Cardinal invariants of the continuum — A survey J¨org Brendle∗ The Graduate School of Science and Technology Kobe University Rokko–dai 1–1, Nada–ku Kobe 657–8501, Japan
[email protected] February 3, 2004 Abstract These are expanded notes of a series of two lectures given at the meeting on axiomatic set theory at Ky¯otoUniversity in November 2000. The lectures were intended to survey the state of the art of the theory of cardinal invariants of the continuum, and focused on the interplay between iterated forcing theory and car- dinal invariants, as well as on important open problems. To round off the present written account of this survey, we also include sections on ZFC–inequalities be- tween cardinal invariants, and on applications outside of set theory. However, due to the sheer size of the area, proofs had to be mostly left out. While being more comprehensive than the original talks, the personal flavor of the latter is preserved in the notes. Some of the material included was presented in talks at other conferences. ∗Supported by Grant–in–Aid for Scientific Research (C)(2)12640124, Japan Society for the Promotion of Science 1 1 What are cardinal invariants? We plan to look at certain basic features of the real line R from the point of view of combinatorial set theory. For our purposes, it is convenient to work with the Cantor space 2ω or the Baire space ωω instead of R itself. Here we put, as usual, 2 = {0, 1} ω = N = {0, 1, 2, 3, ...} = the natural numbers 2ω = the set of functions from ω to 2 ωω = the set of functions from ω to ω.