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On relatively analytic and Borel subsets

Published online by Cambridge University Press:  12 March 2014

Arnold W. Miller
Arnold W. Miller*
Affiliation:
University of Wisconsin-Madison, Department of Mathematics, van Vleck Hall, 480 Lincoln Drive, Madison, Wisconsin 53706-1388, USA, E-mail: [email protected], URL: http://www.math.wisc.edu/~miller
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Abstract

Define to be the smallest cardinality of a function f: X→Y with I, X, Y, ⊆ 2ω such that there is no Borel function g ⊇ f. In this paper we prove that it is relatively consistent with ZFC to have b < where b is, as usual, smallest cardinality of an unbounded family in Ωω. This answers a question raised by Zapletal.

We also show that it is relatively consistent with ZFC that there exists X ⊆ 2ω such that the Borei order of X is bounded but there exists a relatively analytic subset of X which is not relatively coanalytic. This answers a question of Mauldin.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 1 , March 2005 , pp. 346 - 352
Copyright
Copyright © Association for Symbolic Logic 2005

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References

REFERENCES

[1]Bartoszyński, Tomek and Judah, Haim, Set theory. On the structure of the real line, A K Peters, Ltd., Wellesley, MA, 1995.Google Scholar
[2]Miller, Arnold W., On the length of Borel hierarchies, Annals of Mathematical Logic, vol. 16 (1979), no. 3, pp. 233–267.CrossRefGoogle Scholar
[3]Miller, Arnold W., Special subsets of the real line. Handbook of set-theoretic topology (Kunen, K. and , editors), North-Holland, Amsterdam, 1984, pp. 201–233.CrossRefGoogle Scholar
[4]Miller, Arnold W., Some interesting problems. Set theory of the reals (Judah, Haim, editor), Israel Mathematical Conference Proceedings, vol. 6, American Mathematical Society, 1993, continuously updated on my home page, pp. 645–654.Google Scholar
[5]Rothberger, Fritz, Sur les familles indénombrables de suites de nombres naturels et les problèmes concernant la propriété C, Proceedings of the Cambridge Philosophical Society, vol. 37 (1941), pp. 109–126, (in French).CrossRefGoogle Scholar
[6]Zapletal, Jindřich, Descriptive set theory and definable forcing. Memoirs of the American Mathematical Society, vol. 167, American Mathematical Society, 2004.Google Scholar