Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λ-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege’s Grundgesetze der Arithmetik for which Russell applied his famous paradox 1 and this led him to introduce the first theory of types, the Ramified Type Theory (rtt). We present rtt formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from rtt leading to the simple theory of types stt. We present stt and Church’s own simply typed λ-calculus (λ→C 2) and we finish by comparing rtt, stt and λ→C. §1. Introduction. Nowadays, type theory has many applications and is used in many different disciplines. Even within logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. But, before 1903 when Russell first introduced

In 1922, Thoraf Skolem published a paper titled "Some remarks on Axiomatized Set Theory". The paper presents a new proof of... This dissertation focuses almost exclusively on the first half of this project -- i.e., the half which tries to expose an initial tension between Cantor's theorem and the Löwenheim-Skolem theorem. I argue that, even on quite naive understandings of set theory and model theory, there is no such tension. Hence, Skolem's Paradox is not a genuine paradox, and there is very little reason to worry about (or even to investigate) the more extreme consequences that are supposed to follow from this paradox. The heart of my...

Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s

The evolution of types and logic in the 20th

The introduction of a general definition of function was key to Frege’s formalisation of logic. Self-application of functions was at the heart of Russell’s paradox. Russell introduced type theory in order to control the application of functions and hence to avoid the paradox. Since, different type systems have been introduced, each allowing different functional power. Despite the extensive use of types in many applications, there remains many “non believers ” in type theory. In this talk, I will briefly review the evolution of types from the time of Euclid (325 B.C.) to the mid of the 20th century. Then, I will introduce de Bruijn’s formulation of functions and types in Automath, his famous system for automating mathematics. De Bruijn’s formulation is a living example which illustrates that while type theory is useful, there are many other considerations that need to be accommodated when attempting to “computerize ” a system. This talk is of interest for anyone