# Tag: Georg Cantor

• ## Incompleteness in the natural world

A post inspired by Godel, Escher, Bach, Complexity: A Guided Tour, an article in this week’s New Scientist about the clash between general relativity and quantum mechanics and personal humiliation. The everyday incompleteness: This is the personal humiliation bit. For the first time ever I went on a “Parkrun” today – the 5km Finsbury Park…

• ## Not a proof that aleph null and the order of the continuum are the same

One final point from Wheels, Life and Other Mathematical Amusements– this time a “non-proof”. Some argue that the order of the counting numbers, is the same as that of the continuum – in other words that there is no difference in the scale of these two infinities. Here is an argument that is sometimes advanced…

• ## Another way of looking at the alephs

This is another insight gained from Wheels, Life and Other Mathematical Amusements– this time about the transfinite numbers. The smallest transfinite number, so -called is that of the countable infinity, or the counting numbers (the integers). Start at 1 (or 0) and keep going. But how many sets can one make from the counting numbers?…

• ## The wheels of Aristotle

Sometimes you come across a thing where beauty matches simplicity, and Aristotle’s Wheel Paradox is just such a thing. I came across it this afternoon after I returned to London from two too-short weeks away and a second-hand book I’d ordered, Wheels, Life and Other Mathematical Amusements, based on Martin Gardner‘s columns for the Scientific…

• ## “Ultimate L” explained

Well, that is what I wanted to write about – Hugh Woodin‘s theories of infinity, and indeed I found a blog that discussed a lot of this – http://caicedoteaching.wordpress.com/2010/10/19/luminy-hugh-woodin-ultimate-l-i/ But that is somewhat beyond me! And there does not appear to be a wikipedia article on this, so you may have to wait for some…

• ## 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)…

• ## 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…

• ## The importance of Turing’s findings

Yesterday I was speaking to a friend – who knows about computers but not about computer science – who told me this blog was just too difficult: so I thought I’d seek to order my own thoughts after reading The Annotated Turing and assist humanity in general (!) by writing a piece that explains what…