63 research outputs found
Stationary reflection principles and two cardinal tree properties
We study consequences of stationary and semi-stationary set reflection. We
show that the semi stationary reflection principle implies the Singular
Cardinal Hypothesis, the failure of weak square principle, etc. We also
consider two cardinal tree properties introduced recently by Weiss and prove
that they follow from stationary and semi stationary set reflection augmented
with a weak form of Martin's Axiom. We also show that there are some
differences between the two reflection principles which suggest that stationary
set reflection is analogous to supercompactness whereas semi-stationary set
reflection is analogous to strong compactness.Comment: 19 page
Stationary Reflection and the failure of SCH
In this paper we prove that from large cardinals it is consistent that there
is a singular strong limit cardinal such that the singular cardinal
hypothesis fails at and every collection of fewer than
stationary subsets of reflects simultaneously. For uncountable
cofinality, this situation was not previously known to be consistent. Using
different methods, we reduce the upper bound on the consistency strength of
this situation for to below a single partially
supercompact cardinal. The previous upper bound of infinitely many supercompact
cardinals was due to Sharon.Comment: 23 page
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
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