Tracing Internal Categoricity

Informally speaking, the categoricity of an axiom system means that its non‐logical symbols have only one possible interpretation that renders the axioms true. Although non‐categoricity has become ubiquitous in the second half of the twentieth century whether one looks at number theory, geometry or analysis, the first axiomatizations of such mathematical theories by Dedekind, Hilbert, Huntington, Peano and Veblen were indeed categorical. A common resolution of the difference between the earlier categorical axiomatizations and the more modern non‐categorical axiomatizations is that the latter derive their non‐categoricity from Skolem's Paradox and Gödel's Incompleteness Theorems, while the former, being second order, suffer from a heavy reliance on meta‐theory, where the Skolem–Gödel phenomenon re‐emerges. Using second‐order meta‐theory to avoid non‐categoricity of the meta‐theory would only seem to lead to an infinite regress. In this article we maintain that internal categoricity breaks with this traditional picture. It applies to both first‐ and second‐order axiomatizations, although in the first‐order case we have so far only examples. It does not depend on the meta‐theory in a way that would lead to an infinite regress. It also covers the classical categoricity results of early researchers. In the first‐order case it is weaker than categoricity itself, and in the second‐order case stronger. We provide arguments to suggest that internal categoricity is the “right” concept of categoricity.