6 1. COMBINATORIAL GEOMETRIES AND INFINITARY LOGICS ℶ𝛼. (See Chapter 13 of Keisler book or [Mor65b].) Appendix A implies that if a sentence of 𝐿𝜔 1 ,𝜔 has a model with cardinality at least ℶ𝜔 1 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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