6 1. COMBINATORIAL GEOMETRIES AND INFINITARY LOGICS
ℶ𝛼. (See Chapter 13 of Keisler book or [Mor65b].) Appendix A implies that if a
sentence of 𝐿𝜔1,𝜔 has a model with cardinality at least ℶ𝜔1 then it has arbitrarily
large models. Much of the difficulty of Part IV, of this text stems from dealing with
this issue.
Remark 1.2.6 (Set theoretical notation). We use with little reference such
basic notations as cf(𝜆) for cofinality and (cub) for closed unbounded set. For
background on such concepts see a set theory text such as [Kun80].
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