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 ℵ_{1}, 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 ℵ_{1} 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]