= Bounded definition of omega
{title2=$x=\omega$}
In <ZF>, $\omega$ is the least nonzero <limit ordinal>. Ordinality is expressed by a <bounded formula in set theory> asserting transitivity and strict total membership order, with well-foundedness supplied by the <Axiom of foundation>. Put $L(x)=\operatorname{Ord}(x)\land x\ne\varnothing\land\forall y\in x\,\exists z\in x\,(y\in z)$. Then $x=\omega$ is equivalent over <ZF> to $L(x)\land\forall y\in x\neg L(y)$, again bounded. Therefore nested <transitive models> of <ZF> have the same $\omega$.
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