptical argument against the existence of non-empirical things in mathematics, an argument ultimately based on a view of how words get their meaning.

The story of Gentzen’s original consistency proof for first-order number theory [9], as told by Paul Bernays [1, 9], [11, Letter 69, pp. 76–79], is now familiar: Gentzen sent it off to Mathematische Annalen in August of 1935 and then withdrew it in December after receiving criticism and, in particular, the criticism that the proof used the Fan Theo...

We discuss both the historical roots of Skolem’s primitive recursive arithmetic, its essential role in the foundations of arithmetic, its relation to the finitism of Hilbert and Bernays, and its relation to Kant’s philosophy of mathematics.

Menzler-Trott Eckart . Translated by Smoryński Craig and Griffor Edward . Logic's lost genius: The life of Gerhard Gentzen. History of mathematics, vol. 33. American Mathematical Society, Providence, RI, 2007, xxii+441 pp. - Volume 16 Issue 2 - W. W. Tait

2007 Spring Meeting of the Association for Symbolic Logic - Volume 13 Issue 4 - William Tait

Without Abstract

Gödel regarded the Dialectica interpretation as giving constructive content to intuitionism, which otherwise failed to meet reasonable conditions of constructivity. He founded his theory of primitive recursive functions, in which the interpretation is given, on the concept of computable function of finite type. I will (1) criticize this foundation,...

We discuss the semantical categories of base and object implicit in the Curry-Howard theory of types and we derive derive logic and, in particular, the comprehension principle in the classical version of the theory. Two results that apply to both the classical and the constructive theory are discussed. First, compositional semantics for the theory...

The last section of “Lecture at Zilsel's” [9,§4] contains an interesting but quite condensed discussion of Gentzen's first version of his consistency proof for PA [8], reformulating it as what has come to be called the no-counterexample interpretation. I will describe Gentzen's result (in game-theoretic terms), fill in the details (with some correc...

Restricted to first-order formulas, the rules of inference in the Curry-Howard type theory are equivalent to those of first-order predi- cate logic as formalized by Heyting, with one exception: 9-elimination in the Curry-Howard theory, where 9x : A.F(x) is understood as disjoint union, are the projections, and these do not preserve first- orderedne...

this paper are all formulated in terms of a formula #(X), with only X free. For now, the formula is one in the language of basic set theory and X is a second-order variable. The corresponding condition is that #(A) is true in R(#) for some A and, for no # # is #(A R(#)) true in R(#). The formal expression that this condition is an existence conditi...

this paper. Almost all of them have led to what I hope are improvements in the text

this paper to deny the existence of something called `the mind'. But I do mean to call into question appeals to it in analyzing cognitive notions such as understanding and knowing, where its domain is taken to be independent of what one might find out in cognitive science. In this respect, I am expressing the skepticism of Sellars in "Empiricism an...

representability in intuition. (See [2, p. 40].) But our problem is, of course, not the finiteness of a number, but the infinity of numbers. There is, I think, a di#culty with Bernays' notion of formal object, where this is intended to extend to numbers so large as, not only to be beyond processing by the human mind, but possibly to be beyond repre...

This paper contains a defense against anti-realism in mathematics in the light both of incompleteness and of the fact that mathematics is a ‘cultural artifact.’. Anti-realism (here) is the view that theorems, say, of aritltmetic cannot be taken at face value to express true propositions about the system of numbers but must be reconstrued to be abou...

This paper is in honor of my colleague and friend, Leonard Linsky, on the occasion of his retirement. I presented the earliest version in the Spring of 1992 to a reading group, the other members of which were Leonard Linsky, Steve Awodey, Andre Carus and Mike Price. I presented later versions in the autumn of 1992 to the philosophy colloquium at Mc...

This paper was written in honor of Charles Parsons, from whom I have profited for many years in my study of the philosophy of mathematics and expect to continue profiting for many more years to come. In particular, listening to his lecture on "Sets and classes", published in [Parsons, 1974], motivated my first attempts to understand proper classes...

this paper were read in the Spring of 1997 in the Philosophy Colloquium at the University of Chicago, at the Pacific Division meeting of the APA in March, 1998, and in the Philosophy Colloquium at the University of California-Riverside in November 1998. All of these were based on a manuscript composed in 1986. An even earlier version received valua...

