Simpson and Yokoyama (2013) [9] asked whether there exists a characterization of the natural numbers by a second-order sentence which is provably categorical in the theory . We answer in the negative, showing that for any characterization of the natural numbers which is provably true in , the categoricity theorem implies induction.
On the other hand, we show that does make it possible to characterize the natural numbers categorically by means of a set of second-order sentences. We also show that a certain -conservative extension of admits a provably categorical single-sentence characterization of the naturals, but each such characterization has to be inconsistent with .