Melvyn Bragg and guests discuss the nature and existence of mathematical infinity. Jonathan Swift encapsulated the counter-intuitive character of infinity with insouciant style:“So, naturalists observe, a fleaHath smaller fleas on him that preyAnd these hath smaller fleas to bite ‘emAnd so proceed ad infinitum.”Alas, the developing utility mathematicians put to the idea of infinity did not find the English philosopher Thomas Hobbes quite so relaxed. When confronted with a diagram depicting an infinite solid whose volume was finite, he wrote, “To understand this for sense, it is not required that a man should be a geometrician or logician, but that he should be mad”. Yet philosophers and mathematicians have continued to grapple with the unending, and it is a core concept in modern maths.So, what is mathematical infinity? Are some infinities bigger than others? And does infinity exist in nature?With Ian Stewart, Professor of Mathematics at the University of Warwick; Robert Kaplan, co-founder of The Math Circle at Harvard University and author of The Art of the Infinite: Our Lost Language of Numbers; Sarah Rees, Reader in Pure Mathematics at the University of Newcastle.