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 “nonproof”. 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 tooshort weeks away and a secondhand 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/luminyhughwoodinultimateli/ 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 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…

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…