Natural numbers in Haskell

It is a bit surprising that Haskell has no type for natural numbers. The Haskell Report defines only Integer for, well, integers and Int for integers that are reasonably small. The base package then adds Word as the unsigned variant of Int as well as types for signed and unsigned integers with a fixed number of bits. An unsigned variant of Integer is missing. Continue reading

New Foundations

It was around the beginning of my undergraduate studies when I became interested in foundations of mathematics in general and axiomatic set theory in particular. Shortly after, I started reading parts of Abraham Fraenkel’s brilliant book Einleitung in die Mengenlehre. I read the second edition of this book, which had originally been published in the 1920ies. In his book, Fraenkel first introduces set theory in a naïve way, then discusses certain paradoxes arising from the naïve treatment, and finally presents the axiomatic set theory developed by Ernst Zermelo and himself.

A few weeks ago, I stumbled across set theory again when Sergei Tupailo gave two talks at the Institute of Cybernetics on an alternative axiomatic set theory called New Foundations. The first talk was about a classical proof by Ernst Specker, the second one was about a contribution by Tupailo himself. Tupailo’s talks caused me to have a look at these “new foundations”, which were, in fact, completely new to me. In this blog post, I want to tell you a bit about this interesting topic. Continue reading