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A result of relative consistency about the predicate WO(δ, x)

Published online by Cambridge University Press:  12 March 2014

René David
René David*
Affiliation:
Université Toulouse le Mirail, Toulouse, France
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Extract

In [K] Keisler introduces a set theoretical relation WO(δ, x), where δ is an ordinal. This relation is characterized in ZF set theory by the following properties:

(1) WO(0, x) if and only if there is a wellordering on x.

(2) For δ > 0, WO(δ, x) if and only if there is a function ƒ with domain an ordinal λ such that:

We denote WO∞ the collection denned by: WO∞(x) = ∃δ WO(δ, x).

In [K] Keisler shows that countable transitive models of ZF + ∃x ¬ WO∞(x) have transitive uncountable elementary extensions with the same ordinals. Moreover for transitive models, satisfying ∃x ¬ WO∞(x) is also a necessary condition for the existence of an elementary extension with the same ordinals. (See [K bis] and also [K-M] where the connection with forcing is analysed.)

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 45 , Issue 3 , September 1980 , pp. 483 - 492
Copyright
Copyright © Association for Symbolic Logic 1980

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References

REFERENCES

[D]David, R., Résultats de consistence relative concernant les éssencés WO(α, x), Thèse de 3e cycle, Université de Paris VII, 1973.Google Scholar
[D bis]David, R., Note aux Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, vol. 80 (1975), pp. 981–983.Google Scholar
[Gr]Grigorieff, S., Intermediate submodels and generic extensions in set theory, Annals of Mathematics, vol. 101 (1975), pp. 447–490.CrossRefGoogle Scholar
[K]Keisler, H. J., Logic with the quantifier “there exist uncountably many”, Annals of Mathematical Logic, vol. 1 (1970), pp. 1–93.CrossRefGoogle Scholar
[K bis]Keisler, H. J., Forcing and omitting types, Motley Symposium.Google Scholar
[Kr]Krivine, J. L., Cours de théorie des ensembles, notions de forcing, Université Paris VII, 1970.Google Scholar
[K-M]Krivine, J. L. and McAloon, K., Forcing and generalized quantifiers, Annals of Mathematical Logic, vol. 5 (1973), pp. 199–255.CrossRefGoogle Scholar
[S]Shoenfield, J. R., Unramified forcing, Axiomatic set theory, Proceedings of Symposia in Pure Mathematics, vol. 13, part I (Scott, D., Editor), American Mathematical Society, Providence, Rhode Island, 1971, pp. 357–382.CrossRefGoogle Scholar