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Two remarks on the Lifschitz realizability topos

  • Jaap Van Oosten (a1)

The purpose of this note is to clarify two points about the topos Lif, introduced in [13] as a generalization of Lifschitz' realizability ([9, 12]). Lif is a subtopos of Hyland's Effective topos ([1]). The points I want to make are:

Remark 1. Lif is the largest subtopos of satisfying the axiom (O):

where denotes partial recursive application, and “∈ Tot” means that e and f range over codes for total recursive functions. One may read (O) as the statement “The union of two -sets is again a -set”. That is, let be a subtopos of . Then (O) is true in for the standard interpretation (the variables range over the natural numbers object of , etc.) if and only if the inclusion factors through the inclusion Lif ↣ .

Remark 2. Like , Lif contains at least two weakly complete internal full subcategories, thus providing us with more models of polymorphism and other impredicative type theories.

The principle (O) has some standing in the history of constructive mathematics:

- H. Friedman has proved that (O) is equivalent to a formulation of intuitionistic completeness of the intuitionistic predicate calculus for Tarskian semantics; see [8]. This is not to imply that this result is of immediate relevance to Lif: Friedman works in a system of analysis, a theory of lawless sequences with an axiom of “open data” for arithmetical formulas, which at least for the domain of all functions from N to N fails in Lif. However, there might exist a “nonstandard model” of arithmetic and a corresponding system of analysis, in which we may be able to carry out his proof.

- Moreover, Remark 2 entails that Lif should provide us with models of synthetic domain theory (for an exposition, see [3]), and one with the nice property that the dual of one of the axioms (axiom 7 in [3]) comes for free, by (O).

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[1]Hyland, J. M. E., The effective topos, The L. E. J. Brower centenary symposium (Troelstra, and Dalen, Van, editors), North-Holland, 1982, pp. 165216.
[2]Hyland, J. M. E., A small complete category, Annals of Pure and Applied Logic, vol. 40 (1988), pp. 135165.
[3]Hyland, J. M. E., First steps in synthetic domain theory, Cateogry theory, proceedings of the conference in Como 1990 (Carboni, , Pedicchio, , and Rosolini, , editors), Lecture Notes in Mathematics, no. 1488, Springer, 1991, pp. 131156.
[4]Hyland, J. M. E., Johnstone, P. T., and Pitts, A. M., Tripos theory, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 88 (1980), pp. 205232.
[5]Hyland, J. M. E., Robinson, E., and Rosolini, G., Algebraic types in PER models, Mathematical foundations of programming language semantics (Main, , Mislove, , et al., editors), Lecture Notes in Computer Science, no. 442, Springer, 1990, pp. 333350.
[6]Hyland, J. M. E., The discrete objects in the effective topos, Proceedings of the London Mathematical Society, vol. 60 (1990), pp. 136.
[7]Johnstone, P. T., Topos theory, Academic Press, 1976.
[8]Leivant, D., Intuitionistic formal systems, Harvey Friedman's research on the foundations of mathematics (Harrington, et al., editors), North Holland, 1985.
[9]Lifschitz, V., CT0 is stronger than CT0!, Proceedings of the American Mathematical Society, vol. 73 (1979), pp. 101106.
[10]Phoa, W., Relative computability in the effective topos, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 106 (1989), pp. 419422.
[11]Rosolini, G., About modest sets, International Journal on the Foundations of Computer Science, vol. 1 (1990), pp. 341353.
[12]van Oosten, J., Liftschitz' readability, this Journal, vol. 55 (1990), pp. 805821.
[13]van Oosten, J., Extension of Lifschitz' realizability to higher arithmetic, and a solution to a problem of F, Richman, this Journal, vol. 56 (1991), pp. 964973.
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