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Non-existence of universal members in classes of abelian groups
TLDR
In this paper, it was shown that there is no universal reduced separable abelian p-group in lambda if aleph ∆ = lambda ∆ ∆ < 2.Abstract:
We prove that if mu^+< lambda =cf(lambda)< mu^{aleph_0}, then there is no universal reduced torsion free abelian group. Similarly if aleph_0< lambda < 2^{aleph_0}. We also prove that if 2^{aleph_0}< mu^+< lambda =cf(lambda)< mu^{aleph_0}, then there is no universal reduced separable abelian p-group in lambda. (Note: both results fail if lambda = lambda^{aleph_0} or if lambda is strong limit, cf (mu)= aleph_0< mu).read more
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Non-existence of Universal Orders in Many Cardinals
Menachem Kojman,Saharon Shelah +1 more
TL;DR: It is shown that if there is a universal linear order at a regular λ and its existence is not a result of a trivial cardinal arithmetical reason, then λ “resembles” ℵ 1 —a cardinal for which the consistency of having a universal order is known.
References
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Book
Proper and Improper Forcing
TL;DR: In this paper, the authors give a complete presentation of the theory of proper forcing and its relatives, starting from the beginning and avoiding the metamathematical considerations, and show methods which can be used for such independence results.
Journal Article
On what I do not understand (and have something to say): Part I
TL;DR: A non-standard paper, containing some problems in set theory I have in various degrees been interested in this article, contains 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.
Journal ArticleDOI
Combinatorial problems on trees: Partitions, δ-systems and large free subtrees
Matatyahu Rubin,Saharon Shelah +1 more
TL;DR: The partition theorems on trees are proved and generalize to a setting of trees the theoreMS of Erdos and Rado on δ-systems and the theoresms of Fodor and Hajnal on free sets.
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Club Guessing and the Universal Models
TL;DR: 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 is surveyed.