Published online by Cambridge University Press: 13 March 2015
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
$ACA_0^ +$.