444 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
Club guessing and the universal models
We survey the use of club guessing and other pcf constructs in the context of
showing that a given partially ordered class of objects does not have a
largest, or a universal element
Non existence of universals for classes like reduced torsion free abelian groups under non neccessarily pure embeddings
We consider a class K of structures e.g. trees with omega +1 levels, metric
spaces and mainly, classes of Abelian groups like the one mentioned in the
title and the class of reduced separable (Abelian) p-groups. We say M in K is
universal for K if any member N of K of cardinality not bigger than the
cardinality of M can be embedded into M . This is a natural, often raised,
problem. We try to draw consequences of cardinal arithmetic to non--existence
of universal members for such natural classes
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