The classical theory of types in question is essentially the theory of Martin-Löf [1] but with the law of double negation elimination. I am ultimately interested in the theory of types as a framework for the foundations of mathematics and, for this purpose, we need to consider extensions of the theory obtained by adding ‘well-ordered types,’ for ex...

The preface by Parsons begins: “This book contains the most substantial philosophical papers I wrote for publication up to 1977, with one new essay added. … The collection is unified by a common point of view underlying the essays and by certain problems that are approached from different angles in different essays. Most are directly concerned with...

Meeting of the Association for Symbolic Logic, Chicago, 1977 - Volume 43 Issue 3 - Carl G. Jockusch, Robert I. Soare, William Tait, Gaisi Takeuti

Meeting of the Association for Symbolic Logic, Chicago 1975 - Volume 41 Issue 2 - John Baldwin, D. A. Martin, Robert I. Soare, W. W. Tait

This chapter presents prove of Normal Form Theorem for the bar recursive functions of finite type. These functions will be represented by the bar recursive terms, which are built up from constants denoting the basic operators of explicit definition, primitive recursion and bar recursion. Rules of conversion will be introduced to express the action...

This chapter discusses the applications of the cut elimination theorem to some subsystems of classical analysis. The principal result presented in the chapter is a constructive consistency proof for the system (Σ12-ADC) of second order number theory with the Σ12 axiom of dependent choice. It is shown that every derivation in this system of an arith...

The reduction of the lambda calculus to the theory of combinators in (Schonfinkel, 1924) applies to positive implicational logic, i.e. to the typed lambda calculus, where the types are built up from atomic types by means of the operation A −→ B ,t oshow that the lambda operator can be eliminated in favor of combinators K and S of each type A −→ (B...

An attractive format for semantics is that in which composite expressions are built up from atomic ones by means of the operation of concatenation and the concatenation XY expresses the application of a function X to an argument Y . The use of relative pronouns presents an obstacle to this form of compositional semantics, since the reference of a r...

The volumes of Godel's collected papers under review consist almost en- tirely of a rich selection of his philosophical/scientific correspondence, includ- ing English translations face-to-face with the originals when the latter are in German. The residue consists of correspondence with editors (more amusing than of any scientific value) and five le...

This chapter presents the concept of constructive reasoning. Turing's analysis of mechanical computation provides a precise model for the basic constructive concept of operating on the finite configurations of atoms according to rules. It should be noted that the present analysis differs in its conception from Kreisel. Kreisel regards the essential...

T 0 will denote Gödel's theory T[3] of functionals of finite type (f.t.) with intuitionistic quantification over each f.t. added. T 1 will denote T 0 together with definition by bar recursion of type o, the axiom schema of bar induction, and the schema of choice. Precise descriptions of these systems are given below in §4. The main results of this...

Takeuti (3) showed that the consistency of analysis (i.e. second order number theory) is finitisticall y implied by the Hauptsatz for second order logic» i.e. by the proposition that every theorem of this system is derivable without cut.1 We will prove that, conversely, the Hauptsatz for this system follows from a certain generalization of the cons...

This chapter describes infinitely long terms of transfinite type. Functionals of higher type have been introduced into proof theory, which gives an interpretation of first order number theory in terms of the impredicative primitive recursive (p.r.) functionals of finite type. The chapter shows that for the consistency of number theory, Gentzen's us...

This paper deals with Hilbert's substitution method for eliminating bound variables from first order proofs. With a first order system S framed in the ε-calculus [2] the problem is to associate a system S ' without bound variables and an effective procedure for transforming derivations in S into derivations in S ′. The transform of a formula A deri...

This paper deals mainly with quantifier-free second order systems (i.e., with free variables for numbers and functions, and constants for numbers, functions, and functionals) whose basic rules are those of primitive recursive arithmetic together with definition of functionals by primitive recursion and explicit definition. Precise descriptions are...

In [1], it is conjectured that if S is a sentence in the first-order functional calculus with identity, and every subsystem of every finite relational system which satisfies S also satisfies S , then S is finitely equivalent to a universal sentence. (Two sentences are finitely equivalent if and only if they are satisfied by the same finite relation...