The Stratiied Foundations as a Theory Modulo

The Stratiied Foundations are a restriction of naive set theory where the comprehension scheme is restricted to stratiiable propositions. It is known that this theory is consistent and that proofs strongly normalize in this theory. Deduction modulo is a formulation of rst-order logic with a general notion of cut. It is known that proofs normalize in a theory modulo if it has some kind of many-valued model called a pre-model. We show in this note that the Stratiied Foundations can be presented in deduction modulo and that the method used in the original normalization proof can be adapted to construct a pre-model for this theory. The Stratiied Foundations are a restriction of naive set theory where the comprehension scheme is restricted to stratiiable propositions. This theory is consistent 8] and proofs in this theory strongly normalize 2], while naive set theory is contradictory and the consistency of the Stratiied Foundations together with the extensionality axiom-the so-called New Foundations-is open. The Stratiied Foundations extend simple type theory and the normal-ization proof for the Stratiied Foundations, like that of type theory uses Girard's reducibility candidates. These two proofs, like all proofs following the line of Tait and Girard, have some parts in common. This motivates the investigation of general normalization theorems that have normalization theorems for speciic theories as consequences. The normalization theorem for deduction modulo 7] is an example of such a general theorem. It concerns theories expressed in deduction modulo 5] that are rst-order theories with a general notion of cut. According to this theorem, proofs normalize in a theory in deduction modulo if this theory has some kind of many-valued model called a pre-model. For instance, simple type theory can be expressed in deduction modulo 5, 6] and it has a pre-model 7, 6] and hence it has the normalization property. The normalization proof obtained this way is modular: all the lemmas speciic to type theory are concentrated in the pre-model construction while the theorem that the existence of a pre-model implies normalization is generic and can be used for any other theory in deduction modulo. The goal of this note is to show that the Stratiied Foundations also can be presented in deduction modulo and that the method used in the original normalization proof can be adapted to construct a pre-model for this theory. The normalization proof obtained this way is simpler than the original one because it …