Question

(For information on cardinal characteristics of the continuum aka cardinal invariants see Joel David Hamkins' MO answer here; Andreas Blass's handbook article is an excellent reference.)

Problem 2.3 of Shelah's "On What I Do Not Understand (and Have Something to Say), Part I" (published in 2000 in Fundamenta Mathematicae) states, "Investigate cardinal invariants of the continuum showing $\geq 3$ may have prescribed order". One major barrier to such an investigation is the fact that countable support iteration of proper forcings yields models where the continuum is $\aleph_2$. In such models given any three cardinal characteristics at least two will have to be equal.

My question is the following. To what extent has such an investigation been pursued? In either the literature or folklore are there any results proving the consistency of inequalities $\mathfrak{c}_0<\mathfrak{c}_1<\mathfrak{c}_2$ where the $\mathfrak{c}_i$ are cardinal characteristics?

Answer 1

There is a paper of Shelah and Goldstern devoted to the separation of many simple cardinal invariants (this is a technical term). There are more recent papers on this subject by Kellner and Shelah, if I remember correctly.

An easy case that I am very familiar with are the so called localization numbers. A closed set $S\subseteq\omega^\omega$ is $n$-ary if in the tree $T(S)$ of finite initial segments of elements of $S$ every node has at most $n$ immediate successors.
For $n\geq 2$ let $\ell_n$ be the least size of a family of $(n-1)$-ary sets that covers all of $n^\omega$.

Any finitely many $\ell_n$ can be separated from each other simultaneously.
This is shown in [Geschke, Kojman, Convexity numbers of closed subsets in R^n, Proc. Am. Math. Soc. 130, No. 10, 2871-2881 (2002)], which is here.

Proofs of such statements usually involve forcing with large countable support products over a model of GCH rather than iterated forcing. However, there are also some examples that use iterated forcing. See for example this paper by Shelah and Steprans.