Return to Answer

"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

"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

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

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.

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 $add, cov, non, cof$:

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 $add, cov, non, cof$: