ON IDEMPOTENT ULTRAFILTERS IN HIGHER-ORDER REVERSE MATHEMATICS
ALEXANDER P. KREUZER · Journal of Symbolic Logic · 2015
Abstract We analyze the strength of the existence of idempotent ultrafilters in higher-order reverse mathematics. Let $\left( {{{\cal U}_{{\rm{idem}}}}} \right)$ be the statement that an idempotent ultrafilter on ℕ exists. We show that over $ACA_0^\omega$ , the higher-order extension of ACA0, the statement $\left( {{{\cal U}_{{\rm{idem}}}}} \right)$ implies the iterated Hindman’s theorem (IHT) and we show that $ACA_0^\omega + \left( {{{\cal U}_{{\rm{idem}}}}} \right)$ is ${\rm{\Pi }}_2^1$ -conservative over $ACA_0^\omega + IHT$ and thus over $ACA_0^ +$ .