  Simpson and the second author asked whether there exists a characterization
of the natural numbers by a second-order sentence which is provably categorical
in the theory RCA$^*_0$. We answer in the negative, showing that for any
characterization of the natural numbers which is provably true in WKL$^*_0$,
the categoricity theorem implies $\Sigma^0_1$ induction. On the other hand, we
show that RCA$^*_0$ 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 $\Pi^1_2$-conservative extension of RCA$^*_0$ admits a provably
categorical single-sentence characterization of the naturals, but each such
characterization has to be inconsistent with WKL$^*_0$+superexp.
