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A note on iterative arguments in a topos

Published online by Cambridge University Press:  24 October 2008

J. J. C. Vermeulen
J. J. C. Vermeulen
Affiliation:
Department of Mathematics, University of Cape Town, Rondebosch 7700, South Africa
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Extract

Intuitively, transfinite iteration is a repetitive process, which eventually reaches completion, but might need to progress through an infinite chain of steps before finally doing so. But whereas such a chain is always readily at hand in classical set theory in the form of ordinals, iterative arguments involving sets (i.e. objects) in a general topos have to depend on some intrinsic or naturally available inductive structure, say algebraic, which might not be associated with a (well-ordered) chain.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 1 , January 1992 , pp. 57 - 62
Copyright
Copyright © Cambridge Philosophical Society 1992

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References

REFERENCES

1Aczel, P.. An introduction to inductive definitions. In Handbook of Mathematical Logic, Studies in Logic and the Foundations of Math. no. 90 (North-Holland, 1977), p. 739.CrossRefGoogle Scholar
2Borceux, F. and Veit, B.. Continuous Grothendieck Topologies. Ann. Soc. Sci. Bruxelles Sr. I 100 (1986), 3142.Google Scholar
3Johnstone, P. T.. Tychonoff's theorem without the axiom of choice. Fund. Math. 113 (1981), 2135.CrossRefGoogle Scholar
4Joyal, A. and Tierney, M.. An Extension of the Galois theory of Grothendieck. Memoirs Amer. Math. Soc. no. 309 (American Mathematical Society, 1984).CrossRefGoogle Scholar
5Tarski, A.. A lattice-theoretic fixed-point theorem and its applications. Pacific J. Math. 5 (1955), 574.CrossRefGoogle Scholar
6Vermeulen, J. J. C.. Weak compactness in constructive space. Math. Proc. Cambridge Philos. Soc. 110 (1991), 6374.Google Scholar