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Aspects of constructivity in mathematics

DOI: 10.1016/s0049-237x(09)70438-0
  • Logic
  • Mathematics


Publisher Summary This chapter discusses the aspects of constructivity in mathematics, which include continuity of solutions in parameters, topological models for intuitionistic analysis, and functional interpretations of analysis. A classical notion of constructive result is discussed and specifically related to the notion of continuity in parameters. There was a clear need for a systematic treatment of the way in which constructive proofs give rise to continuity in parameters, though the main tool, which is used to make the connection did not begin to emerge until Scott 1968, 1970. In this chapter, notions of continuity in parameters are applied to parameters of various types: natural numbers (N), reals (R), continuous maps R to R (R → R), continuous maps from R → R to R, and so on. All this takes place in a suitable cartesian closed category. The chapter describes the topological models for these types. However sheaves on topological spaces model much more than finite types; there is a cumulative hierarchy of sheaves, which models intuitionistic Zermelo-Fraenkel set theory (IZF) together with Zorn's Lemma (ZL).

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