Wilfred Sieg and Dirk Schlimm. Dedekind's Analysis of Number: Systems and Axioms.

After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those (...) involving the distinction between characterizing a system and axiomatizing the truths of a system. (shrink)

The philosopher Rudolf Carnap, although not himself an originator of mathematical advances in logic, was much involved in the development of the subject. He was the most important and deepest philosopher of the Vienna Circle of logical positivists, or, to use the label Carnap later preferred, logical empiricists. It was Carnap who gave the most fully developed and sophisticated form to the linguistic doctrine of logical and mathematical truth: the view that the truths of mathematics and logic do not describe (...) some Platonistic realm, but rather are artifacts of the way we establish a language in which to speak of the factual, empirical world, fallouts of the representational capacity of language. Carnap was also the thinker who, after Russell, most emphasized the importance of modern logic, and the distinctive advances it enables in the foundations of mathematics, to contemporary philosophy. It was through Carnap's urgings, abetted by Hans Hahn, once Carnap arrived in Vienna as Privatdozent in philosophy in 1926, that the Vienna Circle began to take logic seriously and that positivist philosophy began to grapple with the question of how an account of mathematics compatible with empiricism can be given.A particular facet of Carnap's influence is not widely appreciated: it was Carnap who introduced Kurt Gödel to logic, in the serious sense. Although Gödel seems to have attended a course of Schlick's on philosophy of mathematics in 1925–26, his second year at the University, he did not at that time pursue logic further, nor did the seminar leave much of a trace on him. In the early summer of 1928, however, Carnap gave two lectures to the Circle which Gödel attended, or so I surmise. At these occasions, Carnap presented material from his manuscript treatise, Untersuchungen zur allgemeinen Axiomatik, that is, “Investigations into general axiomatics”, which dealt with questions of consistency, completeness and categoricity. Carnap later circulated this material to various people including Gödel. (shrink)

The common thread running through the logicism of Frege, Dedekind, and Russell is their opposition to the Kantian thesis that our knowledge of arithmetic rests on spatio-temporal intuition. Our critical exposition of the view proceeds by tracing its answers to three fundamental questions: What is the basis for our knowledge of the infinity of the numbers? How is arithmetic applicable to reality? Why is reasoning by induction justified?

In 1929 Carnap gave a paper in Prague on Investigations in General Axiomatics; a briefsummary was published soon after. Its subject lookssomething like early model theory, and the mainresult, called the Gabelbarkeitssatz, appears toclaim that a consistent set of axioms is complete justif it is categorical. This of course casts doubt onthe entire project. Though there is no furthermention of this theorem in Carnap''s publishedwritings, his Nachlass includes a largetypescript on the subject, Investigations inGeneral Axiomatics. We examine this work here,showing (...) that it provides important insights intoCarnap''s development during this critical period, thetransition from Aufbau to Syntax,especially regarding the nature and motivation ofCarnap''s logicism. Moreover, we show how theAxiomatics influenced Carnap''s student Gödel inreaching the fundamental logical results that soonafterwards undermined Carnap''s project. (shrink)

This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...) so as to shed new light on the relevant strengths and limits of higher-order logic. (shrink)

Ernst Friedrich Ferdinand Zermelo transformed the set theory of Cantor and Dedekind in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization setting out generative set-existence principles. Zermelo thereby tempered the ontological thrust of early set theory, initiated the delineation of what is to be regarded as set-theoretic, drawing out the combinatorial aspects from the logical, and established the basic conceptual framework for the development of modern set theory. Two (...) decades later Zermelo promoted a distinctive cumulative hierarchy view of models of set theory and championed the use of infinitary logic, anticipating broad modern developments. In this paper Zermelo's published mathematical work in set theory is described and analyzed in its historical context, with the hindsight afforded by the awareness of what has endured in the subsequent development of set theory. Elaborating formulations and results are provided, and special emphasis is placed on the to and fro surrounding the Schröder-Bernstein Theorem and the correspondence and comparative approaches of Zermelo and Gödel. Much can be and has been written about philosophical and biographical issues and about the reception of the Axiom of Choice, and we will refer and defer to others, staying the course through the decidedly mathematical themes and details. (shrink)

This paper concerns Carnap’s early contributions to formal semantics in his work on general axiomatics between 1928 and 1936. Its main focus is on whether he held a variable domain conception of models. I argue that interpreting Carnap’s account in terms of a fixed domain approach fails to describe his premodern understanding of formal models. By drawing attention to the second part of Carnap’s unpublished manuscript Untersuchungen zur allgemeinen Axiomatik, an alternative interpretation of the notions ‘model’, ‘model extension’ and ‘submodel’ (...) in his theory of axiomatics is presented. Specifically, it is shown that Carnap’s early model theory is based on a convention to simulate domain variation that is not identical but logically comparable to the modern account. (shrink)