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For limit ordinals $\alpha \le \beta$, the structures $V_{\alpha }$ and $V_{\beta }$ agree on whether any shared set is a cardinal number. They agree on being an ordinal. If $\delta <\alpha$ is not a cardinal number, a bijection from some $\gamma <\delta$ onto $\delta$ has rank of a set at most $\delta +3$, hence belongs to $V_{\alpha }$. Its defining properties are bounded formulas in set theory. Both structures therefore see the same failure witness. This full agreement is stronger than downward absoluteness of cardinalhood between arbitrary transitive models.

A forcing collapses a ground-model cardinal $\kappa$ if it forces a bijection from a smaller ordinal onto the ground ordinal $\kappa$. The ordinal itself is preserved; its being an initial ordinal is lost.

Regard the proper class of all ordinals as a poset category with $\alpha \to \beta$ when $\alpha \ge \beta$ in the usual order. It is locally small and complete: the product of any set-indexed family is its ordinary supremum, and equalizers are automatic in a poset. It has no initial object because there is no largest ordinal. Consequently every representable functor from it to sets preserves all small limits but has no left adjoint, since a left adjoint would have to send the empty set to an initial object.

If a forcing is $\kappa$-closed in the ground model, then it adds no functions from a ground ordinal $\alpha <\kappa$ into a ground set $B$. Recursively decide each value inside the ground model and take common stronger bounds at limit stages and after the final step. Conditions deciding a whole ground function are dense below any condition asserting this type of function. The dense-below generic meeting lemma ensures the generic filter meets that dense subset of a forcing order. One arbitrary bound need not be in the filter in an ordered set.

An unbounded class in set theory of ordinals is closed if every nonempty set-sized increasing sequence of its members has its supremum in the class. This is the proper-class analogue of a club set. The reflection theorem for definable hierarchies gives such a class for every finite collection of first-order formulas.

A level $L_{\lambda }$ is a coding level when $\lambda$ is a limit ordinal, the representation of every well-order code in $L_{\lambda }$ is below $\lambda$, and every ordinal below $\lambda$ has a well-order code in $L_{\lambda }$.

A forcing is cofinality-preserving over a transitive model $M$ if every generic extension has the same cofinality for every ground ordinal. The values are compared as ordinals, which forcing preserves.

Two ordinals commute under ordinal multiplication if and only if their squares commute. One implication follows from associativity. For the converse, squaring is strictly increasing: $A<B$ implies $A^2\le BA<B^2$. If $\alpha \beta <\beta \alpha$, monotonicity givesa contradiction to commuting squares. Swap the factors to exclude the reverse inequality; zero factors cause no exception.

Finite canonical terms built from zero, ordinal sums and powers of omega describe exactly the ordinals below epsilon zero. Syntactic validity and comparison are decidable by recursion through the exponent subterms. Enumerating valid term codes transfers their order to a computable well-order on all natural numbers.

If $M$ is a transitive model of ZF, its internally computed $L_{\alpha }$ is the actual $L_{\alpha }$ for each ordinal $\alpha \in M$. At a successor stage use absoluteness of satisfaction for the same set structure, formula codes and parameters; at a limit stage take the union of the already identical earlier stages. Choice is not required.

A countable ordinal has a countable set as its underlying set.

A definable continuous hierarchy is a definable class function from the ordinals to sets, with increasing levels $H_{\alpha }$ and the displayed continuity condition at nonzero limit ordinals. Its union is a definable class in set theory. One often additionally requires transitive sets as levels. Finite collections of first-order formulas reflect along this hierarchy by the reflection theorem for definable hierarchies.

The ordinal ${\epsilon }_0$ is the least fixed point of $\alpha \mapsto {\omega }^{\alpha }$. It is the supremum of $1,\omega ,{\omega }^{\omega },{\omega }^{{\omega }^{\omega }},\dots$, and is countable.

For $f,g:\kappa \to \kappa$, write $f{<}^{\ast }g$ when there is $\gamma <\kappa$ such that $f(\delta )<g(\delta )$ whenever $\gamma <\delta <\kappa$. For an infinite regular cardinal, suprema of fewer than $\kappa$ ordinals below $\kappa$ remain below it, permitting the long chain under eventual domination construction.

With $\kappa$ an uncountable regular cardinal, finite partial functions on $\kappa \times \omega$ assigning a value below the first coordinate collapse every ground ordinal below $\kappa$ to countable size. The Delta-system lemma gives the $\kappa$ chain in a partial order condition; the possible-values lemma for chain-condition forcing then preserves the regularity of $\kappa$. Therefore $\kappa$ becomes the extension ${\omega }_1$. Strong inaccessibility is enough but is not needed for this identification.