Shelah isolates the notion of "$\aleph\_1$-free iteration" in the first two sections of Chapter IX from *Proper and Improper Forcing*, and he proves there that properness is preserved by this sort of iteration. He mentions in his "On what I do not understand" paper that this was his third proof of the preservation of properness, and he goes on to pose a few questions about "nep forcings" (non-elementary proper) and preservation of properties in free limits.


My real questions are: "What are $\aleph\_1$-free iterations good for?" and "Has anyone other than Shelah worked with them?", but I'll settle for an answer to the question "Are there situations where the use of $\aleph\_1$-free iteration rather than countable support iteration has been critical?"




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Edit: I did have a chance to talk with Shelah about this. As far as he is aware, no one else has worked with the idea, and it is still unknown if free limits have "real applications" other than making some proofs of iteration theorems run a little smoother. 



On free limits of iterations
This is more of a comment than an answer, but as I couldn't post a comment, I'm writing here.


If I understood correctly the definitions in chapter IX from Proper and Improper Forcing, it seems that you can find some variants of free limits (not $\aleph\_1$-free, though) in the following work of Shelah:


In the paper "[On CON($\mathfrak{d}\_{\lambda}>cov\_{\lambda}(meagre)$)](http://arxiv.org/pdf/0904.0817.pdf)" (Sh:945) in page 29 there is a construction that resembles the free limit, which is used in order to establish a result that is somewhat analogous to a basic result of Judah and Shelah (JdSh:292) on FS iterations of Suslin forcing (namely the complete embeddability of iterations along a subset of the original ordinal).


It also seems that free-like constructions appear in Shelah's paper "Properness without elementaricity" (this is definition 5.7 in Sh630), but I don't know what is it good for, as I'm still reading this paper.


