Tag: Infinity
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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…
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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…
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Cosmologists’ problems with aleph-null 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. Aleph-null () 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…
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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 self-contained)…
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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 non-enumerability 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…
cartesian product ... stuff about computing mostly 