Crthuky

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 $e:Xrightarrow C(X)$. Then define the tower generated by $X$ by letting $X_0=X$, $X_alpha+1=C(X_alpha)$ and
$X_gamma=varinjlim_alpha<lambdaX_alpha$ 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 $V_gamma[X]=bigcup_alpha<gammaV_alpha[X]$ 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
$G_alpha+1=mathrmAut(G_alpha)$ and let $G_gamma=varinjlim_alpha<gammaG_alpha$. 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. In this case, the mapping $Gmapsto G_alpha$ is not functorial.

Example 2: Frames are the objects that people study in point-free topology. If $L$ is a frame, then let $mathfrakC(L)$ denote the lattice of congruences of the frame $L$. Then $mathfrakC(L)$ is always a frame. Define a mapping $e:LrightarrowmathfrakC(L)$ by letting $(x,y)in e(a)$ if and only if $xvee a=yvee a$. 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 $e:L_betarightarrowmathfrakC(L_beta)$ is a surjection if and only if $betageqalpha$ 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 $e:LrightarrowmathfrakC(L)$ is an isomorphism, then $L$ is a complete Boolean algebra.

Unlike Example 1, Example 2 is functorial.

Consider the following construction of sets of ordinals.

• $X_0=0$,




• $X_alpha+1=$ the closure of $X_alpha$ under $gammamapstogamma+1$ and under countable sums,




• $X_alpha=bigcup_beta<alphaX_beta$ 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 $betageqomega_1$.

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, 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 $0^#$.