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λ-scales, κ-Souslin sets and a new definition of analytic sets1

Published online by Cambridge University Press:  12 March 2014

Douglas R. Busch
Douglas R. Busch*
Affiliation:
School of Historical, Philosophical and Political Studies, Macquarie University, North Ryde, New South Wales. 2113, Australia
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Summary

The notions of sets of reals being κ-Souslin (κ a cardinal) and admitting a λ-scale (λ an ordinal) are due respectively to D. A. Martin and Y.N. Moschovakis. A set is ω-Souslin if and only if it is Σ1 1 (analytic). We show that a set is ω-Souslin if and only if it admits an (ω + l)-scale. Jointly with Martin and Solovay we show that if κ is uncountable and has cofinality ω, then being κ-Souslin is equivalent to admitting a κ-scale. Our results together with those of Kechris give a new simultaneous characterization of Σ1 1 and Δ1 1 (Borel) sets (a set is Σ1 1 if it admits an (ω + 1)-scale and Δ1 1 if it admits an ω-scale) and determine completely the relation between the κ-Souslin sets and the sets admitting λ-scales.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 2 , June 1976 , pp. 373 - 378
Copyright
Copyright © Association for Symbolic Logic 1976

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Footnotes

1

This research was undertaken at the Rockefeller University in New York under the guidance of Professor D. A. Martin, during the academic year 1971–72. It constitutes a portion of the author's Ph.D thesis. The author wishes to express his gratitude to Professor D. A. Martin and to Professor Hao Wang for their guidance and encouragement.

References

REFERENCES

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