 If one accepts a certain open-endedness of the collection of arithmetical predicates, then a categoricity theorem of sorts for arithmetic can be obtained without overstepping the bounds of first-order logic and without appealing to an informal concept of computability. Suppose that there are two mathematicians, A and B, who both assert the first-order Peano-axioms in their own idiolect. Suppose furthermore that A and B regard the collection of predicates for which mathematical induction is permissible as open-ended, and are both willing to accept the other’s induction scheme as true. Then A and B have the wherewithal to convince themselves that both idiolects describe isomorphic structures (Parsons 1990b). Such arguments are called internal categoricity arguments. They are widely debated in contempory philosophy of mathematics: see for instance (Button &amp; Walsh 2019).