 Solution

There is a bounded definition of omega. Transitivity has the bounded definition $\mathrm{Trans}(x)\equiv \forall u\in x\forall v\in u(v\in x)$. An ordinal is a transitive set of transitive sets on which membership is a strict total order; this too has a bounded formula in set theory, since ZF supplies well-foundedness by the Axiom of foundation.

Define the bounded predicateFor an ordinal, this says that it is a nonzero limit ordinal. It follows in ZF thatsince $\omega$ is the least nonzero limit ordinal. The right side is a bounded formula in set theory; consequently it is absolute between the specified transitive models. This also shows directly that those models have the same $\omega$.