It seems that every construction of a model in which the Axiom of Choice fails involves some kind of symmetry. Is there an example of a construction of a model where AC fails but no argument involving symmetry appears? Is there any result that connects the negation of choice (any kind of choice) to some kind of symmetry?
Does negation of Axiom of Choice imply symmetry?
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$\begingroup$
logic
set-theory
axiom-of-choice
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1I find "some kind of symmetry" a bit vague. – 2012-12-08
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0@Hagen: It's not vague at all. Permutation models; symmetric extensions; relative definability -- all those are essentially exploiting the existence of "somewhat indiscernable elements". – 2012-12-08
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0@AsafKaragila OK, then I may not have been acquainted enough with te subject (and I'm not surprised). Then again, if you donÄt consider "somewhat" a bit vague ... :) – 2012-12-08
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1@Hagen: This is a philosophical thing, really, and I came to understand it in full while working on my masters thesis during the summer. The original idea uses atoms and permutations of them, and this translated into forcing and permutations of the generic sets. The reason it works is because atoms are completely indiscernible from one another; and generic sets are indiscernible *enough* over the ground model (i.e. from the point of view of the ground models the generic sets have the same properties and cannot be discerned). – 2012-12-08
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1[cont] and so permutations of these indiscernible sets ensure that nothing which can separate them from one another stays in the model, and so the axiom of choice fails (because well-orders can discern them, obviously... :-)) – 2012-12-08
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0There must be a way to make this question precise. Perhaps something like, any (small) model of $\textrm{ZF}+\lnot\textrm{C}$ must have a non-trivial automorphism group... – 2012-12-08
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0@Zhen: But that's not really the question, is it now? – 2012-12-08
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0In my view, a positive answer to the OP's second question would look something like that. What would you count as a a positive answer? – 2012-12-08
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3@Zhen: Even when there is obvious symmetry in the construction of a model, it often disappears in the final result. For example, Cohen's original models for the failure of AC involved (in modern terms) a Boolean-valued universe invariant under Boolean automorphisms that arbitrarily permute a countable set of generic reals. But the 2-valued models obtained via a generic filter have no automorphisms except the identity. (If I remember correctly, Cohen later built 2-valued models that have nontrivial automorphisms, but this took some additional work.) – 2012-12-08