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Trees and -subsets of ω1ω1

Published online by Cambridge University Press:  12 March 2014

Alan Mekler  and
Jouko Väänänen
Alan Mekler
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, Canada
Jouko Väänänen
Affiliation:
Department of Mathematics, University of Helsinki, Helsinki, Finland
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Abstract

We study descriptive set theory in the space by letting trees with no uncountable branches play a similar role as countable ordinals in traditional descriptive set theory. By using such trees, we get, for example, a covering property for the class of -sets of .

We call a family of trees universal for a class of trees if ⊆ and every tree in can be order-preservingly mapped into a tree in . It is well known that the class of countable trees with no infinite branches has a universal family of size ℵ1. We shall study the smallest cardinality of a universal family for the class of trees of cardinality ≤ ℵ1 with no uncountable branches. We prove that this cardinality can be 1 (under ¬CH) and any regular cardinal κ which satisfies (under CH). This bears immediately on the covering property of the -subsets of the space .

We also study the possible cardinalities of definable subsets of . We show that the statement that every definable subset of has cardinality <ωn or cardinality is equiconsistent with ZFC (if n ≥ 3) and with ZFC plus an inaccessible (if n = 2).

Finally, we define an analogue of the notion of a Borel set for the space and prove a Souslin-Kleene type theorem for this notion.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 3 , September 1993 , pp. 1052 - 1070
Copyright
Copyright © Association for Symbolic Logic 1993

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References

REFERENCES

[1]Baumgartner, J., Harrington, L., and Kleinberg, E., Adding a closed unbounded set, this Journal, vol. 41 (1976), pp. 481–482.Google Scholar
[2]Blackwell, D., Borel sets via games, The Annals of Probability, vol. 9 (1981), pp. 321–322.CrossRefGoogle Scholar
[3]Halko, A., mimeographed notes, University of Helsinki, 1988.Google Scholar
[4]Hyttinen, T. and Tuuri, H., Constructing strongly equivalent nonisomorphic models for unstable theories, Annals of Pure and Applied Logic, vol. 52 (1991), pp. 203–248.CrossRefGoogle Scholar
[5]Hyttinen, T. and Väänänen, J., On Scott- and Karp-trees of uncountable models, this Journal, vol. 55 (1990), pp. 897–908.Google Scholar
[6]Jech, T., Set theory, Academic Press, San Diego, California, 1978.Google Scholar
[7]Juhász, I., Cardinal functions II, Handbook of set-theoretic topology (Kunen, K. and Vaughan, J. E., editors), North-Holland, Amsterdam, 1984, pp. 63–109.CrossRefGoogle Scholar
[8]Levy, A., Definability in axiomatic set theory II, Mathematical logic and foundations of set theory (Bar-Hillel, Y., editor), North-Holland, Amsterdam, 1970.Google Scholar
[9]Mekler, A. and Shelah, S., The Canary tree, Canadian Mathematical Bulletin (to appear).Google Scholar
[10]Nadel, M. and Stavi, J., L∞λ equivalence, isomorphism, and potential isomorphism, Transactions of American Mathematical Society, vol. 236 (1978), pp. 51–74.Google Scholar
[11]Shelah, S., Classification theory, revised edition, North-Holland, Amsterdam, 1990.Google Scholar
[12]Shelah, S., A weak generalization of MA to higher cardinals, Israel Journal of Mathematics, vol. 30 (1978), pp. 297–306.CrossRefGoogle Scholar
[13]Shelah, S., Tuuri, H., and Väänänen, J., On the number of automorphisms of uncountable models, this Journal (to appear).Google Scholar
[14]Solovay, R., A model of set-theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, vol. 92 (1970), pp. 1–56.CrossRefGoogle Scholar
[15]Todorčević, S., Stationary sets, trees, and continuums, Publications de l’Institut Mathématique, Nouvelle Série, vol. 29 (43), Institut Mathématique, Belgrade, 1981, pp. 249–262.Google Scholar
[16]Tuuri, H., Relative separation theorems for Lκ+κ, Notre Dame Journal of Formal Logic, vol. 33 (1992), pp. 383–401.CrossRefGoogle Scholar
[17]Väänänen, J., A Cantor-Bendixson theorem for the space , Fundamenta Mathematicae, vol. 137 (1991), pp. 187–199.CrossRefGoogle Scholar
[18]Väänänen, J., Games and trees in infinitary logic: A survey, Quantifiers (Krynicki, M., Mostowski, M., and Szczerba, L., editors), Kluwer Academic Publishers, Dordrecht (to appear).Google Scholar