Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-13T01:45:19.971Z Has data issue: false hasContentIssue false
  • English
  • Français

More about relatively lawless sequences

Published online by Cambridge University Press:  12 March 2014

Joan Rand Moschovakis
Joan Rand Moschovakis*
Affiliation:
Department of Mathematics, Occidental College, Los Angeles, California 90041-3392
Get access Rights & Permissions [Opens in a new window]

Abstract

In the author's Relative lawlessness in intuitionistic analysis [this Journal, vol. 52 (1987), pp. 68–88] and An intuitionistic theory of lawlike, choice and lawless sequences [Logic Colloquium ’90, Springer-Verlag, Berlin, 1993, pp. 191–209] a notion of lawlessness relative to a countable information base was developed for classical and intuitionistic analysis. Here we simplify the predictability property characterizing relatively lawless sequences and derive it from the new axiom of closed data (classically equivalent to open data) together with a natural principle of invariance under finite translation. We characterize relative lawlessness in terms of a notion of forcing. Finally, we study relative lawlessness on an arbitrary fan and show that the collection of lawless binary sequences (which is comeager in the sense of Baire) has probability measure zero. The reasoning is predominantly constructive.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 3 , September 1994 , pp. 813 - 829
Copyright
Copyright © Association for Symbolic Logic 1994

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.)

References

REFERENCES

[1]Kleene, S. C. [1952], Introduction to metamathematics, North-Holland, Amsterdam.Google Scholar
[2]Kleene, S. C. and Vesley, R. E. [1965], The foundations of intuitionistic mathematics, especially in relation to recursive functions, North-Holland, Amsterdam.Google Scholar
[3]Kreisel, G. [1968], Lawless sequences of natural numbers, Compositio Mathematica, vol. 20. pp. 222–248.Google Scholar
[4]Kreisel, G. and Troelstra, A. S. [1970], Formal systems for some branches of intuitionistic analysis, Annals of Mathematical Logic, vol. 1, pp. 229–387.CrossRefGoogle Scholar
[5]Levy, A. [1970], Definablity in axiomatic set theory, Mathematical logic and foundations of set theory (Proceedings, Jerusalem 1968), North-Holland, Amsterdam, pp. 129–145.Google Scholar
[6]Moschovakis, J. R. [1987], Relative lawlessness in intuitionistic analysis, this Journal, vol. 52. pp. 68–88.Google Scholar
[7]Moschovakis, J. R. [1993], An intuitionistic theory of lawlike, choice and lawless sequences, Logic Colloquium ’90, Lecture Notes in Logic 2 (Oikkonen, J. and Väänänen, J., editors), Springer-Verlag, pp. 191–209.Google Scholar
[8]Myhill, J. [1967], Notes towards an axiomatization of intuitionistic analysis, Logique et Analyse, vol. 9. pp. 280–297.Google Scholar
[9]Troelstra, A. S. [1977], Choice sequences, a chapter of intuitionistic mathematics, Clarendon Press, Oxford.Google Scholar
[10]Troelstra, A. S. and van Dalen, D. [1988], Constructivism in mathematics: an introduction, 1, 2, North-Holland, Amsterdam.Google Scholar
[11]van Dalen, D. [1978], An interpretation of intuitionistic analysis, Annals of Mathematical Logic, vol. 13. pp. 1–43.CrossRefGoogle Scholar