Cohen on the continuum hypothesis


Set Theory and the Continuum Hypothesis, by Paul Cohen (pictured above) was one of the great mathematical breakthroughs of the 1960s, and one of the few works in the history of mathematics to introduce a whole new method of proof (“forcing”). Cohen was just 30 when he wrote the book. There’s a copy in 2B6. Cohen died in 2007.

When we first think of “infinity”, we probably assume all infinities are the same. In the late 19th century Georg Cantor came up with a precise definition of when two sets A and B are equally big – if you can pair off all the elements of the two sets, one element of A with each one element of B – and showed that while some infinities are the same, others are different.

The set of even counting numbers (2, 4, 6, 8…) is just as big as the set of all counting numbers (1, 2, 3, 4….) The set of all rational numbers (fractions, including ½, ⅓, etc., as well as whole numbers) is also the same size. So is the set of all numbers including surds. But the set of all real numbers also including numbers like π and e which are not “algebraic” (are not the solution to any polynomial equation with whole-number coefficients) – that set is bigger than the set of all counting numbers.

Looking at it that way, there are different “sizes” of infinity. ℵ0, the “countable” infinity of all counting numbers 1, 2, 3, 4…. is the smallest.

The size of the set of all real numbers can be shown to be 20.

The continuum hypothesis claims that 20 = ℵ1. Or, in other words, that the infinity of all real numbers is the “next biggest” infinity after the countable infinity of all counting numbers.

For a long time the continuum hypothesis was thought to be obviously true, but difficult to prove. Paul Cohen showed, surprisingly, that with the usual assumptions of mathematical reasoning there is no way to tell whether the continuum hypothesis is true or false. A “model” in which it is false is just as logically sound as one in which it is false.

Look at the excerpt from Cohen below. Again, it shows us in real maths there are quite a lot of words. It’s about explaining concepts. Doing calculations is important, but secondary.