"Paradox" with the Axiom of Dependent Choice.

Question

Here "paradox" only refers to the fact that it conflicts with our real-life intuition. For example, one is able to prove the Banach–Tarski paradox as a theorem under ZF+AC(Axiom of Choice).

There are weaker versions of choice axioms, and among them, DC(Axiom of Dependent Choice) is heavily used in, for example, recursive defining sequences. There are a number of scenarios under which one must deal with a countable number of objects and elements in it, and in my opinion, DC is more basic and more "reasonable" than AC. So I start to wonder: Is there a "paradox" under ZF+DC that conflicts our geometric intuition or other types of common sense?

Answer 1

I would argue that the answer to your question is (according to current knowledge) no.

The theory ZF + DC + "Every set of reals is Lebesgue measurable (and has the property of Baire and the perfect set property)" is consistent$^1$ by a theorem of Solovay, and this rules out the whole Banach-Tarski "flavor." It's inherently difficult to answer a question like yours negatively, but I think this is a strong argument against the possibility of DC-only geometric paradoxes.

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$^1$Well, under a mild assumption: we need the theory ZFC + "There is an inaccessible cardinal" to be consistent (and in fact Shelah showed that this consistency assumption is optimal for both measurability and for the perfect set property, while no assumption beyond the consistency of ZFC is needed for the Baire property).

Answer 2

While $\sf ZF+DC$ does not prove the Division Paradox, it is certainly consistent with it, and in the models that we know of where the Banach–Tarski Paradox is false, the Division Paradox holds.

So, what is the Division Paradox? Your geometric intuition tells you that if you break a stick into five parts, then the amount of "stuff" on the stick is at least $5$. In more mathematical words, if you partition a line into non-empty parts, then the number of parts is less or equal to the number of points on the line.

In the models where all sets are measurable, which are the models where Banach–Tarski fails, we can partition the real line into more parts than points. We can also partition the real line into a completely incomparable number of parts, so there's no comparison between the number of parts and the points on the real line.

Arguably, this is not a geometric intuition. And arguably, this is not an answer since $\sf ZF+DC$ does not outright prove the Division Paradox. But in our current understanding of the universe of sets, one of the two paradoxes hold.

Of course, if you want to be more "grown up" about geometry, you might argue that the Hahn–Banach theorem is intuitive, or at least that if $B$ is a Banach space, then its dual space (of continuous functionals) is nontrivial, i.e. we can separate points using functionals, which is a statement equivalent to the Hahn–Banach theorem. And in that case, you're in trouble. Hahn–Banach is sufficient to prove the Banach–Tarski paradox. So $\sf ZF+DC$ proves that you have to choose between Banach–Tarski or Banach spaces with trivial duals.

_{(Caveat lector: paradoxes and intuition are subjective and over time we deconstruct and reconstruct our intuition. What is intuitive to one might as well be paradoxical to another.)}