(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?

  • $\begingroup$ Regarding the major barrier mentioned above, there has been some work done by Shelah on iterated forcing making the continuum > aleph2. For example: arxiv.org/pdf/0707.1818v2 $\endgroup$ – Haim Aug 30 '10 at 22:33

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 (Wayback Machine).

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.

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Bumping an old question, because the situation has changed dramatically in the last year or so:

The most well-known cardinal characteristics of the continuum are those appearing in Cichoń's Diagram, which also presents all $ZFC$-provable relations between pairs of these characteristics: besides the dominating number $\mathfrak{d}$ and the bounding number $\mathfrak{b}$, these are all of the form $inv(\mathcal{I})$, for $\mathcal{I}$ the ideal of meager sets $(\mathcal{M})$ or null sets $(\mathcal{N})$, and $inv$ one of $add, cov, non, cof$:

  • $add$ of an ideal $\mathcal{I}$ is the least cardinality of a family of sets from $\mathcal{I}$ whose union is not in $\mathcal{I}$;

  • $cov$ of an ideal $\mathcal{I}$ is the least cardinality of a family of sets from $\mathcal{I}$ covering all of $\mathbb{R}$;

  • $non$ of an ideal $\mathcal{I}$ is the least cardinality of a set of reals not in $\mathcal{I}$; and

  • $cof$ of an ideal $\mathcal{I}$ is the cofinality of the partial order $\langle\mathcal{I}, \subseteq\rangle$.

As far as I know, the first results separating more than two cardinal characteristics from Cichon's diagram came out in the last two years:

  • In late 2012, Mejia constructed several examples of models separating multiple characteristics at once (see section 6).

  • In a paper arxived today(!), Fischer/Goldstern/Kellner/Shelah produced a model of $\aleph_1=\mathfrak{d}=cov(\mathcal{N})<non(\mathcal{M})<non(\mathcal{N})<cof(\mathcal{N})<2^{\aleph_0}$.

Now, I don't understand the proofs at all, but it seems the proofs by Mejia and by F/G/K/S are fundamentally different. The F/G/K/S paper has this to say about the two different approaches to separating multiple characteristics simultaneously:

"We cannot use the two best understood methods [to separate $\ge 3$ characteristics simultaneously], countable support iterations of proper forcings (as it forces $2^{\aleph_0}\le\aleph_2$) and, at least for the "right hand side" of the diagram, we cannot use finite support iterations of ccc forcings in the straightforward way (as it adds lots of Cohen reals, and thus increases $cov(\mathcal{M})$ to $2^{\aleph_0}$).

There are ways to overcome this obstacle. One way would be to first increase the continuum in a "long" finite support iteration, resulting in $cov(\mathcal{M})=2^{\aleph_0}$, and then "collapsing" $cov(\mathcal{M})$ in another, "short" finite support iteration. In a much more sophisticated version of this idea, Mejia [Mej13] recently constructed several models with many simultaneously different cardinal characteristics in Cichon's Diagram (building on work of Brendle [Bre91], Blass-Shelah [BS89] and Brendle-Fischer [BF11]).

We take a different approach, completely avoiding finite support, and use something in between a countable and finite support product (or: a form of iteration with very "restricted memory").

This construction avoids Cohen reals, in fact it is $\omega^\omega$-bounding, resulting in $\mathfrak{d}=\aleph_1$. This way we get an independence result "orthogonal" to the ccc/finite-support results of Mejia.

The fact that our construction is $\omega^\omega$-bounding is not incidental, but rather a necessary consequence of the two features which, in our construction, are needed to guarantee properness: a "compact" or "finite splitting" version of pure decision, and fusion . . ."

[Creature forcing and five cardinal characteristics in Cichon's diagram], pg. 2

I can't resist bumping this to add one fascinating result (from this morning!) of Goldstern, Kellner, and Shelah: that as many as possible of the cardinal characteristics in Cichon's diagram can be separated simultaneously! Now granted, this isn't a consistency result relative to ZFC - at present, it takes four strongly compact cardinals, which is no small assumption - but still, this is absolutely amazing.

EDIT: And as of a couple weeks ago, the other shoe has dropped (thanks to Jakob for bringing this to my attention below): Goldstern, Kellner, Mejia, and Shelah got it in ZFC.

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    $\begingroup$ Thank you for advertising. Suggestions/corrections/comments are welcome. $\endgroup$ – Goldstern Feb 4 '14 at 12:34
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    $\begingroup$ I am not so familiar with historic iteration (front.math.ucdavis.edu/math.LO/9607227), but it seems to me that this is quite different from our paper. Our forcings are not ccc, and our forcing looks more like a product than an iteration, in the sense that there is no obvious linear order on the coordinates; each "iterand" produces a real which is somewhat generic over all the others. Also (bug or feature?) we do not increase $\mathfrak d$. At the moment. $\endgroup$ – Goldstern May 4 '14 at 16:33
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    $\begingroup$ Another update: arxiv.org/abs/1904.02617 : we (Goldstern Mejia Shelah and myself) add m, p and h; so now we have 13 characteristics (maybe it is rude to advertise your own work, but I'm too lazy to be polite and create a sock puppet account.) $\endgroup$ – Jakob Apr 13 '19 at 18:44
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    $\begingroup$ And yet another update: Now without large cardinals arxiv.org/abs/1906.06608 $\endgroup$ – Jakob Jul 7 '19 at 20:26
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    $\begingroup$ Since Jakob is too modest, I'll also note that the paper from 2017 was accepted to Annals of Mathematics. $\endgroup$ – Asaf Karagila Jul 8 '19 at 10:25

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