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Semantics and Soundness

  • Patrick Schultz
  • David I. Spivak
Chapter
Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 29)

Abstract

In this chapter, we prove that the temporal type theory developed in Chaps. 4 and 5 is sound in the topos \(\mathcal {B}\). To do so, we begin in Sect. 6.1 by recalling the Kripke–Joyal semantics by which to interpret type-theoretic formulas in the topos \(\mathcal {B}\). Then in Sect. 6.3 we discuss the sheaf of real numbers and Time. We then proceed to our main goal: proving that our type signature—i.e., the one atomic predicate symbol and the ten axioms presented in Chap. 5—are sound in \(\mathcal {B}\). This is done in Sect. 6.5, which begins with a Table 6.2 summarizing the type signature.

References

  1. [Jac99]
    Jacobs, B.: Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics, vol. 141, pp. xviii+ 760. North-Holland, Amsterdam (1999). ISBN:0-444-50170-3Google Scholar
  2. [MM92]
    MacLane, S., Moerdijk, I.: Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer, New York (1992). ISBN:0387977104Google Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Patrick Schultz
    • 1
  • David I. Spivak
    • 1
  1. 1.Massachusetts Institute of TechnologyCambridgeUSA

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