An Elementary Construction of an Ultrafilter on $\aleph_1$ Using the Axiom of Determinateness
Marco R. Vervoort
Abstract:
An elementary construction of an ultrafilter on AlephOne
using the Axiom of Determinateness
Marco R. Vervoort
In this article we construct a free and scomplete ultrafilter on the set
\omega_1, using AD.
First we define for each V \subset \omega_1 a game G(V). From the axiom
AD we have that for each V \subset \omega_1 , either the first or the
second player has a winning strategy in G(V). We then show, in several
lemma's, how to obtain winning strategies in G(V) for several different
constructions of V from other sets. Finally, we show that the collection
{ V \subset \omega_1 | the first player has a winning strategy in G(V) }
has several closure properties corresponding to the lemma's just proved,
and that this set is in fact a free and \sigmacomplete ultrafilter.