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Given aleph-one cubes with no common volume in 3D space, replacing each cube with the largest sphere that fits in it will give you aleph-one spheres with no common volume in 3D space.
Unfortunately we don't know how to fill aleph-one cubes into 3D space in a way, that they don't overlap with 3D intersections.
We can easily do so with aleph-zero cubes, but have no known way to do it with aleph-one cubes.
Others are telling me, not to even try, because it's surely impossible and I will suffer some great misfortune, if I try.
I am not sure, if it's really impossible.You can surely fil N dimensional hyperspace with aleph-one N-1 dimensional hypercubes. That IS possible.
Now, what if we h... (read more)