this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his well-known paradox, an early expression of our motif. The motif becomes fully manifest through the study of functions f :

This paper deals with aspects of my doctoral dissertation 1

. After giving a brief overview of the renewal of interest in logic and the foundations of mathematics in G ottingen in the period 1914-1921, I give a detailed presentation of the approach to the foundations of mathematics found in Behmann's doctoral dissertation of 1918, Die Antinomie der transfiniten Zahl und ihre Auflosung durch die Theorie von Russell und Whitehead. The dissertation was written under the guidance of David Hilbert and was primarily intended to give a clear exposition of the solution to the antinomies as found in Principia Mathematica. In the process of explaining the theory of Principia, Behmann also presented an original approach to the foundations of mathematics which saw in sense perception of concrete individuals the Archimedean point for a secure foundation of mathematical knowledge. The last part of the paper points out an important numbers of connections between Behmann's work and Hilbert's foundational thought. 1. Logic and Foundations of Mathematics in G ...

This manual describes Isabelle’s formalizations of many-sorted first-order logic (FOL) and Zermelo-Fraenkel set theory (ZF). See the Reference Manual for general Isabelle commands, and Introduction to Isabelle for an overall tutorial. This manual is part of the earlier Isabelle documentation, which is somewhat superseded by the Isabelle/HOL Tutorial [11]. However, the present document is the only available documentation for Isabelle’s versions of firstorder

Classical logic is the study of ”safe ” formal reasoning. Western Philosophers de-veloped classical logic over a period of thirty-three centuries after its introduction in the form of syllogistic by Aristotle [1] in the third century B. C. Beginning in the nineteenth century with De Morgan [2] and Boole [3], responsibility for the develop-ment of classical logic moved from the philosophical to the mathematical community.

The accompanying thesis is part of a long-term project to enable computers to do mathematics in the same way that humans do. I will sketch something of the nature of mathematics and the project, and then turn to role of the thesis. Mathematics Mathematics arises from the interaction of two dissimilar modes of reasoning: a ‘soft ’ side, dealing with ideas and analogies, and a ‘hard ’ side, dealing with verification. The ‘hard ’ side is easier to pin down. It consists primarily of formal ‘proofs’, each consisting of a series of assertions. A mathematician can verify that a proof is correct by following it, step by step, checking that each step follows from previous ones via facts already proved to be correct. The ‘soft ’ side is less easily described. It consists of intuitions about the formal objects constructed in mathematical proofs; ideas that one piece of mathematics may analogically correspond to another piece of mathematics; or even analogies between mathematics and objects in the physical world.

In "Principia Mathematica " [17], B. Russell and A.N. Whitehead propose a type system for higher order logic. This system has become known under the name "ramified type theory". It was invented to avoid the paradoxes, which could be conducted from Frege's "Begriffschrift" [7]. We give a formalization of the ramified type theory as described in the Principia Mathematica, trying to keep it as close as possible to the ideas of the Principia. As an alternative, distancing ourselves from the Principia, we express notions from the ramified type theory in a lambda calculus style, thus clarifying the type system of Russell and Whitehead in a contemporary setting. Both formalizations are inspired by current developments in research on type theory and typed lambda calculus; see e.g. [3]. In these formalizations, and also when defining "truth", we will need the notion of substitution. As substitution is not formally defined in the Principia, we have to define it ourselves. Finally, the reaction by Hilbert and Ackermann in [10] on the

In Russell's Ramified Theory of Types rtt, two hierarchical concepts dominate: orders and types. The use of orders has as a consequence that the logic part of rtt is predicative. The concept of order however, is almost dead since Ramsey eliminated it from rtt. This is why we find Church's simple theory of types (which uses the type concept without the order one) at the bottom of the Barendregt Cube rather than rtt. Despite the disappearance of orders which have a strong correlation with predicativity, predicative logic still plays an influential role in Computer Science. An important example is the proof checker Nuprl, which is based on Martin-Löf's Type Theory which uses type universes. Those type universes, and also degrees of expressions in Automath, are closely related to orders. In this paper, we show that orders have not disappeared from modern logic and computer science, rather, orders play a crucial role in understanding the hierarchy of modern systems. In order to achieve our goal, we concentrate on a subsystem of Nuprl. The novelty of our paper lies in: 1) a modest revival of Russell's orders, 1 2) the placing of the historical system rtt underlying the famous Principia Mathematica in a context with a modern system of computer mathematics (Nuprl) and modern type theories (Martin-Löf's type theory and PTSs), and 3) the presentation of a complex type system (Nuprl) as a simple and compact PTS.

. Developing a tool kit for program analysis requires a general metalanguage (or user interface) in which to specify the program analyses, and many past and current approaches are semantics-directed in the sense that they attempt to exploit the structure of the semantics of the program. In this paper we take a tool-maker's perspective at an approach based on two-level semantics, focusing on the flexible way to incorporate and combine a repertoire of program analyses. We conclude by identifying a number of key considerations for the design of semantics-directed frameworks or tool kits for program analysis. Keywords. Program Analysis, Abstract Interpretation, Denotational Semantics, Two-Level Metalanguages, Tools for Program Analysis. 1 Introduction The predominant use of program analysis is to enable compilers to generate better code: to supply information about the context in order to generate more specialised code or in order to validate program transformations. While this is by no ...

This paper presents some fundamental aspects of the design and the implementation of an automated prover for Zermelo-Fraenkel set theory within the well-known Theorema system. The method applies the “Prove-Compute-Solve”-paradigm as its major strategy for generating proofs in a natural style for statements involving constructs from set theory.