Logic Seminar: An Introduction to Combinatorial Cardinal Characteristics of the Continuum

In many contexts, countable sets behave very differently from sets of the cardinality c of the continuum. If we do not assume the continuum hypothesis, there may be cardinal numbers strictly between countable and c. So it makes sense to ask which of these cardinals behave like countable and which behave like c. To any such context, we associate a cardinal characteristic, namely the smallest cardinal that behaves like c. I plan to introduce a few of the most widely studied cardinal characteristics and describe connections between them. I'll include a few proofs, intended as easy illustrations of more generally applicable methods.