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In almost every graduate set theory text there are some parts about equivalences of $AC$, its consequences, some axioms like $AD$ which imply $\neg AC$, some well-known axiomatic systems which $AC$ fails to be true in them, some forcing construction like Solovay's model which force some mathematical property in $ZF$ via violating $AC$ , etc.

Let us define all information like what stated in above paragraph as general knowledge of any working set theorist about $AC$.

On the other hand there are some more advanced material about $AC$. Such material are just of interest of those who want to work specifically on different aspects of the axiom of choice or in the context of choice-less set theory.

Let us call all those set theorists who have no information more than what every working set theorist knows about $AC$, as dummies (with respect to choice-less set theory of course!)

My question simply is:

Question: What are nice references on choice-less set theory for dummies from introductory to advanced level? Which are more essential to become familiar with this field in the professional class?

Remark: I work on interactions of large cardinals and forcing and my main motivation for being interested in choice-less set theory is based on study of Reinhardt and Berkeley cardinals which are very large cardinal assumptions inconsistent with $AC$, with a (hopefully) rich theory within $ZF$.

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    $\begingroup$ I am unaware of any such thing. $\endgroup$ – Asaf Karagila Feb 9 '15 at 7:15
  • $\begingroup$ @AsafKaragila (+1) Hi! ... and possibly this means that I should delete this question because nobody else is aware of such a thing! Is Felgner's book good or it is old fashioned? $\endgroup$ – user180918 Feb 9 '15 at 7:22
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    $\begingroup$ Felgner's book is nice, but it's quite old. Jech's book is very nice but quite old. John Bell's book seems nice, but I have not read it in detail. Herrlich's book is nice, but doesn't cover set theoretic techniques, just consequences. I don't really know anything serious in mathematics which is "for dummies". $\endgroup$ – Asaf Karagila Feb 9 '15 at 7:25
  • $\begingroup$ @AsafKaragila Please see my definition of "dummies" it includes most of main stream set theorists! A question: minorities are usually well-organized and well-connected communities.A nice example in set theory, is the community of "foundation-less" set theorists, those who work on Quine's new foundation $NF$ or other non-well founded systems. One can find lists of non-well founded set theorists and their publications around the net. What about choice-less set theorists? Is there any list of those who work on this topic? I don't know to whom should I contact when I have a question on this topic! $\endgroup$ – user180918 Feb 9 '15 at 7:41
  • $\begingroup$ I read your definition; but I don't think that any one book suffices. Jech focuses too much effort on atoms and transfer theorems; and within the field of choiceless set theory, you can find different professionals with different approaches which will tell you that different things are the important basis (I'd tell you that symmetric extensions are important, someone else might tell you that they are less important because of atoms and the embedding theorems, others will tell you that $\mathsf{HOD}(X)$ and $L(X)$ are more important constructions here). $\endgroup$ – Asaf Karagila Feb 9 '15 at 7:44

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