Internal categoricity and the natural numbers
The simple conclusion of the preceding chapters is that moderate modelism fails. But this leaves us with a choice between abandoning moderation and abandoning modelism. The aim of this chapter, and the next couple of chapters, is to outline a speculative way to save moderation by abandoning modelism. The idea is to do metamathematics without semantics, by working deductively in a higher-order logic. In this chapter, the focus is on the internal categoricity of arithmetic. After formalising an internal notion of a model of the Peano axioms, we show how to internalise Dedekind’s Categority Theorem. The resulting “intolerance” of Peano arithmetic provides internalists with a way to draw the distinction between algebraic and univocal theories. In the appendices, we discuss how this relates to Parsons’ important work, and establish a certain dependence of the internal categoricity theorem on higher-order logic.