6 research outputs found
Saturated filters at successors of singulars, weak reflection and yet another weak club principle
Suppose that lambda is the successor of a singular cardinal mu whose
cofinality is an uncountable cardinal kappa. We give a sufficient condition
that the club filter of lambda concentrating on the points of cofinality kappa
is not lambda^+-saturated. The condition is phrased in terms of a notion that
we call weak reflection. We discuss various properties of weak reflectio
On squares, outside guessing of clubs and I_{<f}[lambda]
Suppose that lambda = mu^+. We consider two aspects of the square property on
subsets of lambda. First, we have results which show e.g. that for aleph_0 <=
kappa =cf (kappa)< mu, the equality cf([mu]^{<= kappa}, subseteq)= mu is a
sufficient condition for the set of elements of lambda whose cofinality is
bounded by kappa, to be split into the union of mu sets with squares. Secondly,
we introduce a certain weak version of the square property and prove that if mu
is a strong limit, then this weak square property holds on lambda without any
additional assumption
On what I do not understand (and have something to say): Part I
This is a non-standard paper, containing some problems in set theory I have
in various degrees been interested in. Sometimes with a discussion on what I
have to say; sometimes, of what makes them interesting to me, sometimes the
problems are presented with a discussion of how I have tried to solve them, and
sometimes with failed tries, anecdote and opinion. So the discussion is quite
personal, in other words, egocentric and somewhat accidental. As we discuss
many problems, history and side references are erratic, usually kept at a
minimum (``see ... '' means: see the references there and possibly the paper
itself).
The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The
other half, concentrating on model theory, will subsequently appear