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?
The reference is to a result of Harrington and Shelah, who showed that a Borel linear order does not contain an $\omega_1$-chain. Since every uncountable wellordering contains an initial copy of $\omega_1$, it follows that a Borel linear order cannot contain an uncountable increasing wellordered chain. By reversing the linear order and applying the theorem again, we see that there are no uncountable decreasing wellordered chains either.
I conclude from this that the author probably meant for "chain" to be "wellordered chain" (which also explains the otherwise unusual phrase "increasing or decreasing").