Why here (Almost Free Modules: Set-theoretic Methods by P.C. Eklof, A.H. Mekler), on the page 181, in the 3rd line there is
$$2^{\aleph_\omega}<\aleph_{\omega_2}$$? How the index $2$ in the r.h.s. was created?
The notation "pcf" stands for possible cofinality. Shelahintroduced a powerful tool into the study of cardinal arithmetic when he considered (for an infinite set $A$ of regular cardinals) the set $\operatorname{pcf}(A)$, defined to be the set of all cardinals $\lambda$ such that for some ultrafilter $U$ on $A$, $\lambda$ is the cofinality of the ultraproduct of ordered sets $\prod_{\kappa\in A} \kappa/U$. Among the striking results he proved (in ZFC) are: if $2^{\aleph_n}<\aleph_\omega$ for all $n\in\omega$ then $2^{\aleph_\omega}<\min\{\aleph_{(2^{\aleph_0})^+},\aleph_{\omega_4}\}$; in particular, if GCH holds below $\aleph_\omega$, then $2^{\aleph_\omega}<\aleph_{\omega_2}$. (See Shelah 1992, Shelah 1994 or Jech 1995 or Burke-Magidor 1990.)
Best Answer
The $\aleph_{\omega_2}$ arises because under the assumption of GCH below $\aleph_\omega$, we have:
$$(2^{\aleph_0})^+ = \aleph_1^+ = \omega_2$$
so that the minimum $\min\{\aleph_{(2^{\aleph_0})^+},\aleph_{\omega_4}\}$ is actually not $\aleph_{\omega_4}$, but the other argument, which evaluates to $\aleph_{\omega_2}$.