A universal coanalytic linear ordering. (English) Zbl 0980.03052
The author constructs a linear ordering \((P,{<_P})\) such that \(P\) is a \(\Pi^1_1\) set of reals and \(<_P\) is a \(\Pi^1_1\) relation, and for any linear ordering \((A,{<_A})\) where \(A\) is a set of reals and \(<_A\) is a \(\boldsymbol\Pi^1_1\) relation, \((A,{<_A})\) is order embeddable into \((P,{<_P})\). In the proof he uses results on Borel embeddings proved by L. Harrington and S. Shelah [“Counting equivalence classes for co-\(\kappa\)-Souslin relations”, D. van Dalen et al. (eds.), Logic Colloquium 1980. Amsterdam etc.: North-Holland, Stud. Logic Found. Math. 108, 147-152 (1982; Zbl 0513.03024)] and by L. Harrington, D. Marker and S. Shelah [“Borel orderings”, Trans. Am. Math. Soc. 310, No. 1, 293-302 (1988; Zbl 0707.03042)].
Reviewer: Miroslav Repický (Košice)
References:
| [1] | Leo Harrington and Saharon Shelah, Counting equivalence classes for co-\?-Souslin equivalence relations, Logic Colloquium ’80 (Prague, 1980) Stud. Logic Foundations Math., vol. 108, North-Holland, Amsterdam-New York, 1982, pp. 147 – 152. · Zbl 0513.03024 |
| [2] | Leo Harrington, David Marker, and Saharon Shelah, Borel orderings, Trans. Amer. Math. Soc. 310 (1988), no. 1, 293 – 302. · Zbl 0707.03042 |
| [3] | A. S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1994. · Zbl 0805.54035 |
| [4] | Yiannis N. Moschovakis, Descriptive set theory, Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland Publishing Co., Amsterdam-New York, 1980. · Zbl 0433.03025 |
| [5] | Joseph G. Rosenstein, Linear orderings, Pure and Applied Mathematics, vol. 98, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. · Zbl 0488.04002 |
| [6] | S. M. Srivastava, A course on Borel sets, Graduate Texts in Mathematics, vol. 180, Springer-Verlag, New York, 1998. · Zbl 0903.28001 |
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