The transition from PA to T(PA) can be imagined as an act of reflection on the truth of \(\mathcal{L}\)-sentences in PA. Similarly, the step from the typed \(T\)-sentences to the compositional axioms is also tied to a reflection principle, specifically the uniform reflection principle over the typed uniform \(T\)-sentences. This is the collection of sentences  \(\forall x\, Bew_{S} (\ulcorner \phi(\underline{x})\urcorner) \rightarrow \phi(x) \) where \(\phi\) ranges over formulas in \(\mathcal{L}_T\) with one free variable and S is the theory of the uniform typed T-sentences. Uniform reflection exactly captures the difference between the two theories: the reflection principle is both derivable in T(PA) and suffices to derive the six compositional axioms (Halbach 2001). Moreover, the equivalence extends to iterations of uniform reflection, in that for any ordinal \(\alpha , 1 + \alpha\) iterations of uniform reflection over the typed \(T\)-sentences coincides with T(PA) extended by transfinite induction up to the ordinal \(\varepsilon_{\alpha}\), namely the \(\alpha\)-th ordinal \(\kappa\) with the property that \(\omega^{\kappa} = \kappa \) (Leigh 2016). 