The previous chapter gestured towards a proof of incompleteness which relies on constructing a Gödel sentence which is true if and only if it is unprovable. This proof idea is ingenious. Too ingenious? Some, when they first encounter it, worry that an illegitimate trick is being played.
That’s wrong, as hopefully will become quite clear in later chapters. It might help, though, to counter any temptation to think that there’s something fishy about the incompleteness theorem if we pause to look at a different sort of proof — one that doesn’t depend on the construction of a tricksy Gödel sentence.
Chapter 4, ‘Undecidability and incompleteness’, therefore gives a lovely proof which I first learnt from Timothy Smiley a very long time ago. The argument is one of my favourite ones in all logic — it’s so elegant and easy to understand that every student should know it!