 Definition 2 (Curry-complete theory) A theory \(\mathcal{T}\) is Curry-complete provided that for every sentence \(\pi\) in the language of \(\mathcal{T}\), there is some \(\pi'\) such that (i) there is a Curry sentence for \(\pi'\) and \(\mathcal{T}\) and (ii) if \(\vdash_{\mathcal{T}} \pi'\) then \(\vdash_{\mathcal{T}} \pi\).