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51:30 Eidgenössische Technische Hochschule (ETH) Zürich English 2012

Foundations of Unlimited Category Theory

Following a discussion of various forms of set-theoretical foundations of category theory and the controversial question of whether category theory does or can provide an autonomous foundation of mathematics, this lecture concentrates on the question whether there is a foundation for “unlimited” or “naïve” category theory. I proposed four criteria for such some years ago. This lecture describes how much had previously been accomplished on one approach to meeting those criteria, then takes care of one important obstacle that had been met in that approach, and finally explains what remains to be done if one is to have a fully satisfactory solution.
  • Published: 2012
  • Publisher: Eidgenössische Technische Hochschule (ETH) Zürich
  • Language: English
1:27:55 Eidgenössische Technische Hochschule (ETH) Zürich English 2012

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.
  • Published: 2012
  • Publisher: Eidgenössische Technische Hochschule (ETH) Zürich
  • Language: English
21:42 Eidgenössische Technische Hochschule (ETH) Zürich English 2012

Paul Bernays Lectures 2012 - Opening

The Paul Bernays Lectures are a new, annual and three-part honorary lecture series about the philosophy of the exact sciences. The series will alternate 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 who was engaged in teaching and research at the ETH from 1933 to 1959. Within the framework of the Paul Bernays lectures outstanding philosophers of logic, mathematics or physics will be invited to speak about their pioneering research. The first lecture will present the research topics for a wider audience, the second lecture will address the research community, and a final seminar will offer to the members of the research community the opportunity to discuss with the invited speaker selected research topics and ideas.
  • Published: 2012
  • Publisher: Eidgenössische Technische Hochschule (ETH) Zürich
  • Language: English
1:10:43 Eidgenössische Technische Hochschule (ETH) Zürich English 2012

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.
  • Published: 2012
  • Publisher: Eidgenössische Technische Hochschule (ETH) Zürich
  • Language: English
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