infinity
What is infinity? Does it even exist? And how many infinities are there? 
Are there more irrational numbers than rational numbers, or more rational numbers than irrational numbers? Well, there are infinitely many of both, so the question doesn't make sense. It turns out, however, that the set of rational numbers is infinite in a very different way from the set of irrational numbers. 
Is the Universe finite or infinite? Is there infinity inside a black hole? Is space infinitely divisible or is there a shortest length? Can infinity occur at all in the cosmos or is it a mathematical construct? Find out more in our podcast with Anthony Aguirre, John D. Barrow and George Ellis. 
An infinite set is called countable if you can count it. In
other words, it's called countable if you can put its members into
onetoone correspondence with the natural numbers 1, 2, 3, ... .

Does infinity exist? In the latest online poll of our Science fiction, science fact project you told us that you'd like an answer to this question. So we went to speak to cosmologist John D. Barrow to find out more. We also bring you a range of other Plus articles on the subject of infinity, as well as an article from FQXi who are our partners on this project. Happy reading! 
Infinity is a pain. Its paradoxes easily ensnare the unsuspecting
reasoner. So over the centuries,
mathematicians have carefully constructed
bulwarks against its predations.
But now cosmologists have developed
theories that put them squarely outside
the mathematicians' "green zone" of
safety.

In the latest poll of our Science fiction, science fact project you told us that you wanted to know if infinity exists. In this interview the cosmologist John D. Barrow gives us an overview on the question, from Aristotle's ideas to Cantor's neverending tower of mathematical infinities, and from shock waves to black holes. 
Quantum mechanics and general relativity are incompatible — and this has led to a decadeslong search for a theory of quantum gravity that could combine the two. But the particle physicist Richard Woodard thinks that the mismatch between the two could be nothing more than an illusion, created by the complicated maths techniques used in attempts to unite them. 
John Barrow gives us an overview, from Aristotle's ideas to Cantor's neverending tower of mathematical infinities, and from shock waves to black holes. 
We all take for granted that mathematics can be used to describe the world, but when you think about it this fact is rather stunning. This article explores what the applicability of maths says about the various branches of mathematical philosophy. 