Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On Tarski's fixed point theorem


Author: Giovanni Curi
Journal: Proc. Amer. Math. Soc. 143 (2015), 4439-4455
MSC (2010): Primary 03G10, 03E70; Secondary 03F65, 18B35
DOI: https://doi.org/10.1090/proc/12569
Published electronically: June 11, 2015
MathSciNet review: 3373943
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Peter Aczel, Frege structures and the notions of proposition, truth and set, The Kleene Symposium (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1978), Stud. Logic Foundations Math., vol. 101, North-Holland, Amsterdam-New York, 1980, pp. 31-59. MR 591874 (82e:03045)
  • [2] 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 (85g:03085), https://doi.org/10.1016/S0049-237X(09)70120-X
  • [3] 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 (88a:03149), https://doi.org/10.1016/S0049-237X(09)70683-4
  • [4] Peter Aczel, Aspects of general topology in constructive set theory, Ann. Pure Appl. Logic 137 (2006), no. 1-3, 3-29. MR 2182096 (2007b:03106), https://doi.org/10.1016/j.apal.2005.05.016
  • [5] 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 (2011a:54001), https://doi.org/10.1016/j.apal.2009.03.005
  • [6] P. Aczel and M. Rathjen, Notes on Constructive Set Theory, Mittag-Leffler Technical Report No. 40, 2000/2001.
  • [7] 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 (2012c:03175), https://doi.org/10.1016/j.tcs.2010.12.019
  • [8] T. Coquand, A Topos Theoretic Fix Point Theorem, Unpublished note, June 1995.
  • [9] 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 (2012e:03137), https://doi.org/10.1002/malq.200910037
  • [10] Giovanni Curi, On the existence of Stone-Čech compactification, J. Symbolic Logic 75 (2010), no. 4, 1137-1146. MR 2767961 (2012g:54049), https://doi.org/10.2178/jsl/1286198140
  • [11] Giovanni Curi, Topological inductive definitions, Ann. Pure Appl. Logic 163 (2012), no. 11, 1471-1483. MR 2959656, https://doi.org/10.1016/j.apal.2011.12.005
  • [12] 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 (82d:03061)
  • [13] Nicola Gambino, Heyting-valued interpretations for constructive set theory, Ann. Pure Appl. Logic 137 (2006), no. 1-3, 164-188. MR 2182102 (2007e:03104), https://doi.org/10.1016/j.apal.2005.05.021
  • [14] Edward Griffor and Michael Rathjen, The strength of some Martin-Löf type theories, Arch. Math. Logic 33 (1994), no. 5, 347-385. MR 1308848 (95j:03097), https://doi.org/10.1007/BF01278464
  • [15] J. M. E. Hyland, The effective topos, The L.E.J. Brouwer Centenary Symposium (Noordwijkerhout, 1981) Stud. Logic Foundations Math., vol. 110, North-Holland, Amsterdam-New York, 1982, pp. 165-216. MR 717245 (84m:03101)
  • [16] 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 (2005g:18007)
  • [17] A. Joyal and I. Moerdijk, Algebraic set theory, London Mathematical Society Lecture Note Series, vol. 220, Cambridge University Press, Cambridge, 1995. MR 1368403 (96k:03127)
  • [18] Robert S. Lubarsky, CZF and second order arithmetic, Ann. Pure Appl. Logic 141 (2006), no. 1-2, 29-34. MR 2229927 (2007a:03094), https://doi.org/10.1016/j.apal.2005.07.002
  • [19] 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 (2004f:03103), https://doi.org/10.1002/malq.200310054
  • [20] 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 (2006m:03095)
  • [21] 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 (92b:68047), https://doi.org/10.1142/S0129054190000242
  • [22] T. Streicher, Realizability models for CZF+ $ \neg $ Pow, unpublished note.
  • [23] Paul Taylor, Intuitionistic sets and ordinals, J. Symbolic Logic 61 (1996), no. 3, 705-744. MR 1412506 (97j:03102), https://doi.org/10.2307/2275781
  • [24] A. Troelstra and D. van Dalen, Constructivism in mathematics, an introduction. Vol. I. North-Holland, 1988.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 03G10, 03E70, 03F65, 18B35

Retrieve articles in all journals with MSC (2010): 03G10, 03E70, 03F65, 18B35


Additional Information

Giovanni Curi
Affiliation: Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Trieste, 63 - 35121 Padova, Italy
Email: giovanni.curi@email.it

DOI: https://doi.org/10.1090/proc/12569
Keywords: Tarski's fixed point theorem, inductive definitions, constructive set theories, uniform objects
Received by editor(s): March 25, 2013
Received by editor(s) in revised form: November 5, 2013, March 13, 2014, and May 30, 2014
Published electronically: June 11, 2015
Dedicated: To Orsola
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society