2,164 research outputs found
Dense ideals and cardinal arithmetic
From large cardinals we show the consistency of normal, fine,
-complete -dense ideals on for
successor . We explore the interplay between dense ideals, cardinal
arithmetic, and squares, answering some open questions of Foreman
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
Pcf theory and cardinal invariants of the reals
The additivity spectrum ADD(I) of an ideal I is the set of all regular
cardinals kappa such that there is an increasing chain {A_alpha:alpha<kappa\}
in the ideal I such that the union of the chain is not in I.
We investigate which set A of regular cardinals can be the additivity
spectrum of certain ideals.
Assume that I=B or I=N, where B denotes the sigma-ideal generated by the
compact subsets of the Baire space omega^omega, and N is the ideal of the null
sets.
For countable sets we give a full characterization of the additivity spectrum
of I: a non-empty countable set A of uncountable regular cardinals can be
ADD(I) in some c.c.c generic extension iff A=pcf(A).Comment: 9 page
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