Two results on borel orders

Journal of Symbolic Logic 54 (3):865-874 (1989)

We prove two results about the embeddability relation between Borel linear orders: For $\eta$ a countable ordinal, let $2^\eta$ (resp. $2^{<\eta}$) be the set of sequences of zeros and ones of length $\eta$ (resp. $<\eta$), equipped with the lexicographic ordering. Given a Borel linear order $X$ and a countable ordinal $\xi$, we prove the following two facts. (a) Either $X$ can be embedded (in a $\triangle^1_1(X,\xi)$ way) in $2^{\omega\xi}$, or $2^{\omega\xi + 1}$ continuously embeds in $X$. (b) Either $X$ can embedded (in a $\triangle^1_1(X,\xi)$ way) in $2^{\omega\xi}$, or $2^{\omega\xi}$ continuously embeds in $X$. These results extend previous work of Harrington, Shelah and Marker
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DOI 10.2307/2274748
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Borel Quasi-Orderings in Subsystems of Second-Order Arithmetic.Alberto Marcone - 1991 - Annals of Pure and Applied Logic 54 (3):265-291.
Linearization of Definable Order Relations.Vladimir Kanovei - 2000 - Annals of Pure and Applied Logic 102 (1-2):69-100.

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