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I have a certain set-theoretic axiom (WISC) which follows from Choice (this is a nuking a fly BTW), but which I suspect is independent of ZF. To show this I need to show that a certain set does not exist. I have no experience with forcing, but I'm willing to learn. I have a plan of attack, but would like to know if it is what I should be doing, or whether some other approach is needed.

Here are the details, and apologies for the vagueness of the questions.

WISC - for every set $S$, there is a set $C_S\subset \{ p:A\to S |\ p\ \mbox{is onto}\}$ such that $$ \forall\ \mbox{surjections }f:B\to S\ \exists (p:A\to S)\in C_S,\ s:A\to B \mbox{ such that } f\circ s = p $$

Clearly, to show this is independent of ZF (or other set-theoretic foundations), we just need a set $S$ such that $C_S$ doesn't exist. We might as well try as a first attempt $S = \mathbb{N}$. If we had a countable model of ZF, if we could force that $C_\mathbb{N}$ was bounded below in size by some uncountable set, and still have a countable model, then I believe we would be done.

Question 1: Is this approach likely to work, or is it too naive?

Notice that one implication of $\lnot$WISC is that there are a proper class of non-split surjections, but this is not sufficient to conclude $\lnot$WISC. Or in the above approach, we would have an uncountable collection of non-split surjections.

Failing the approach above, and in the absence of other options for $\mathbb{N}$, one could step it up a notch and try $C=\mathbb{R}$, but would obviously need a different strategy. Given that I'm familiar in principle with forcing to the level covered in MacLane-Moerdijk (a topos theoretic approach, but not very in-depth),

Question 2: What are good references to get a feel for what one can force along these lines (violations of Choice)?

I'm thinking Blass' paper on SVC might be a good start, but I may need to read something else before that.

I'm trying to show that a particular class is not a set, so perhaps I need to work with NBG + $\lnot$C, and use conservativity over ZF, but given this question: Forcing over models without the axiom of choice, one might suppose that forcing over foundations without Choice other than ZF is even rarer.

The alternative is to consider a quantitative version of WISC, where the set $C_S$ must have cardinality less than a given (regular) cardinal $\kappa$. In the case that $\kappa$ is inaccessible, we can take a Grothendieck universe and so recover the original version of WISC. The presentation axiom/COSHEP is the case when this $\kappa = 2$. Conversely, to show independence, we could just force that $|C_S| \ge \kappa$ for inaccessible $\kappa$, and so show that there is no small set $C_S$ (relative to the universe given by $\kappa$).

Question 3: is this a better way to go about it?

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The problem with forcing is that it preserves AC, so you can't force ¬WISC starting with a model of AC. You can make AC fail by passing to a symmetric submodel, but then the technologically simpler permutation models would be the first thing to try. Unfortunately most of these will fail too since set-forcing extensions and set-based permutation models will preserve SVC. So one needs use a symmetric submodel of a class-forcing extension in order to arrive at a model of ¬WISC. There are very few of those around... –  François G. Dorais Oct 31 '11 at 0:20
    
@François Well, I mean to start from ZF + $\lnot$C, but I infer from your comment that something in my question means I am secretly accepting Choice somewhere. Is it in the countable model? –  David Roberts Oct 31 '11 at 0:35
    
By the way, from the categorical point of view, it is harder distinguish between pure forcing extensions, symmetric models, and in-between options. See Blass and Scedrov, Freyd's models for the independence of the axiom of choice (Mem. Amer. Math. Soc. 79, 1989). –  François G. Dorais Oct 31 '11 at 0:37
    
@Francois: Do permutation models over a class of atoms act like symmetric submodels of class-forcing extensions? –  Asaf Karagila Oct 31 '11 at 0:42
    
@David: No, it's that it's customary to start with a model of AC (e.g. L) and then take a symmetric model to do something different. There are very few handy models of ¬AC to start with (and all those I can think of satisfy SVC). Theoretically, you could start with a model where AC fails very badly, but then there is a good chance WISC already fails there without doing anything. –  François G. Dorais Oct 31 '11 at 0:44

2 Answers 2

This is just to answer the reference request part of the question, I'm making this community wiki so that you can add your favorite references.

  • The best place to learn about permutation models and symmetric models is Jech's The Axiom of Choice. It's not recent, but all the classic models are there and it's packed with tons of useful information.

  • For the bridge between category-theoretic and set-theoretic methods, the main resource should be Blass and Scedrov Freyd's models for the independence of the axiom of choice (Mem. Amer. Math. Soc. 79, 1989).

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Jech's book has a current reprint, so it can be bought.

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Best 12 USD I spent in the passing year! (shipping not included, sadly...) –  Asaf Karagila Oct 31 '11 at 13:40

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