TITLE: Are there interesting examples of theorems proved using ‘height’ extensions?
QUESTION [13 upvotes]: It's well known that forcing is more than a tool for proving independence: We can prove theorems and formulate axioms in theories like $\mathsf{ZFC}$ by moving to forcing extensions (e.g. $\mathfrak{p}=\mathfrak{t}$, remarkable cardinals, Todorčević and Farah's book "Some applications of the method of forcing"). Are there nice examples of theorems/axioms that use "height" extensions (i.e. where we use "ordinals" longer than $\mathit{Ord}$ or where the resources required are strictly greater than second-order etc.) to prove some result in $\mathsf{ZFC}$ (or an extension thereof) or about the ground model? (From here on, let "height extension of $M$" denote any model $M'$ such that $\mathit{Ord}^M \in M'$).
Some uses that occur to me:

Uses of $\mathsf{ETR}$ in class theory (e.g. using iterated truth predicates and the connection to determinacy for class games, cf. Gitman and Hamkins "Open Determinacy for Class Games").

#-generation. This is a technical axiom to state, so I won't do so here, the core point is that we capture reflection properties of some model $M$ by taking it to be an initial segment of a model $M'$ generated by an ultrapower construction (this ultrapower, in turn, may be longer than $\mathit{Ord}^M$). See Honzik and Friedman "On Strong Forms of Reflection in Set Theory" for details.

There are useful definable well-order longer than $\mathit{Ord}$ (e.g. the ordering on mice).


I am curious as to whether the use of such "height" extensions pops up a lot in set-theoretic practice, and whether its as useful/ubiquitous as the use of forcing in proving theorems and formulating axioms (beyond relative consistency).

REPLY [6 votes]: Here is another instance, which appears in my recent paper with Bokai Yao on second-order reflection in the context of KMU with abundant urelements.

Joel David Hamkins and Bokai Yao, Reflection in second-order set theory with abundant urelemets bi-interprets a supercompact cardinal, 2022, arXiv:2204.09766.

The following theorem is an immediate consequence of the main theorem.
Theorem. Assume Kelley-Morse set theory with urelements KMU, with the abundant atom axiom and second-order reflection. Then there is a stationary proper class of measurable cardinals, partially supercompact cardinals, and more.
I mention the theorem because the main proof method is to undertake the unrolling construction, which produces sets at ranks higher than Ord.

Starting at lower right in the model $\langle V(A),\in,\mathcal{V}\rangle$ in which the hypothesis is satisfied, we undertake unrolling to produce the models $W$ and $\bar V$, in which there is a supercompact cardinal. In fact, it is the cardinal $\kappa=\text{Ord}^V$ that becomes supercompact in these taller models. And the supercompactness of $\kappa$ in $\bar V$ and $W$ reflects to diverse consequences in the original universe $V(A)$ and its class of pure sets $V$, such as a stationary proper class of measurable and partially supercompact cardinals, as much supercompactness as desired.