Let $T$ be a normal Suslin tree, so among other things for each $x \in T$, there is some $y > x$ at each higher level less than $\omega_1$, and every branch in $T$ is at most countable. My question is, how can both of these be true? Can we not construct a branch of length $\omega_1$ by induction starting at a root and choosing an element in the level above?
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