Higher cardinal characteristics

The field of set theory, a subfield of mathematical logic, began in late 19th century with Georg Cantor`s work on trigonometric series, which soon led him to the concepts of transfinite ordinals and his famous "Continuum Hy- pothesis, a conjecture about the sizes of subsets of the real line. The Con- tinumm Hypothesis was the first of David Hilbert`s legendary list of un- solved problems published in 1900, which had a marked influence on 20th century mathematics. Subsets of the real line, both "regular" subsets such as open sets or Borel sets, as well as pathological sets such as sets that do have a Lebesgue meas- ure (leading to functions that cannot be integrated) have been a central theme of set theory ever since. Real numbers (such as 1, -13.8, pi=3.14..., etc) can naturally be coded by subsets of the natural numbers (0,1,2,...) using binary expansions; this al- lows us to define "higher" analogues of the real numbers, so-called "higher reals", by replacing subsets of the natural numbers by subsets of larger infi- nite sets. Questions about subsets of the real numbers - such as: how large must a non-measurable set be? - can naturally be generalized to questions about these higher reals. Basic questions about the cardinality (or "size") of these infinite sets serve as a touchstone for our understanding of the internal structure of these sets. In our project we will investigate cardinalities of various kinds of infinite subsets of the "higher reals", typically of very complicated subsets. While it is well known that there are set-theoretic universes in which all such sets have the same size, we will construct set-theoretic universes in which there are many kinds of such complicated sets, with several different cardi- nalities.