Tag: Infinity

The beauty of the Riemann sphere
Reading Elliptic Tales: Curves, Counting, and Number Theory today, I came across the Riemann Sphere – a thing of beauty: a closed surface of infinite extent. I will explain the maths of the sphere in a moment, but I am left wondering if one of two things apply: (a) We are terrible at teaching maths…

The maths of the spirograph… with the drawings
Well, I sat down and thought this was going to be easy, but it has taken me three hours to work the maths of a smaller inner wheel rolling around inside a large outer wheel: mainly because for the first two of those I neglected the basic insight that the inner wheel rolls in the…

Cosmologists’ problems with alephnull and the multiverse
This is another insight from Brian Greene’s book The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos – well worth reading. Alephnull () is the order (size) of the set of countably infinite objects. The counting numbers are the obvious example: one can start from one and keep on going. But any…

Goedel’s Incompeteness theorem surpassed?
Gödel’s Incompleteness Theorems are one of the cornerstone’s of modern mathematical thought but it is also a major blot on the mathematical landscape – as it establishes an inherent limit on the ability of mathematicians to describe the mathematical world: the first theorem (often thought of as the theorem) states that no consistent (ie selfcontained)…

The diagonal proof
I am just reading Computability and Logic and it (or at least the old, 1980, edition I am reading) has a rather laboured explanation of Cantor’s 1891 proof of the nonenumerability of real numbers (the diagonal proof) so to reassure myself and because it is interesting I thought I’d set out a version here (closely…