To infinity… and beyond
So says Buzz Lightyear in Toy Story and YES! there is something beyond infinity. In fact, there is an infinity of infinities of different sizes. This was the discovery of Georg Cantor (1845-1919). The idea of the size of infinity goes back, at least, to Galileo who puzzled over the paradox that there are fewer perfect squares (1, 4, 9, 16…) than natural numbers (1,2,3,4…) as some numbers are not included (2,3,5,6, 7, 8, 10…) but the set of perfect squares is the same size as the natural numbers as you can match them up (1 to 1, 2 to 4, 3 to 9, 4 to 16…). Galileo resolved this by arguing that, as far as infinite sets go, ‘size doesn’t matter’ and ideas of bigger and smaller don’t work.
Cantor rejected this and called infinite sets, such as the perfect squares, countable sets if they can be matched to natural numbers. He went on to show, by an ingenious proof, that the set of fractions is countable and so is the ‘same size’ as the natural numbers. He also proved that the set of real numbers was not countable and so is a different size. This makes sense of the infinite divisibility of the real number line: between any two real numbers there is an infinity of other numbers, and so the infinity of the real number line feels to be of a different size to the countable infinity of natural numbers. In fact, there are an infinity of infinities – called transfinite numbers.
Cantor was not celebrated in his lifetime. Poincaré wrote of the ‘grave disease infecting mathematics’ from Cantor’s work on set theory. His work was not published and his advancement was blocked. There were those who recognised the significance of his ideas. The German mathematician David Hilbert (1862-1943), in a celebrated lecture in 1900, set out the main mathematical problems for that century. The first was the Cantor Continuum Hypothesis. Cantor thought there was no set with size between that of the natural numbers and that of the real numbers but he didn’t prove this. It wasn’t until 1963 that Paul Cohen proved a surprising result: The continuum hypothesis can’t be proved either true or false – it is undecidable!