## What is infinity?

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Does infinity exist? Does it exist in the physical world or just in our heads? How many infinities are there? And are we allowed to do sums with them? Find out in these Plus articles.

 Is the Universe finite or infinite? Is there infinity inside a black hole? Is space infinitely divisible or is there a shortest length? We talk to philosophers and physicists to find out. 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. 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 one-to-one correspondence with the natural numbers 1, 2, 3, ... . John Barrow gives us an overview, from Aristotle's ideas to Cantor's never-ending tower of mathematical infinities, and from shock waves to black holes. 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 never-ending tower of mathematical infinities, and from shock waves to black holes. Many people like mathematics because it gives definite answers. Things are either true or false, and true things seem true in a very fundamental way. But it's not quite like that. You can actually build different versions of maths in which statements are true or false depending on your preference. So is maths just a game in which we choose the rules to suit our purpose? Or is there a "correct" set of rules to use? We find out with the mathematician Hugh Woodin.