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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 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.

Use the standard convention that a strongly inaccessible cardinal is uncountable, regular and strong limit. The printed explanatory definition omits uncountability: literally it also admits $\kappa =\omega$, for which no finite cardinal is a worldly cardinal. Thus that omission makes the requested conclusion false under the literal abbreviated definition. The proof below applies to the usual, intended strongly inaccessible cardinal convention $\kappa >\omega$.

First $|V_{\gamma }|<\kappa$ for every $\gamma <\kappa$. At successor stages this follows from the strong limit cardinal property, and at limits from the regular cardinal property. Consequently $V_{\kappa }\models \mathsf{ZFC}$. In particular, for Axiom schema of replacement, a domain $a\in V_{\kappa }$ has size less than $\kappa$, so its functional image has fewer than $\kappa$ elements, so the supremum of the rank of a set over its elements is below $\kappa$ by regular cardinal structure. All other axioms, including axiom of choice, have their witnesses at bounded ranks below $\kappa$; uncountability supplies axiom of infinity.

Enumerate the first-order formulas as ${\phi }_0,{\phi }_1,\dots$. For each finite collection, reflection inside the set structure $V_{\kappa }$ gives a closed unbounded subset of $\kappa$ of agreeing ranks. To see why the witness bounds stay below $\kappa$, there are fewer than $\kappa$ parameter tuples at each $V_{\gamma }$, and the supremum of their least witness ranks remains below $\kappa$ by regular cardinal structure. Closing under these bounds and taking increasing countable limits gives the usual reflection theorem for definable hierarchies argument. The countable intersection of these closed unbounded subsets is still closed unbounded because $\kappa$ is regular and uncountable. At each resulting $\alpha$,The infinite cardinal numbers below $\kappa$ also form a closed unbounded subset: they are unbounded because ${\lambda }^{+}\le 2^{\lambda }<\kappa$ for infinite $\lambda <\kappa$, and a supremum of increasing cardinal numbers is a cardinal number. Intersect the two closed unbounded subsets. Every member of the intersection is a worldly cardinal, and an unbounded subset of a regular cardinal $\kappa$ has size $\kappa$. HenceIn fact this proves the stronger worldly cardinals below an inaccessible cardinal result that these worldly cardinals contain a closed unbounded subset of $\kappa$.

A definable continuous hierarchy is a definable class function $\alpha \mapsto H_{\alpha }$ on the ordinals, with set-valued levels, such that $H_{\alpha }\subseteq H_{\beta }$ for $\alpha <\beta$ andfor nonzero limit ordinals. Its union is a definable class in set theory $H$. Often the definition additionally requires every level to be a transitive set; the following statement also works without that requirement.

The reflection theorem for definable hierarchies says that for any finite collection $\mathcal{F}$ of first-order formulas, there is a closed unbounded class of ordinals $\alpha$ such that, for every $\phi \in \mathcal{F}$ and every parameter tuple from $H_{\alpha }$,A useful justification is the witness-closure proof. Close $\mathcal{F}$ under subformulas. At each level and for each existential subformula, bound the least level containing a witness for each parameter tuple for which a witness exists in $H$. Axiom schema of replacement bounds these indices. Iterating the finitely many bounds through $\omega$ produces a limit level containing all required witnesses. Induction on the first-order formulas, equivalently the finite-formula version of the Tarski-Vaught test, yields agreement. Continuity gives closedness of the reflecting class, and starting above any prescribed ordinal gives unboundedness. This is finite reflection, not a claim that all first-order formulas reflect simultaneously in an arbitrary hierarchy.