Primitive recursive reverse mathematics

Abstract

We use a second-order analogy ${\mathrm{PRA}}^2$ of $\mathrm{PRA}$ to investigate the proof-theoretic strength of theorems in countable algebra, analysis, and infinite combinatorics. We compare our results with similar results in the fast-developing field of primitive recursive (‘punctual’) algebra and analysis, and with results from ‘online’ combinatorics. We argue that ${\mathrm{PRA}}^2$ is sufficiently robust to serve as an alternative base system below ${\mathrm{RCA}}_0$ to study the proof-theoretic content of theorems in ordinary mathematics. (The most popular alternative is perhaps ${\mathrm{RCA}}_0^{⁎}$.) We discover that many theorems that are known to be true in ${\mathrm{RCA}}_0$ either hold in ${\mathrm{PRA}}^2$ or are equivalent to ${\mathrm{RCA}}_0$ or its weaker (but natural) analogy $2^N$-${\mathrm{RCA}}_0$ over ${\mathrm{PRA}}^2$. However, we also discover that some standard mathematical and combinatorial facts are incomparable with these natural subsystems.