This is the 3D version of the countable antichain condition (commonly known as c.c.c.).
c.c.c. is implied by a property called separability, which is part of the definition of the real line (the unique linear complete separable order).
So you can fit aleph1 "cubes" only if you operate in a modified notion of space which is not c.c.c.
On the other hand, the real line contains aleph1 points only in some model of set theory. The precise quantity is 2^aleph0.
It's just a magnificent toy, this ZF construction. And those others set theories as well. No wonder some people here don't want it to be broken. With passion, may I add?
Let us take the aleph-zero dimensional R. Countably infinite dimensional Euclidean space, in other words. Then take a point T and all those points which are finitely far away from this point T. By the standard metrics of sqrt(dx1^2+dx2^2+...).
This space is separable into continuum many such subspaces. Where in every such subspace every two points are close. Close means only a finite distanc...
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