The world between aleph1 and continuum: from Martin's Axiom to Cichoń's Maximum

02.12.2021 15:00 - 15:45

M. Goldstern (TU Wien)

Georg Cantor's "Continuum Hypothesis" (CH) postulates that the continuum (the cardinality of the set of real numbers) is equal to \(\aleph_1\), the smallest uncountable cardinal. Martin's Axiom (MA) is a weakening of CH; it implies that all infinite cardinals below the continuum are similar to \(\aleph_0\), the cardinality of a countable set. For example, MA implies that not only every countable union of null (measure zero) sets is still null, but even every union of fewer than continuum many such sets. This motivates the definition of a so-called cardinal characteristic, the additivity number of the measure zero sets - the answer to the question "how many null sets do we have to join together to get a non-null set". There is a whole zoo of such cardinal characteristics (some of them defined long before the advent of forcing); whenever you know that any countable set of objects with property \(X\) will never have property \(Y\), you may ask how many such objects you need to get to \(Y\).

Accepting CH or just MA as an axiom gives a picture that is on the one hand very clean, but on the other hand also rather poor: most cardinal characteristics can then be shown to be equal to the continuum.

In my talk I will discuss - or at least hint at - some recent (and some old) techniques for constructing "anti-MA" universes, where many cardinals between \(\omega_1\) and the continuum appear as cardinal characteristics (defined by some natural properties \(X\) and \(Y\)).

I will try to hide all technical details, so that my talk will hopefully be understandable also for non-set-theorists.

Organiser:

KGRC

Location:
Zoom Meeting