Is cut-free logic fit for unrestricted abstraction?

Abstract

Without cut the paradoxes of unrestricted abstraction do not yield the empty sequent, so would it be a viable option to give up cut in order to be able to accommodate unrestricted abstraction without the risk of triviality? My answer will be in the negative and that by applying a system LKλ− of cut-free logic with unrestricted abstraction as a meta theory to a rudimentary formal system Σc (“C” for Curry) which is tailored to prove a version of Curry's paradox with a minimum of technicalities. Now, a proof predicate for Σc can be simulated in LKλ− confirming the existence of a deduction of ⊥ in Σc while, on the other hand, a kind of reflection principle can be established for Σc which asserts that every deduction in Σc actually implies the wff of which it is a deduction. This then leads to the result that the deduction of ⊥ is not a deduction in Σc. In view of the primitive recursive character of the proof predicate for Σc we thus arrive at a provable contradiction in LKλ− far more devastating than those established by Russell's or Curry's paradox because it concerns (simulated) primitive recursive predicates, not just a naive notion of set.