Complicated index models and Boolean algebras/ Shelah/ Unclear step in the proof

Question

Here on the page $10$ in the $5$th line (the proof of lemma $1.10$), Shelah defines $n_*$ as $\omega$: $$n_*=\omega,$$ and then he continues: be such that $n_*\geq\text{max}\{n(0),...,n(m-1)\}<\omega.$

Does it make sense?

Answer 1

The paper you are quoting does not say

$n_*\geq\{\text{max}(n(0),...,n(m-1)\}<\omega.$

it says

$n_*\geq\text{max}\{n(0),...,n(m-1)\}<\omega.$

and certainly if the middle $<\omega$, it is going to be less than $n_\ast=\omega$.

I can't speak for the rest of the paper, but I don't see anything particularly odd about the line.