Combinatorial principles in second-order theories of bounded arithmetic

Abstract

An attempt is made to study the mathematical strength of the weak second order theories of Bounded Arithmetic U$\sbsp{2}{i}$ and V$\sbsp{2}{i}$, i $\geq$ 0, introduced by S. Buss. It is first shown that U$\sbsp{2}{1}$ can $\Sigma\sbsp{1}{1,b}$-define the functions in the second class of Grzegorczyk, $\varepsilon\sp2$, or, equivalently, the class of $\Sigma\sbsp{1}{1,b}$-definable functions in U$\sbsp{2}{1}$ is closed under bounded recursion.

It is shown next that U$\sbsp{2}{1}$ proves the $\Delta\sbsp{1}{1,b}$-pigeonhole principle. Two general combinatorial principles, the $\Delta\sbsp{1}{1,b}$-partition principle and the $\Delta\sbsp{1}{1,b}$-equipartition principle, are obtained from it thereby demonstrating that the $\Delta\sbsp{1}{1,b}$-PHP embodies a strong notion of cardinality. The introduction of these principles is motivated with some examples, notably by showing that Euler's theorem is provable in U$\sbsp{2}{1}$. The provability of Euler's theorem in weak first order fragments of Peano Arithmetic is an open problem.

A theory of polynomials is developed in U$\sbsp{2}{1}$ and it is proved that U$\sbsp{2}{1}$ + B $\vdash$ "existence of primitive roots" where formula B asserts that a nontrivial polynomial of degree n can have at most n solutions modulo p if p is a prime. By an essential use of the $\Delta\sbsp{1}{1,b}$-PHP and the $\Delta\sbsp{1}{1,b}$-partition principle it is shown that V$\sbsp{2}{1}\/\vdash$ B, hence V$\sbsp{2}{1}\/\vdash$ "existence of primitive roots".