Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T17:54:28.906Z Has data issue: false hasContentIssue false
  • English
  • Français

ABSTRACT INDUCTIVE AND CO-INDUCTIVE DEFINITIONS

Published online by Cambridge University Press:  01 August 2018

GIOVANNI CURI
GIOVANNI CURI*
Affiliation:
VIA BOITO, 23 - 35122 PADOVA, ITALYE-mail:giovanni.curi@email.it
Get access Rights & Permissions [Opens in a new window]

Abstract

In [G. Curi, On Tarski’s fixed point theorem. Proc. Amer. Math. Soc., 143 (2015), pp. 4439–4455], a notion of abstract inductive definition is formulated to extend Aczel’s theory of inductive definitions to the setting of complete lattices. In this article, after discussing a further extension of the theory to structures of much larger size than complete lattices, as the class of all sets or the class of ordinals, a similar generalization is carried out for the theory of co-inductive definitions on a set. As a corollary, a constructive version of the general form of Tarski’s fixed point theorem is derived.

Keywords

Primary 03G1003E70Secondary 03F6518B35inductive definitionsco-inductive definitionscomplete latticesTarski’s fixed point theorem
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 2 , June 2018 , pp. 598 - 616
Copyright
Copyright © The Association for Symbolic Logic 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Dedicated to Peter Aczel, on the occasion of his 75th birthday.

References

REFERENCES

Aczel, P., An introduction to inductive definitions, Handbook of Mathematical Logic (Barwise, J., editor), Studies in Logic and the Foundations of Mathematics, Vol. 90, North-Holland, Amsterdam, 1977, pp. 739–782.CrossRefGoogle Scholar
Aczel, P., The type theoretic interpretation of constructive set theory: Inductive definitions, Logic, Methodology, and Philosophy of Science VII (Marcus, R. B., Dorn, G. J. W., and Weingartner, P., editors), Studies in Logic and the Foundations of Mathematics, Vol. 114, North-Holland, Amsterdam, 1986, pp. 17–49.Google Scholar
Aczel, P., Aspects of general topology in constructive set theory. Annals of Pure and Applied Logic, vol. 137 (2006), no. 1–3, pp. 3–29.CrossRefGoogle Scholar
Aczel, P., The relation reflection scheme. Mathematical Logic Quarterly, vol. 54 (2008), no. 1, 511.CrossRefGoogle Scholar
Aczel, P. and Rathjen, M., Notes on constructive set theory, Mittag-Leffler, Technical report no. 40, 2000/2001.Google Scholar
Aczel, P. and Rathjen, M., Constructive Set Theory, Book Draft, 2010.Google Scholar
Cousot, P. and Cousot, R., Constructive versions of Tarski’s fixed point theorems. Pacific Journal of Mathematics, vol. 82 (1979), no. 1, pp. 43–57.CrossRefGoogle Scholar
Curi, G., Topological inductive definitions. Annals of Pure and Applied Logic, vol. 163 (2012), no. 11, pp. 1471–1483. Kurt Gödel Research Prize Fellowships 2010.CrossRefGoogle Scholar
Curi, G., On Tarski’s fixed point theorem. Proceedings of the American Mathematical Society, vol. 143 (2015), pp. 4439–4455.CrossRefGoogle Scholar
Gambino, N., Heyting-valued interpretations for constructive set theory. Annals of Pure and Applied Logic, vol. 137 (2006), no. 1–3, pp. 164–188.CrossRefGoogle Scholar
Myhill, J., Constructive set theory, this Journal, vol. 40 (1975), no. 3, pp. 347–382.Google Scholar
Tarski, A., A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics, vol. 5 (1955), pp. 285–309.CrossRefGoogle Scholar
Troelstra, A. and van Dalen, D., Constructivism in Mathematics, an Introduction, Vol. I, North-Holland, Amsterdam, 1988.Google Scholar