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On the consistency of some partition theorems for continuous colorings, and the structure of \(\aleph _ 1\)-dense real order types. (English) Zbl 0585.03019

The authors describe and apply various methods of constructing c.c.c. posets. Of these the club method is probably best-known: if one wants to build an object of size \(\omega_ 1\) with certain properties, then very often the obvious poset of finite approximations is not c.c.c.; the club method first chooses a club set C in \(\omega_ 1\), one can then take the poset of finite approximations which are separated by C, i.e. between any two elements of the approximation there is an element of C. If C is chosen careful enough then this poset will be c.c.c.
Usually the club method requires CH; the authors also show how to remove CH in many constructions. This opens the way to proving statements like ”all \(\omega_ 1\)-dense subsets of \({\mathbb{R}}\) are isomorphic” consistent with \(2^{\omega}>\omega_ 1\). The CH-requirement prevented this at first.
The paper contains much more than is described here and is recommended reading for anyone who wants to prove consistency results about sets of size \(\omega_ 1\).
Reviewer: K.P.Hart

MSC:

03E05 Other combinatorial set theory
03E35 Consistency and independence results
03E50 Continuum hypothesis and Martin’s axiom
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