Categorical characterizations of the natural numbers require primitive recursion

TL;DR

This paper investigates the logical foundations of characterizing natural numbers categorically within second-order arithmetic, revealing limitations and possibilities depending on the base theory and extensions used.

Contribution

It demonstrates that provably categorical characterizations of natural numbers require strong extensions or lead to inconsistency, clarifying the logical boundaries of such characterizations.

Findings

Categorical characterizations in RCA$^*_0$ are possible with multiple sentences.

Single-sentence characterizations in certain extensions are inconsistent with stronger theories.

Any provably categorical second-order characterization in WKL$^*_0$ implies $ ext{Σ}^0_1$ induction.

Abstract

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.