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Combinatorial Cardinal Characteristics of the Continuum

In: Handbook of Set Theory

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  • Andreas Blass

    (University of Michigan, Department of Mathematics)

Abstract

The combinatorial study of subsets of the set N of natural numbers and of functions from N to N leads to numerous cardinal numbers, uncountable but no larger than the continuum. For example, how many infinite subsets X of N must I take so that every subset Y of N or its complement includes one of my X’s? Or how many functions f from N to N must I take so that every function from N to N is majorized by one of my f’s? The main results about these cardinal characteristics of the continuum are of two sorts: inequalities involving two (or sometimes three) characteristics, and independence results saying that other such inequalities cannot be proved in ZFC. Other results concern, for example, the cofinalities of these cardinals or connections with other areas of mathematics. This survey concentrates on the combinatorial set-theoretic aspects of the theory.

Suggested Citation

  • Andreas Blass, 2010. "Combinatorial Cardinal Characteristics of the Continuum," Springer Books, in: Matthew Foreman & Akihiro Kanamori (ed.), Handbook of Set Theory, chapter 6, pages 395-489, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4020-5764-9_7
    DOI: 10.1007/978-1-4020-5764-9_7
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