In a paper placed on the arXiv today Shelah references theorem 0.9 from this paper (also Shelah) that uses $\aleph_{736}$ as an upper bound. This strikes me as analogous to Skewes' number. Are there any other examples where explicit mentions of 'large alephs' are used in proofs or theorems? Large in this instance is $\aleph_n$ for $n > 3$, say.

Edit: The first version of this question asks for $n$ a natural number, because clearly there are uses of $\aleph_\omega$ that are not all that uncommon, I imagine. But if there are uses of alephs - in isolation, not as part of a general transfinite induction scheme - that have $n$ an infinite ordinal like $\omega^2\cdot 5 + 45$ or some specific polynomial in $\omega$ which is 'not boring' (e.g. $\omega, \omega+1$), then I'd like to hear those as well.

Bonus points, even though this is a separate question: where does the 736 come from? Is it due to some sort of Goedelian numbering scheme that encodes the statement “G is a free abelian group”, which is part of the theorem?