On Tarski’s fixed point theorem

Curi, Giovanni

doi:10.1090/proc/12569

On Tarski’s fixed point theorem
HTML articles powered by AMS MathViewer

by Giovanni Curi PDF

Proc. Amer. Math. Soc. 143 (2015), 4439-4455 Request permission

Abstract:

A notion of abstract inductive definition on a complete lattice is formulated and studied. As an application, a constructive version of Tarski’s fixed point theorem is obtained.

References

Peter Aczel, Frege structures and the notions of proposition, truth and set, The Kleene Symposium (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1978), Studies in Logic and the Foundations of Mathematics, vol. 101, North-Holland, Amsterdam-New York, 1980, pp. 31–59. MR 591874
Peter Aczel, The type theoretic interpretation of constructive set theory: choice principles, The L. E. J. Brouwer Centenary Symposium (Noordwijkerhout, 1981) Stud. Logic Found. Math., vol. 110, North-Holland, Amsterdam, 1982, pp. 1–40. MR 717236, DOI 10.1016/S0049-237X(09)70120-X
Peter Aczel, The type theoretic interpretation of constructive set theory: inductive definitions, Logic, methodology and philosophy of science, VII (Salzburg, 1983) Stud. Logic Found. Math., vol. 114, North-Holland, Amsterdam, 1986, pp. 17–49. MR 874778, DOI 10.1016/S0049-237X(09)70683-4
Peter Aczel, Aspects of general topology in constructive set theory, Ann. Pure Appl. Logic 137 (2006), no. 1-3, 3–29. MR 2182096, DOI 10.1016/j.apal.2005.05.016
Peter Aczel and Giovanni Curi, On the $T_1$ axiom and other separation properties in constructive point-free and point-set topology, Ann. Pure Appl. Logic 161 (2010), no. 4, 560–569. MR 2584733, DOI 10.1016/j.apal.2009.03.005
P. Aczel and M. Rathjen, Notes on Constructive Set Theory, Mittag-Leffler Technical Report No. 40, 2000/2001.
Benno van den Berg and Ieke Moerdijk, Aspects of predicative algebraic set theory, II: Realizability, Theoret. Comput. Sci. 412 (2011), no. 20, 1916–1940. MR 2814767, DOI 10.1016/j.tcs.2010.12.019
T. Coquand, A Topos Theoretic Fix Point Theorem, Unpublished note, June 1995.
Giovanni Curi, On some peculiar aspects of the constructive theory of point-free spaces, MLQ Math. Log. Q. 56 (2010), no. 4, 375–387. MR 2681341, DOI 10.1002/malq.200910037
Giovanni Curi, On the existence of Stone-Čech compactification, J. Symbolic Logic 75 (2010), no. 4, 1137–1146. MR 2767961, DOI 10.2178/jsl/1286198140
Giovanni Curi, Topological inductive definitions, Ann. Pure Appl. Logic 163 (2012), no. 11, 1471–1483. MR 2959656, DOI 10.1016/j.apal.2011.12.005
M. P. Fourman and D. S. Scott, Sheaves and logic, Applications of sheaves (Proc. Res. Sympos. Appl. Sheaf Theory to Logic, Algebra and Anal., Univ. Durham, Durham, 1977) Lecture Notes in Math., vol. 753, Springer, Berlin, 1979, pp. 302–401. MR 555551
Nicola Gambino, Heyting-valued interpretations for constructive set theory, Ann. Pure Appl. Logic 137 (2006), no. 1-3, 164–188. MR 2182102, DOI 10.1016/j.apal.2005.05.021
Edward Griffor and Michael Rathjen, The strength of some Martin-Löf type theories, Arch. Math. Logic 33 (1994), no. 5, 347–385. MR 1308848, DOI 10.1007/BF01278464
J. M. E. Hyland, The effective topos, The L.E.J. Brouwer Centenary Symposium (Noordwijkerhout, 1981) Studies in Logic and the Foundations of Mathematics, vol. 110, North-Holland, Amsterdam-New York, 1982, pp. 165–216. MR 717245
Peter T. Johnstone, Sketches of an elephant: a topos theory compendium. Vol. 2, Oxford Logic Guides, vol. 44, The Clarendon Press, Oxford University Press, Oxford, 2002. MR 2063092
A. Joyal and I. Moerdijk, Algebraic set theory, London Mathematical Society Lecture Note Series, vol. 220, Cambridge University Press, Cambridge, 1995. MR 1368403, DOI 10.1017/CBO9780511752483
Robert S. Lubarsky, CZF and second order arithmetic, Ann. Pure Appl. Logic 141 (2006), no. 1-2, 29–34. MR 2229927, DOI 10.1016/j.apal.2005.07.002
Michael Rathjen and Robert S. Lubarsky, On the regular extension axiom and its variants, MLQ Math. Log. Q. 49 (2003), no. 5, 511–518. MR 1998402, DOI 10.1002/malq.200310054
Michael Rathjen, Realizability for constructive Zermelo-Fraenkel set theory, Logic Colloquium ’03, Lect. Notes Log., vol. 24, Assoc. Symbol. Logic, La Jolla, CA, 2006, pp. 282–314. MR 2207359
G. Rosolini, About modest sets, Internat. J. Found. Comput. Sci. 1 (1990), no. 3, 341–353. Third Italian Conference on Theoretical Computer Science (Mantova, 1990). MR 1097022, DOI 10.1142/S0129054190000242
T. Streicher, Realizability models for CZF+ $\neg$ Pow, unpublished note.
Paul Taylor, Intuitionistic sets and ordinals, J. Symbolic Logic 61 (1996), no. 3, 705–744. MR 1412506, DOI 10.2307/2275781
A. Troelstra and D. van Dalen, Constructivism in mathematics, an introduction. Vol. I. North-Holland, 1988.