7 Added reference to a paper by Shelah and Steprans.

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.

6 Improved notation

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 intitial 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 (n-1)$-ary sets that covers all of$n^\omega$. Any finitely many$\ell_n$can be separated from each other simultaneously. This is shown in [Geschke, Kojman, Convexity numbers of closed subsets in R^n, Proc. Am. Math. Soc. 130, No. 10, 2871-2881 (2002)] 2002)], which is here. Proofs of such statements usually involve forcing with large countable support products over a model of GCH rather than iterated forcing. However, there are also some examples that use iterated forcing. 5 added 11 characters in body There is a paper of Shelah and Goldstern devoted to the separation of many simple cardinal invariants (this is a technical term). There are more recent papers on this subject by Kellner and Shelah, if I remember correctly. An easy case that I am very familiar with are the so called localization numbers. A closed set$S\subseteq\omega^\omega$is$n$-ary if in the tree of finite intitial segments of elements of$S$every node has at most$n$immediate successors. Let For$n\geq 2$let$\ell_n$be the least size of a family of$n$-ary n-1$-ary sets that covers all of $\omega^\omega$.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.

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