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Models of intuitionistic TT and NF

  • Daniel Dzierzgowski (a1)

Abstract

Let us define the intuitionistic part of a classical theory T as the intuitionistic theory whose proper axioms are identical with the proper axioms of T. For example, Heyting arithmetic HA is the intuitionistic part of classical Peano arithmetic PA.

It's a well-known fact, proved by Heyting and Myhill, that ZF is identical with its intuitionistic part.

In this paper, we mainly prove that TT, Russell's Simple Theory of Types, and NF, Quine's “New Foundations,” are not equal to their intuitionistic part. So, an intuitionistic version of TT or NF seems more naturally definable than an intuitionistic version of ZF.

In the first section, we present a simple technique to build Kripke models of the intuitionistic part of TT (with short examples showing bad properties of finite sets if they are defined in the usual classical way).

In the remaining sections, we show how models of intuitionistic NF2 and NF can be obtained from well-chosen classical ones. In these models, the excluded middle will not be satisfied for some non-stratified sentences.

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References

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[1] Boffa, Maurice, The point on Quine's NF (with a bibliography), Teoria, vol. IV/2 (1984), pp. 313.
[2] Boffa, Maurice and Crabbé, Marcel, Les théorèmes 3-stratifiés de NF3, Comptes Rendus de l’ Académie des Sciences de Paris, Série A, vol. 280 (1975), pp. 16571658.
[3] Fitting, Melvin Chris, Intuitionistic logic model theory and forcing, Studies in Logic, North Holland, Amsterdam, London, 1969.
[4] Forster, Thomas E., On a problem of Dzierzgowski, Bulletin de la Société Mathématique de Belgique, série B, vol. 44 (1992), no. 2, pp. 207214.
[5] Forster, Thomas E., Set theory with a universal set. Exploiting an untyped universe, Oxford Logic Guides, vol. 20, Clarendon Press. Oxford University Press, 1992.
[6] Körner, Friederike, Cofinal indiscernibles and their application to New Foundations, Mathematical Logic Quaterly, vol. 40 (1994), no. 3, pp. 347356.
[7] Lavendhomme, René and Lucas, Thierry, A note on intuitionistic models of ZF, Notre Dame Journal of Formal Logic, vol. 24 (1983), no. 1, pp. 5466.
[8] Myhill, John, Embedding classical type theory in “intuitionistic” type theory, Axiomatic set theory (Scott, Dana S., editor), Proceedings of Symposia in Pure Mathematics, vol. XIII, Part I, American Mathematical Society, Providence, R.I., 1971, Proceedings of the Summer Institute in Set Theory, University of California, Los Angeles, 1967, pp. 267270.
[9] Myhill, John, Some properties of intuitionistic Zermelo-Fraenkel set theory, Cambridge summer school in mathematical logic, Lecture Notes in Mathematics, vol. 337, Springer-Verlag, 1973, pp. 206231.
[10] Myhill, John, Embedding classical type theory in “intuitionistic” type theory: a correction, Axiomatic set theory (Jech, Thomas, editor), Proceedings of Symposia in Pura Mathematics, vol. XIII, Part II, American Mathematical Society, Providence, R.I., 1974, Proceedings of the Summer Institute in Set Theory, University of California, Los Angeles, 1967, pp. 185188.
[11] Powell, William C., Extending Gödel's negative interpretation to ZF, this Journal, vol. 40 (1975), pp. 221229.
[12] Scott, Dana, Quine's individuals, Logic, methodology and philosophy of science (Stanford, California) (Nagel, Suppes, and Tarski, editors), Stanford University Press, 1962, Proceedings of the International Congress, Stanford, California, 1960, pp. 111115.

Models of intuitionistic TT and NF

  • Daniel Dzierzgowski (a1)

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