I am looking for examples of constructions for transfinite towers $(X\_{\alpha})\_{\alpha}$ generated by structures $X$ where the problem of determining whether the tower $(X\_{\alpha})\_{\alpha}$ stops growing is a non-trivial problem or the problem of determining the ordinal in which $X\_{\alpha}$ stops growing is a non-trivial problem.


In particular, I want the tower $(X\_{\alpha})\_{\alpha}$ to be generated by the following construction. Suppose that for each object $X$, there is a new object $C(X)$ and a morphism $e:X\rightarrow C(X)$. Then define the tower generated by $X$ by letting $X\_{0}=X$, $X\_{\alpha+1}=C(X\_{\alpha})$ and
$X\_{\gamma}=\varinjlim\_{\alpha<\lambda}X\_{\alpha}$ for limit ordinals $\gamma$ where the direct limit is taken in the category that $X$ belongs to.


I want all of the objects $X$ and each $X\_{\alpha}$ to be set sized.


**Non-Example:** The hierarchy of sets $(V\_{\alpha}[X])\_{\alpha}$ where
$V\_{0}[X]=X,V\_{\alpha+1}[X]=P(V\_{\alpha}[X])$ and $V\_{\gamma}[X]=\bigcup\_{\alpha<\gamma}V\_{\alpha}[X]$ does not count as an example of what I am looking for since the tower $(V\_{\alpha}[X])\_{\alpha}$ never stops growing and therefore whether $(V\_{\alpha}[X])\_{\alpha}$ terminates is now a trivial mathematics problem.


**Example 1:** Suppose that $G$ is a group. Let $G\_{0}=G$, and let
$G\_{\alpha+1}=\mathrm{Aut}(G\_{\alpha})$ and let $G\_{\gamma}=\varinjlim\_{\alpha<\gamma}G\_{\alpha}$. The transition mapping from $G\_{\alpha}$ to $G\_{\alpha+1}$ is the mapping $e$ where $e(g)(h)=ghg^{-1}$. Then $(G\_{\alpha})\_{\alpha}$ is the automorphism group tower generated by $G$. [The automorphism group tower always terminates.](http://www.ams.org/journals/proc/1998-126-11/S0002-9939-98-04797-2/home.html) In this case, the mapping $G\mapsto G\_{\alpha}$ is not functorial.


**Example 2:** Frames are the objects that people study in point-free topology. If $L$ is a frame, then let $\mathfrak{C}(L)$ denote the lattice of congruences of the frame $L$. Then $\mathfrak{C}(L)$ is always a frame. Define a mapping $e:L\rightarrow\mathfrak{C}(L)$ by letting $(x,y)\in e(a)$ if and only if $x\vee a=y\vee a$. Then the function $e$ is a frame homomorphism.


There are frames $L$ where the congruence tower generated by $L$ never terminates. However, for ordinals $\alpha$, it is a difficult open problem to determine whether there is a frame $L$ where $e:L\_{\beta}\rightarrow\mathfrak{C}(L\_{\beta})$ is a surjection if and only if $\beta\geq\alpha$ since in all known examples, the congruence tower either terminates before the fourth step or so or it never terminates.


If $L$ is a frame and $e:L\rightarrow\mathfrak{C}(L)$ is an isomorphism, then $L$ is a complete Boolean algebra.


Unlike Example 1, Example 2 is functorial.



Examples of transfinite towers
Consider the following construction of sets of ordinals.


* $X\_0=\{0\}$,
* $X\_{\alpha+1}=$ the closure of $X\_{\alpha}$ under $\gamma\mapsto\gamma+1$ and under *countable* sums,
* $X\_\alpha=\bigcup\_{\beta<\alpha}X\_\beta$ for $\alpha$ limit.


We can consider in ZF the question of whether this tower stabilizes.


It is definitely consistent that the answer is *yes*, since in ZFC $\omega\_1$ is regular, so $X\_\beta=\omega\_1$ for all $\beta\geq\omega\_1$.


The question of whether this tower can be non-stabilizing is much harder, and is equivalent to asking whether it is consistent that all ordinals have countable cofinality. This was shown to be consistent [by Gitik in 1980](https://link.springer.com/article/10.1007/BF02760939), assuming consistency of a proper class of strongly compact cardinals. Let me note that large cardinals cannot be avoided, because even making $\omega\_1$ and $\omega\_2$ singular has consistency at least as high as $0^\#$.


