The motivation for this question is that I am working through an exercise to force the GCH (generalized continuum hypothesis) over a model of ZFC and obtain a model of ZFC where GCH holds. 


The forcing is an ORD length iteration of $Add(\gamma^+, 1)$ at every cardinal $\gamma$ ($\gamma$ is a cardinal in $V^{P\_{\gamma}})$, with Easton support. I'll just say here what Easton support is:


1) at regular $\gamma$, $P\_{\gamma}$ is the direct limit of $P\_{\alpha}$, $\alpha < \gamma$, meaning that for all $p \in P\_{\gamma}$ there is a $\beta < \gamma$ such that $p\restriction \beta \in P\_{\beta}$ and for all $\xi \ge \beta$, $p(\xi) = 1$


2) otherwise, $P\_{\gamma}$ is an inverse limit, meaning for all $p \in P\_{\gamma}$, the restriction of $p$ to any smaller cardinal is in that stage of the forcing, and there is no imposition that the coordinates of $p$ be trivial past some smaller stage. 


Note that at successor cardinals $\gamma$, $P\_{\gamma}$ is automatically a direct limit if we force at cardinal stages. 


My question is about forcing at singular cardinal stages. These mathoverflow questions, answers, and the surrounding comments ([what goes wrong](https://mathoverflow.net/questions/112736/what-goes-wrong-in-easton-forcing-if-we-dont-just-use-regular-cardinals), [definition of easton product](https://mathoverflow.net/questions/116564/ill-admit-it-i-dont-understand-the-definition-of-the-easton-product)), were very helpful to me, but I still have a confusion. 


Let me first explain what I do understand regarding forcing the GCH at singular cardinals and then my question.


Since $\gamma^+$ is regular for all $\gamma$, the important arguments to show that when we factor $P \cong P^{\le \gamma}\*P^{> \gamma}$, the first factor satisfies the $\gamma^+$-chain condition and the second is $<\gamma^+$-closed, all go through, whether $\gamma$ is regular or singlular. 


However, part of the above claim rests on a $\Delta$-system argument to show that $P^{\le \gamma}$ satisfies the $\gamma^+$-chain condition. And it is here where we use that any condition $p \in P^{\le \gamma}$ has a small domain compared to $\gamma^+$, and so a $\gamma^+$ sized set of domains of conditions in $P^{\le \gamma}$ forms a $\Delta$-system. 


What can I say in the case of singular cardinals? How can the $\gamma^+$ sized set of domains of conditions in $P^{\le \gamma}$ form a $\Delta$-system if we used an inverse limit to support the iteration at that stage? Why not impose a direct limit at every stage of forcing?



The canonical forcing of the GCH and direct limits.  
Let me take the case of $\aleph\_\omega$ as a specific example. Similar considerations apply at other singulars.


If you want $\sigma$-closure above $\omega\_1$, you need the full support at $\aleph\_\omega$; bounded support (i.e., direct limit) will not be closed enough. 


Also: You need not worry about collapsing $\aleph\_\omega$ (or changing its cofinality). If the elements of an unbounded set of cardinals below $\aleph\_\omega$ stay cardinals, then $\aleph\_\omega$ will stay a cardinal as well (and if a cobounded set of cardinals below $\aleph\_\omega$ is collapsed to $\lambda$, then $\aleph\_\omega$ will certainly not become $\lambda^+$ in the extension, and hence collapse as well. If $\kappa$ is a singular cardinal of cofinality $\lambda < \kappa$, then the cofinality of $\kappa$ will only change if the cofinality of $\lambda$ changes. This will be true no matter what support you take. 


