Paul Bernays Lectures 2012
Solomon Feferman, Stanford University, USA
Solomon Feferman is the Patrick Suppes Professor of Humanities and Sciences, and Professor of Mathematics and Philosophy at Stanford University. His main interests are in mathematical logic (especially proof theory and the theory of computation), the foundations of mathematics (especially constructive and semi-constructive approaches), the philosophy of mathematics and the history of modern logic. He was awarded the Rolf Schock prize in logic and philosophy in 2003.
Programme
Opening address
Professor Ernst Engeler
Tuesday, 11 September 2012, 4.30 p.m.
Lecture 1: Bernays, Hilbert and Gödel
Tuesday, 11 September 2012, 5.00 p.m.
Abstract: Paul Bernays was brought from Zürich to Göttingen in 1917 by David Hilbert – the leading mathematician of the time – to assist him in developing his consistency programme for the foundations of mathematics. The major exposition of that work appeared in the 1930s in the two volume opus by Hilbert and Bernays, Grundlagen der Mathematik, whose preparation was due entirely to Bernays. In the meantime, Kurt Gödel, a precocious doctorate in Vienna, had discovered his remarkable incompleteness theorems which threatened to undermine Hilbert’s programme. Though Hilbert refused to accept that, Bernays undertook to absorb the significance of those theorems through correspondence with Gödel. This led to a lifelong deep personal and intellectual relationship between the two of them whose high points will be traced in the lecture.
Lecture 2: Is the continuum hypothesis a definite mathematical problem?
Wednesday, 12 September 2012, 2.15 p.m.
Abstract: Georg Cantor established the modern theory of sets with his theory of transfinite cardinal and ordinal numbers, which began with his proof that the set of real numbers has greater cardinality than the set of natural numbers; Cantor’s Continuum Hypothesis (CH) states that there is no intermediate cardinal number. The call to establish CH was the first in the famous list of twenty-three challenging mathematical problems that Hilbert posed at the International Congress of Mathematicians in 1900. Yet, a century later, it did not appear on the list of the seven Millennium Prize Problems worth a million dollars each, despite the fact that no solution to it has been found in the mean time (or since). In this lecture I will discuss the evidence for my view (contrary to Gödel above all) that CH is not a definite mathematical problem, despite the fact that it is formulated in terms of concepts that have become an established part of mathematics.
Lecture 3: Theories of operations, sets and classes
Wednesday, 12 September 2012, 4.30 p.m.
Abstract: For the axiomatic system OST of Operational Set Theory the usual language of set theory is expanded to allow one to talk about (possibly partial) operations applicable both to sets and to operations. It is related both to von Neumann’s theory of sets and functions and Bernays’ theory of sets and classes. OST is equivalent in strength to admissible set theory, and a natural extension of OST is equivalent in strength to ZFC. The language of OST provides a framework in which to express "small" large cardinal notions – such as those of being an inaccessible cardinal, a Mahlo cardinal, and a weakly compact cardinal – in terms of operational closure conditions that specialize to the analogue notions on admissible sets. This illustrates a wider programme whose aim is to provide a common framework for analogues of large cardinal notions that have appeared in admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics, and systems of recursive ordinal notations that have been used in proof theory.