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IKP and friends

Published online by Cambridge University Press:  12 March 2014

Robert S. Lubarsky
Robert S. Lubarsky*
Affiliation:
1755 Ne 18TH St. Ft. Lauderdale, Fl 33305, USA, E-mail: [email protected]@att.net
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Extract

There has been increasing interest in intuitionistic methods over the years. Still, there has been relatively little work on intuitionistic set theory, and most of that has been on intuitionistic ZF. This investigation is about intuitionistic admissibility and theories of similar strength.

There are several more particular goals for this paper. One is just to get some more Kripke models of various set theories out there. Those papers that have dealt with IZF usually were more proof-theoretic in nature, and did not provide models. Furthermore, the inspirations for many of the constructions here are classical forcing arguments. Although the correspondence between the forcing and the Kripke constructions is not made tight, the relationship between these two methods is of interest (see [6] for instance) and some examples, even if only suggestive, should help us better understand the relationship between forcing and Kripke constructions. Along different lines, the subject of least and greatest fixed points of inductive definitions, while of interest to computer scientists, has yet to be studied constructively, and probably holds some surprises. Admissibility is of course the proper set-theoretic context for this study. Finally, while most of the classical material referred to here has long been standard, some of it has not been well codified and may even be unknown, so along the way we'll even fill in a gap in the classical literature.

The next section develops the basics of IKP, including some remarks on fixed points of inductive definitions.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 4 , December 2002 , pp. 1295 - 1322
Copyright
Copyright © Association for Symbolic Logic 2002

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References

REFERENCES

[1]Barwise, Jon, Admissible sets and structures, Springer-Verlag, 1975.CrossRefGoogle Scholar
[2]Beeson, Michael J., Foundations of constructive mathematics, Springer-Verlag, 1985.CrossRefGoogle Scholar
[3]Friedman, Harvey and Scedrov, Andrej, The lack of definable witnesses and provably recursive functions in intuitionistic set theories, Advances in Mathematics, (1985), pp. 1–13.Google Scholar
[4]Jech, Thomas, Set theory, Academic Press, 1978.Google Scholar
[5]Jech, Thomas, Multiple forcing, Cambridge University Press, 1986.Google Scholar
[6]Lipton, Jim, Realizability, set theory, and term extraction, The Curry-Howard isomorphism (de Groote, Ph., editor), Cahiers du Centre de logique (Université catholique de Louvain), vol. 8, Academia, 1995, pp. 257–364.Google Scholar
[7]Lubarsky, Robert, Sacks forcing sometimes needs help to produce a minimal upper bound, this Journal, (1989), pp. 490–498.Google Scholar
[8]Lubarsky, Robert, Intuitionistic L, Logical methods in computer science: The Nerode Conference (Crossley, et al., editor), Birkhauser, 1993, pp. 555–571.Google Scholar
[9]Powell, William, Extending Gödel's negative interpretation to ZF, this Journal, (1975), pp. 221–229.Google Scholar