A universal coanalytic linear ordering (Q2750870)

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 \textit{L. Harrington} and \textit{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 \textit{L. Harrington, D. Marker} and \textit{S. Shelah} [``Borel orderings'', Trans. Am. Math. Soc. 310, No. 1, 293-302 (1988; Zbl 0707.03042)].