I am trying to make sense of what this theorem from C.I. Steinhorn, Borel Structures and Measure and Category Logics, says.
Theorem 1.3.3. A Borel linear order cannot have an uncountable increasing or decreasing chain.
I didn't find definitions in the paper, but I am quite sure that a pair $(A,\leq)$ is a Borel linear order iff $A$ is a Borel set in a standard Borel space $X$ and $\leq$ is Borel as a subset of $X^2$. I presume that an increasing chain is just a linear order with no greatest element and similarly for decreasing.
Here is the problem. Intuitively, I would guess that the lexicographic order on Baire space is a Borel linear order. But then Baire space itself should be an uncountable increasing chain under this order.
What am I misunderstanding?