Possibly this is already too far away to qualify as 'answer', but it is long for a comment and might be of interest.

There is a classical construction of descending (not ascending), so rather matching artinian than noetherian, subgroups of abelian groups (or, to artificial stay a bit closer let's say submodules of Z-modules, but then this got generalized to modules over other rings so it is not totally artificial), where indexing by ordinals is done and relevant. They are called Ulm subgroups.

Let $G$ be an abelian group and $p$ some (fixed) prime. Than the (classical) p-height of an element $g$ is the supremum of all naturals such that the equation $p^nx = g$ has a solution $x$ in $G$.

One can express this differently by defining
$p^nG$ to be the set, in fact it is a subgroup, of all $h\in G$ of the form $p^ng$ and define the p-height as the supremum of all $n$ such that $g \in p^nG$.

Now, one can continue and say $p^{\omega}G$ is the inersection of all the $p^nG$ and so on.

Or formally, $p^0G = G$, $p^{\alpha+1}G= p(p^{\alpha}G)$ and $p^{\beta}G =\bigcap_{\alpha \lt \beta}$ for $\beta$ a limit ordinal.

This forms a descending chain of subgroups that (thus) eventually stabilzes; the ordinal where it stabilzes is called Ulm length. One can now also generalize the notion of p-height of g by defnining it as the ordinal $\sigma$ such that $g \in p^{\sigma}G$ yet not in $g \in p^{\sigma + 1}G$ and $\infty$ if such an ordinal does not exist.

And this notion is quite useful. For example, a classical result of Ulm (1930s) then says that two countable abelian $p$-groups (ie, order of each element a prime power) $G$ and $H$ are isomophic if and only if $p^{\alpha}/p^{\alpha+1}G$ and $p^{\alpha}H/p^{\alpha+1}H$ are isomorphic for each $\alpha$ (these are called the Ulm factors) and the stable parts are isomorphic.

And conversely for any ordinal one can construct a group of that Ulm length (and in addition one can prescribe to a considerable extent the Ulm factors at each point until there).

Ordinal-valued invariants. You can find an abstract of his talk here: websupport1.citytech.cuny.edu/faculty/lghezzi/Ghezzi/… If you see him around you can ask him, because I think this is related to what he was talking about in his talk. $\endgroup$ – Mahdi Majidi-Zolbanin Mar 13 '12 at 0:35