 At the same time, the Löwenheim-Skolem theorem says that every first-order formal theory that has at least one model with an infinite domain, must have models with domains of all infinite cardinalities. Since the principles of arithmetic, analysis and set theory had better possess at least one infinite model, the Löwenheim-Skolem theorem appears to apply to them. Is this not in tension with Dedekind’s categoricity theorems?