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## On the Proof-Theoretic Foundations of Set Theory
## IntroductionFoundations of set theory relates to answers of the following two main questions: (A) What is a set? (B) What does it mean to reason with sets? With respect to (A) Cantor's informal definition of the notion of a set seems perfectly intuitive. It is natural to think of It might be an issue of confusing extensional and intensional perspectives. The idea of a set as a gathering of given objects into a whole paints a picture of sets as collections Russell's antinomy came as a veritable shock to those few thinkers who occupied themselves with foundational problems at the turn of the century. [4, p. 2] There is something strange about this reaction. Why do we expect that such a, very general, more intensional characterisation will capture just sets as collections of objects in an intuitive extensional sense, i.e., as bracketing a given collection of objects? There is no reason to think that these two notions and perspectives should coincide, i.e., that the intensional characterisation would produce just the definition, i.e., the defining property mentary one. Its proof-theoretic behaviour can, for example, be observed already in intuitionistic propositional logic [3]. So if we accept the idea of abstraction with respect to any given defining property, i.e., full comprehension, as a foundation for set theory, we have an answer to question (A), that is, what a set is. But how should we then understand the paradoxes? The Russell paradox for instance seems to show that something is wrong with respect to question (B). The paradoxical argument builds on several basic assumptions, where one of the most important ones is the assumption that ' The solutions offered by Zermelo-Fraenkel set theories, von Neumann-Bernays set-class theories and type theories follow the strategy of retirement behind more or less safe boundaries (see [4]). There are several ideas about proof-theoretically founded restrictions on the comprehension scheme [5], [9]. Compare further the set theory of Fitch (see [4], [9]), the notion of a Frege structure [1] and notions of structural rules in relation to paradoxes [14]. Now what if we revisit the original idea without making strong assumptions on closure properties of the theoretical notion of a set? That is, take the basic definitions for what they are without confounding intensional and extensional perspectives. |

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