Cardinals below the continuum

Abstract: Georg Cantor's "Continuum Hypothesis" (CH) postulates that every infinite set S of reals is either countable or equinumerous with the set of all reals. Using the axiom of choice this means that the "continuum" (the cardinality of the set of reals) is equal to aleph1, the smallest uncountable cardinal.

David Hilbert's first problem asked if CH is true; we know now that neither CH nor non-CH can be proved from the usual axioms of set theory (ZFC). Paul Cohen's method of forcing allows us to build universes (structures satisfying ZFC) where the continuum is arbitrarily large.

There are many relatives of the continuum, such as the answers to these questions: How many nulls sets (Lebesgue measure zero) do we need to cover the real line? How many points do we need to get a non-null set? How many sequences (or convergent series) do we need to eventually dominate all sequences (convergent series)? etc. All these cardinals are located in the closed interval between aleph1 and the continuum.

In my talk I will present some of these cardinals and hint at the methods used to construct universes where these cardinals have prescribed values, or satisfy strict inequalities.

computational complexitycategory theorylogic

Audience: researchers in the topic

PALS Panglobal Algebra and Logic Seminar

Series comments: The PALS seminar is a research and learning seminar organized by the algebra and logic research group of the Department of Mathematics at the University of Colorado at Boulder. The scope of the seminar includes all topics with links to algebra, logic, or their applications, like general algebra, logic, model theory, category theory, set theory, set-theoretic topology, or theoretical computer science. Please contact one of the organizers for the Zoom password, to join the mailing list or if you want to speak.