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This question follows up on a comment I made on Joseph O'Rourke's recent question, one of several questions here on mathoverflow concerning surprising geometric partitions of space using the axiom of choice.

  1. Joseph O'Rourke: Filling $\mathbb{R}^3$ with skew lines

  2. Zarathustra: Is it possible to partition $\mathbb{R}^3$ into unit circles?

  3. BenoƮt Kloeckner: Is it still impossible to partition the plane into Jordan curves without choice?

The answers to these questions show that the axiom of choice implies, by rather easy and flexible arguments, that space admits several surprising geometrical decompositions. To give a few examples:

  • $\mathbb{R}^3$ is the union of disjoint circles, all with the same radius. Proof: Well-order the points in type $\frak{c}$. At stage $\alpha$, consider the $\alpha^{\rm th}$ point. If it is not yet covered, we may find a unit circle through this point that avoids the fewer-than-$\frak{c}$-many previously chosen circles. QED

  • $\mathbb{R}^3$ is the union of disjoint circles, with all different radii. Proof: at stage $\alpha$, find a circle of a previously unused radius that avoids the fewer-than-$\frak{c}$-many previously chosen circles. QED

  • $\mathbb{R}^3$ is the union of disjoint skew lines, each with a different direction. Proof: at stage $\alpha$, put a line through the $\alpha^{\rm th}$ point with a different direction than any line used previously. QED (Edit: it appears that one might achieve this one constructively by using the $z$-axis and a nested collection of hyperboloids, which unless I am mistaken would give lines of different directions.)

  • $\mathbb{R}^3$ is the union of disjoint lines, each of which pierces the unit ball.

  • And so on.

The excellent article M. Jonsson, J. Wästlund, "Partitions of $\mathbb{R}^3$ into curves" Math. Scand. 83 (1998) 192-204 proves among other things that $\mathbb{R}^3$ can be partitioned into unlinked unit circles, either all with the same radius or all with different radii, and they have a very general theorem about partitioning $\mathbb{R}^3$ into isometric copies of any family of continuum many real algebraic curves.

So the general situation is that the axiom of choice constructions are both easy and flexible and not entirely ungeometrical.

Meanwhile, there are also a few concrete constructions, which do not use the axiom of choice. For example, Szulkin (see theorem 1.1 in the Jonsson, Wästlund article) shows that $\mathbb{R}^3$ is the union of disjoint circles by a completely explicit method. And there are others. But my question is not about these cases where there is a concrete construction. Rather, my question is:

Question 1. Can we prove, in any of the cases where there is no concrete construction, that there is none?

For example, taking Borel as the measure of "explicit", the question would be: can we prove, for any of the kinds of decompositions I mention above or similar ones, that there is no Borel such decomposition? This would mean a decomposition of $\mathbb{R}^3$ such that the relation of "being on the same circle" would be a Borel subset of $\mathbb{R}^6$, and in this case, the function that maps a point to the nature of the circle on which it was to be found (the center, radius, and so on) would also be a Borel function.

A closely related question would be:

Question 2. Can we prove for any of the decomposition types that it requires the axiom of choice?

For example, perhaps there is an argument that in one of the standard models of $\text{ZF}+\neg\text{AC}$ that there is no such kind of decomposition.

My only idea for an attack on question 1 is this: if there were a Borel decomposition of $\mathbb{R}^3$ into circles or lines of a certain kind, then the assertion that this particular Borel definition was such a partition would be $\Pi^1_2$ and hence absolute to all forcing extensions. So the same Borel definition would work to define such a partition even in forcing extensions of the universe having additional reals. Furthermore, the ground model partitions would be a part of that new partition. And this would seem to be a very delicate geometric matter to fit the new reals into the partition in between the ground model objects. But I don't know how to push an argument through.

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Shooting from the hip, what happens when the real numbers are a countable union of countable sets? Should you require DC as well, as it ensures that most mathematics behaves normally, and at least ensures that there are more sets than Borel sets? –  Asaf Karagila Apr 9 '12 at 22:27
Do we know that these things can't happen constructively? –  Andrej Bauer Apr 10 '12 at 4:54
By the way, theorem 1.1 in Jonsson & Wästlund article is not constructive as far as I can tell. There is a problem with "each sphere intersects one of the circles in exactly two points". –  Andrej Bauer Apr 10 '12 at 5:13
Andrej, I was using the term "construction" loosely, in classical logic, but could you explain why the argument isn't constructive in your sense? You have a circle in the $xy$ plane with center at $(4k+1,0,0)$ and a sphere of radius $r$ at the origin. So you are intersecting two circles; you can solve for the intersection points, and they vary step-wise continuously with $r$. (I am guesing that this "step-wise" step is the issue for constructive logic...) –  Joel David Hamkins Apr 10 '12 at 6:38
And no, I don't think we know that we can't do all these things constructively. Perhaps there are nice explicit decompositions of each type. When there is, then we can prove that there is just by exhibiting it. But when there isn't, I don't see how we could prove that there isn't, and this is what my question is about. –  Joel David Hamkins Apr 10 '12 at 8:43
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