65 research outputs found
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
Distributive Aronszajn trees
Ben-David and Shelah proved that if is a singular strong-limit
cardinal and , then entails the
existence of a normal -distributive -Aronszajn tree. Here,
it is proved that the same conclusion remains valid after replacing the
hypothesis by .
As does not impose a bound on the order-type
of the witnessing clubs, our construction is necessarily different from that of
Ben-David and Shelah, and instead uses walks on ordinals augmented with club
guessing.
A major component of this work is the study of postprocessing functions and
their effect on square sequences. A byproduct of this study is the finding that
for regular uncountable, entails the existence of a
partition of into many fat sets. When contrasted with a
classic model of Magidor, this shows that it is equiconsistent with the
existence of a weakly compact cardinal that cannot be split into two
fat sets.Comment: 45 pages; improved and generalized some results, and streamlined the
presentatio
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