What the collapse fixes

Collapse of elementary membership submodels: Let $M$ be a set with $(M,\in )\models \mathrm{E}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}$ and let $X\prec (M,\in )$. In ambient ZF, $\in$ restricted to $X$ is well-founded and extensional. It has a unique transitive collapse $\pi :X\to \bar{X}$, and the inverse collapse followed by inclusion is an elementary embedding $\bar{X}\to M$. Countability is preserved by $\pi$.

Ordinal (von Neumann): A set $\alpha$ is an ordinal when both of the following hold.

1. $\alpha$ is a transitive set: every element of $\alpha$ is also a subset of $\alpha$, that is $x\in \alpha \Rightarrow x\subseteq \alpha$.
2. The membership relation restricted to $\alpha$, namely $\{(x,y)\in \alpha \times \alpha :x\in y\}$, is a strict well-order of $\alpha$ (def-well-order): it is irreflexive, transitive as a relation, trichotomous on $\alpha$, and every nonempty subset of $\alpha$ has an $\in$-least element.

Ordinals are written with lowercase Greek letters, and for ordinals we set

$\alpha <\beta :⟺\alpha \in \beta ,\alpha \le \beta :⟺(\alpha \in \beta \text{ or }\alpha =\beta )\mathrm{.}$

Write $0:=\mathrm{\varnothing }$, which is an ordinal because both clauses hold vacuously, and write ${\alpha }^{+}:=\alpha \cup \{\alpha \}$ for the successor of $\alpha$.