Euclidean Arithmetic: The Finitary Theory of Finite Sets

Abstract

There is a central fallacy that underlies all our thinking about the foundations of arithmetic. It is the conviction that the mere description of the natural numbers as the “successors of zero” (i.e., as what you get by starting at 0 and iterating the operation x ↦ x + 1) suffices, on its own, to characterise the order and arithmetical properties of those numbers absolutely. This is what leads us to suppose that the dots of ellipsis in 0, 1, 2, 3, 4, 5, · · · or in 0, 1, 2, · · · , n are somehow explanatory.