I read a proof that no model of a certain f.o. theory $T_1$ is definable in $(Q,<)$ and I have a problem with understanding the very end of Lemma 3.2 here.
"The distance from $\alpha_i$ to $\alpha_k$ is $k-i$. This is a contradiction"
My question is:A contradiction with what?
The answer will be short,and the setting is on a single page in the link above,so I should not restate it here.
$(\alpha_i, \alpha_j)$ is carried to $(\alpha_i, \alpha_k)$, where $i < j < k$. Apparently (I'd have to see more of the development to be sure), this mapping should preserve distance. But the first distance is $j - i$, while the second is $k - i$.