Postscript II: Infinitary Combinatorics

Abstract

The topics of the last chapter (Chap. 11) naturally lead to the area of Infinitary Combinatorics, which is beyond the scope of this text. This postscript to Part II is intended to be a link for the reader to begin further study in the area. We indicate how the obvious generalizations of three separate topics of the last chapter, namely short orders, König’s Infinity Lemma, and Ramsey’s Theorem, converge naturally to the notion of a weakly compact cardinal, an example of a large cardinal. In addition, it is shown how Suslin’s Problem is equivalent to the existence of Suslin trees. Finally, we briefly mention Martin’s Axiom and Jensen’s Diamond principle $\diamond$, and their implications for the Suslin Hypothesis.