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Are there any books about the axiom of dependent choice? There are books about the axiom of choice, e.g. Herrlich or Jech. But I can't seem to find any on the axiom of dependent choices.

As far as I understand DC is at least as interesting as AC. What am I missing? Thanks for your help.

So the question is: are there any books and if not, why not.

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I would've liked to tag this axiom-of-dependency but there is no such tag. –  Rudy the Reindeer Dec 16 '12 at 18:43
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Do you mean the axiom of dependent choice? –  Brian M. Scott Dec 16 '12 at 18:45
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Herrlich’s book deals with many of the main weakenings of AC including DC. You might want to look at Paul Howard & Jean Rubin, Consequences of the Axiom of Choice. –  Brian M. Scott Dec 16 '12 at 18:51
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I hope you don't mind that I edited the question. –  Asaf Karagila Dec 16 '12 at 18:55
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@Brian: It also has a reasonable website, which is relatively more up-to-date than the book. –  Asaf Karagila Dec 16 '12 at 18:57

1 Answer 1

up vote 10 down vote accepted

I don't know any book that deals specifically with $\mathsf{DC}_\kappa$ principles.

I can tell you that Jech's book has a chapter devoted for this, but half the chapter is built around independence proofs, and not implications. You cannot really find many "applications" of these axioms in Jech's book.

In Herrlich's book you can find more applications of Countable Choice rather than $\mathsf{DC}$, but those are implied by $\mathsf{DC}$ so there is some merit to those in the aspect of $\mathsf{DC}$.

What you may want to do, if you are inclined to learn about $\mathsf{DC}_\kappa$ principles is to check the Consequences of the Axiom of Choice site$^1$ (or book) and find all sort of papers which describe and detail implications and equivalents of these axioms.

But then you will have to to hunt down specific papers, many of them are rather easy to find, but some are not as easy to find and not at all well-known. Furthermore, many of them deal with consistency results (e.g. I have submitted such paper just today). All of these are interesting, but they often provide negative results or partially-positive results, that is what we can't do with $\mathsf{DC}_\kappa$, rather than what we can.

I do suggest Jech's book if only for the first part of the chapter where he proves various implications from $\mathsf{DC}_\kappa$. However you may also be interested in finding those in other papers or books instead.

Edit: I wanted to add a personal remark on the point of "interest". I am very interested in $\mathsf{DC}_\kappa$ personally because I often see it as the "true form of limited choice". However one can classify choice principles to several families, and the one which is actually interesting is not that of $\mathsf{DC}$ but rather that of the Ultrafilter Lemma/Boolean Prime Ideal Theorem. Those principles often make mathematics very useful (filters, compactness, Tychonoff for Hausdorff spaces, Hahn-Banach, etc.), which says something about mathematics. It's not actually the amount of actual choice, but rather the ability to "glue things together" which generates interest.

I will also add on that and say that there is a third major family of choice principles which I classify as those not implied by $\mathsf{DC}$ nor by BPIT. Those are often cardinal-forms (e.g. cardinal representatives; antichains of cardinals; and so on) which in my opinion is actually the most bizarre and pathologically behaving family I have seen since the last time I watched The Addams Family.


Footnotes:

  1. $\mathsf{DC}_{\aleph_\alpha}$ appears as Form 87($\alpha$). Also see Form 43 for $\mathsf{DC}_{\aleph_0}$ and Form 44 for $\mathsf{DC}_{\aleph_1}$.
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I'm sorry -- would you mind editing in a link to "Consequences of the AC site"? I have never heard of it before. It sounds promising. –  Rudy the Reindeer Dec 16 '12 at 18:57
    
Aces! Thank you. –  Rudy the Reindeer Dec 16 '12 at 18:59
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@Matt: I was already editing when you left the comment; I just wanted to find the appropriate form number as well. :-) –  Asaf Karagila Dec 16 '12 at 18:59
    
And a silly-ish add on question: what about books about countable choice? –  Rudy the Reindeer Dec 16 '12 at 19:12
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@Matt: For general consistency and location within the hierarchy of choice principles the same chapter in Jech applies. However for applications you can look at Herrlich's book. He proves quite a few equivalents of countable choice in topology. –  Asaf Karagila Dec 16 '12 at 19:13

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