The Active Role of Language Extensions in Mathematical Reasoning

Extending the language of a theory T by new predicate and function symbols is usually not considered to be an essential component of the reasoning from T, but a matter of convenience, justified by the extension-by-definitions procedure or sometimes by the process of skolemization. In this talk we argue that actually there are important cases in mathematics in which a systematic process of repeatedly extending the base language of T is an essential ingredient of the reasoning from T. A particularly important case of this sort is that of predicative set theory. We show that the systematic use of predicatively justified introduction of new predicate and function symbols allows us to go well beyond Feferman-Schuette ordinal $\Gamma_0$, which is usually taken to be “the limit of predicativity”.