## Journal of Symbolic Logic

### Two Results on Borel Orders

Alain Louveau

#### Abstract

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.

#### Article information

Source
J. Symbolic Logic, Volume 54, Issue 3 (1989), 865-874.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183743023

Mathematical Reviews number (MathSciNet)
MR1011175

Zentralblatt MATH identifier
0687.03028

JSTOR