On the essence and identity of numbers

Abstract

Taking as premises some reasonable principles about the essences of natural numbers, pluralities and sets, the paper offers two types of argument for the conclusions that the natural numbers could not be the Zermelo numbers, the von Neumann numbers, the "Kripke numbers", or the positions in the ω-structure, among other things. These conclusions are thus Benacerrafian in form, but it is emphasized that the two kinds of argument offered in the paper are anti-Benacerrafian in substance, as they are perfectly compatible and in fact congenial with some views on which the numbers could be things of certain other kinds.