Model Theory as Peacock’s Revenge

The diagram represents the real numbers (in other words the numbers with decimal expansions), arranged in order from smaller to greater. The more strenuous textbooks add some other numbers to the diagram, for example e a little to the left of 3 and π a little to the right of it. These two numbers are irrational; in other words they can’t be written as m/n where m and n are integers. Probably the first irrational number to be discovered was √ 2, the square root of 2, somewhere between 1 and 2 in the diagram. Medieval mathematicians regarded irrational numbers as substandard and used insulting names for them, such as ‘surd’, which is the Latin for deaf-mute. A long time ago people realised that negative real numbers don’t have real square roots; for example no real number is a solution of the equation x2 + 1 = 0. In 1545 Girolamo Cardano published a formula for calculating solutions of cubic equations. The cubic equation