Set Theory of the Reals and Large Continuum

We will investigate new forcing frameworks: oracle/preparatory constructions and continuous reading creature construction (bounding and non-bounding), and apply them to questions on cardinal characteristics, measure theory and general topology.

Mathematics operates in a natural and useful way with infinite sets. The size (number of elements) of such sets is called cardinality. In particular, the cardinality of the natural numbers is called "countable", that of the real numbers "continuum". It was already shown by Cantor that continuum is bigger than countable (and in fact that for each cardinality there is a bigger one). The Conitnuum Hypothesis (CH) states that there are not cardinalities between countable and continuum. Gödel and Cohen showed that CH is nether provable not refutable. Mathematics uses several important notions of "negligible small set of reals", for example: "Of Lebesgue measure zero", also called "null set". One can ask: What is the smallest size of a non-null set? The answer to this question is a cardinality, called non(N). It is easy to see that non(N) is bigger than countable and at most continuum. Another question would be: What is the smallest cardinality of a family of null sets with a non-null union? The answer is called add(N). non(N) and add(N) are so-called cardinal characteristics. For other notions of "small" one can define analogous characteristics; e.g. for "countable", for "meager" or for "sigma compact set of irrationals". The most important of the resulting characteristics are summarized in Cichon's diagram, which also shows the provable less-or-equal relations between them (e.g., it is easy to see that non(N) is at least as big as add(N)). The diagram contains twelve entries (including aleph1 and continuum), but two of them can be directly calculated from the rest, leaving ten independent ones. As mentioned, under CH all less-or-equal relations are actually equal (as all entries are continuum). It has been known for a long time that each individual less-or-equal relation can consistently be as less-than (e.g., that consistently add(N) is smaller than non(N)). In the paper "Cichon's Maximum", supported by the FWF, published in the Annals of Mathematics in 2019, we show that consistently all ten entries can take simultaneously different values. (The proof assumes the consistency of certain large cardinals.)