OS Math. Logik, Mengenlehre und Modelltheorie: Cardinals between $\aleph_1$ and continuum

Montag, 03. Dezember 2018
15:15 – 16:15 Uhr

F 426

C. Antos-Kuby, S. Kuhlmann

Martin Goldstern (TU Wien)

Diese Veranstaltung ist Teil der Veranstaltungsreihe „Oberseminar Mathematische Logik, Mengenlehre und Modelltheorie“.

Georg Cantor's "continuum hypothesis (CH)" (1877) states that all infinite subsets of the real line are
-  either countable (equinumerous with the set of natural numbers)
-  or "of size continuum" (equinumerous with the set of all reals),
or in other words: the equivalence relation of equinumerosity (=equal cardinality) divides the infinite subsets of the reals into only 2 classes.

Based on Paul Cohen's forcing method (invented or discovered in 1963), set theorists have constructed many set-theoretical universes where CH fails, often in strong way.  For example, it is well known that  the number of equinumerosity classes may be uncountable, or even equal to the continuum itself.

While proofs that an explicitly given set S of reals is uncountable often exhibit a perfect subset of S, thus showing that S is in fact equinumerous with the real line,  there are also many cardinal numbers in the interval from $\aleph_1$ (the first uncountable cardinal) to continuum which appear as the answers to natural questions about the minimal size of a "pathological" set, such as
-  "How many points must a non-measurable set have?"
-   or "How many meager sets are needed to cover the real line?".

In my talk I will present some methods that can be used to construct set-theoretic universes where many nicely definable cardinals between $\aleph_1$ and continuum  (such as the cardinals in Cichońs diagram) have different values. In particular, I will talk a bit about using the method of Boolean ultrapowers in such constructions.