About ten years ago I conceived the idea of writing a book on the natural numbers because I thought that what had appeared up till then seemed to have reached a point where there was a certain amount of completeness – of course there never will be absolute completeness – and this is one of the attractions of the subject. Anyway it was not until I had retired that I had the time to get down to the task properly. The result is a book which begins with an account of formal languages including the two most basic, namely, the propositional calculus and the predicate calculus, and then goes on to arithmetic; beginning with a very simple arithmetic; finding this inadequate; extending it to overcome this inadequacy; finding the resulting system, though richer in modes of expression, still, but for a different reason, inadequate; extending this in turn to remedy this inadequacy; finding the resulting system has lost some of the ‘nice’ qualities of its predecessor, but is again, for a new reason, inadequate; extending this and so on.

Before I come to develop arithmetic formally, it is convenient to have a primitive notation for the natural numbers (mainly to avoid lengthy circumlocutions) from which the concept of order and the operations of addition and multiplication can easily be obtained. I use sequences of tally marks, this is sufficient for our purposes. The real difficulty with arithmetic, as with other things, enters with the universal quantifier, when we want to make statements about all natural numbers.