Tue, 7 Dec 2021, 1 pm MST
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 ℵ1, 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 ℵ1 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.
[video]