For an ultrafilter
, consider the ultrapower NN/
. γanγ/
is in the top sky of NN/
if there exists a sequence γbnγ β NN such that
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0022481200020648/resource/name/S0022481200020648_eqnU1.gif?pub-status=live)
and
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0022481200020648/resource/name/S0022481200020648_eqnU2.gif?pub-status=live)
In [M2] we showed, assuming the Continuum Hypothesis, that there are exactly 10 c/p's (where c is the ring of real convergent sequences and p is a prime ideal of c). To get the lower bound we showed that there will be at least 10 c/p's in any model of ZFC where there exist both of the following kinds of ultrafilter:
(i) nonprincipal P-points,
(ii) non-P-points
such that when the top sky is removed from NN/
, the remaining model has countable cofinality.
In [M2] we showed that the Continuum Hypothesis implies the existence of the ultrafilter in (ii). In this paper we show that its existence is implied by an axiom weaker than the Continuum Hypothesis, in fact weaker than Martin's Axiom, namely,
(*) If
is a subset of NN such that for any f: N β N there exists g β
such that g(n) > f(n) for all n, then β£
β£ =
.