set theory - Consistency Results Separating Three Cardinal Characteristics Simultaneously

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 $langlemathcal{I}, subseteqrangle$.

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 http://arxiv.org/abs/1211.5209 constructed several examples of models separating multiple characteristics at once (see section 6).




• In a paper arxived today(!), Fischer/Goldstern/Kellner/Shelah http://arxiv.org/abs/1402.0367 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 $2^{aleph_0}lealeph_2$) 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 . . ."

- http://arxiv.org/pdf/1402.0367.pdf, pg. 2