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A boundedness principle for the Hjorth rank

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Abstract

Hjorth (Variations on Scott, 1998; The fine structure and Borel complexity of orbits, 2010) introduced a Scott analysis for general Polish group actions, and asked whether his notion of rank satisfies a boundedness principle similar to the one of Scott rank—namely, if the orbit equivalence relation is Borel, then Hjorth ranks are bounded. We answer Hjorth’s question positively. As a corollary we prove the following conjecture of Hjorth—for every limit ordinal \(\alpha \), the set of elements whose orbit is of complexity less than \(\alpha \) is a Borel set.

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Acknowledgements

This paper is based on the master thesis of the author [2], written under the supervision of Menachem Magidor. The author wishes to thank him for his patient guidance, insightful views, and outstanding intuition, all invaluable for this work. The author thanks the referee for careful reading of the manuscript and many insightful comments and suggestions.

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  1. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

    Ohad Drucker

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  1. Ohad Drucker
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Correspondence to Ohad Drucker.

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This paper is part of the author’s master thesis written in the Hebrew University of Jerusalem under the supervision of Prof. Menachem Magidor.

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Drucker, O. A boundedness principle for the Hjorth rank. Arch. Math. Logic 61, 223–232 (2022). https://doi.org/10.1007/s00153-021-00788-1

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  • DOI: https://doi.org/10.1007/s00153-021-00788-1

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