Ordinals and omega in transitive models

Bounded formulas are absolute for transitive sets: If $M\subseteq N$ are nonempty transitive sets, every ${\mathrm{\Delta }}_0$ formula is absolute between them on parameter tuples from $M$. No internal set-theory axioms are required. The analogous assertion for definable transitive classes is a formula-by-formula scheme.

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 $\alpha$.

Absolute basic set operations and relations: The graphs of empty set, subset, unordered pair, singleton, union, intersection (with $\bigcap \mathrm{\varnothing }=\mathrm{\varnothing }$), difference, Kuratowski ordered pair, Cartesian product, relation domain/range, functionhood, evaluation and injection have ${\mathrm{\Delta }}_0$ definitions. Thus their values agree between transitive membership structures whenever the input and output sets are in the smaller domain. This is graph agreement, not an assertion that an arbitrary transitive domain is closed under these operations.