# Solomon Feferman delivered the first Paul Bernays Lectures at the ETH in Zurich, Switzerland on September 11-12, 2012

The Paul Bernays Lectures are a new, annual and three-part honorary lecture series about the philosophy of the exact sciences.

The series alternates between the philosophy of logic or mathematics, and the philosophy of physics. This lecture series is established in honor of the eminent logician, mathematician and philosopher of logic and mathematics Paul Bernays (1888-1977) who was engaged in teching and research at the ETH from 1933-1959.

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On Tuesday, Sept. 11, 2012, and Wednesday, Sept. 12, 2012, Professor Solomon Feferman delivered the first Paul Bernays Lectures at the ETH in Zurich, Switzerland.

**Lecture 1 (Tuesday, 9/11/12): Bernays, Gödel and Hilbert's consistency program**

Paul Bernays was brought from Zurich to Göttingen in 1917 by David Hilbert - the leading mathematician of the time - to assist him in developing his consistency program 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 program. 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 (Wednesday, 9/12/12): Is the Continuum Hypothesis a definite mathematical problem?**

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. 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.