Explicit uses of alephs above 'small ones'

Question

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?

Answer 1

David:

1. As for the "736" in Shelah's paper, this is just his way of saying "for every $n>1$, there is a group as in the theorem which is free iff the continuum is below $\aleph_n$", there is nothing privileged or special about $\aleph_{736}$ in this result, other than it is ``random''.

2. Of course, what should be by far the most famous occurrence of a "large" $\aleph_n$ is $\aleph_4$, in a result due to Shelah, which is a cornerstone of modern cardinal arithmetic. The result is usually stated about exponentiation, although it is really about the pcf structure. For example:

If $2^{\aleph_0}<\aleph_\omega$, then $\aleph_\omega^{\aleph_0}<\aleph_{\aleph_4}$.

This is quite strange, really. We do not know at the moment that $\aleph_4$ is needed. The best results (due, once again, to Shelah) show that it cannot be anything below $\aleph_1$. Some people suspect $\aleph_2$ should be the right number, but at the moment $\aleph_4$ is what we get. And we do not understand it well. When Shelah found the result, he told Leo Harrington about it, and Leo asked, quite naturally "Why the HELL is 4?" So, of course, when Shelah wrote his "Cardinal Arithmetic" book, that was the title of the relevant section.

3. Though they are not strictly $\aleph_n$s, you get an interesting collection of possible cardinalities as the uncountable spectrum of a countable theory. Given a consistent complete first-order theory $T$ in a countable language which does not admit finite models, let $I(T,\kappa)$ denote the number of isomorphism classes of models of $T$ of size $\kappa$.

The possible values of $I(T,\aleph_0)$ are not known (Vaught's conjecture). But for $\kappa$ uncountable, the possibilities for $I(T,\kappa)$ are known. This is essentially due to Shelah, but the last pieces of the result were obtained in Bradd Hart, Ehud Hrushovski, Michael C. Laskowski, "The uncountable spectra of countable theories", Ann. of Math. (2) 152 (2000), no. 1, 207--257: