In the vast majority of papers forcing is always developed over ZFC.
Not surprisingly too, since infintary combinatorial principles are often used to prove results based on properties such as chain conditions, closure, and so on.
I am looking for a good start on forcing over models of ZF. I have before me two papers which I have yet to read thoroughly, however may not be as useful for this purpose as I am hoping.
- Grigorieff, S. Intermediate Submodels and Generic Extensions in Set Theory. The Annals of Mathematics, Second Series, Vol. 101, No. 3 (May, 1975), pp. 447-490
- Monro, G. P. On Generic Extensions Without the Axiom of Choice. The Journal of Symbolic Logic, Vol. 48, No. 1 (Mar., 1983), pp. 39-52
While I do intend to read them either way, it seems that neither develops the theory of forcing in the absolute absence of choice. I am currently looking for references which deal with such situation, or with the relation between forcing theorems proved in ZFC and the amount of choice needed for them to hold.
Edit: I probably should have mentioned that I am quite familiar with permutation models of ZFA+embedding theorems and transfer theorems (Jech-Sochor, Pincus' theorem) as well with symmetric extensions.
I am not looking for ways to develop forcing extensions of ZF without the axiom of choice; rather I am looking for theorems such as c.c.c forcing does not collapse cardinals and similar theorems extended to the choiceless contexts if possible, or the strength of choice needed for these theorems to hold.
Consider two examples:
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Suppose a model of ZF in which the axiom of choice does not hold. Can we, by set forcing add the axiom of choice? If not, can it be done using a machinery similar to a symmetric extension? If we can in fact find such extension, does that mean the model without choice is a symmetric extension between two larger models?
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Suppose A is an infinite Dedekind-finite set, what can we say on a forcing poset based on A (either domain of functions are partial to A or the range is in A)? Can we "collapse" amorphous sets onto ordinals? Can we collapse one amorphous set onto another? And so on.
Best Answer
Friedman's book "Fine structure and class forcing" develops forcing over ZF, rather than ZFC, in chapter 2. Although chapter 1 is about fine structure, it is not used in chapter 2. Although the rest of his book is well above my level, I find Friedman's exposition of forcing quite helpful.