Thin ultrafilters, P-hierarchu and MArtin Axiom

Abstract

Under MA we prove that for the ideal $\mathcal{I}$ of thin sets on $\omega$ and for any ordinal $\gamma \le {\omega }_1$ there is an $\mathcal{I}$-ultrafilter (in the sense of Baumgartner), which belongs to the class ${\mathcal{P}}_{\gamma }$ of P-hierarchy of ultrafilters. Since the class of ${\mathcal{P}}_2$ ultrafilters coincides with a class of P-points, out result generalize theorem of Flašková, which states that there are $\mathcal{I}$-ultrafilters which are not P-points. It is also related to theorem which states that under CH for any tall P-ideal $\mathcal{I}$ on $\omega$ there is an $\mathcal{I}$-ultrafilter, however the ideal of thin sets is not P-ideal.

12 pages. arXiv admin note: substantial text overlap with arXiv:1108.1818