A maximal almost disjoint family of block subspaces is a collection of infinite-dimensional subspaces of a countable-dimensional vector space, any two of which intersect in only finite dimension, and which cannot be extended — no further subspace is almost disjoint from all of them.
A full MAD family of block subspaces exists in ZFC.
A full MAD family of block subspaces does not exist in Solovay's model over the two-element field.
Both statements are proved in the same paper. The first uses a direct ZFC construction, answering Smythe's question affirmatively. The second uses abstract Mathias forcing to show that collapsing a Mahlo cardinal produces a model of set theory where no such families can exist.
The object is finitely describable. It concerns finite-dimensional intersections of infinite-dimensional subspaces of a countable vector space — concrete linear algebra scaled up. Yet its existence depends on which set-theoretic universe you inhabit. In the standard mathematical universe (ZFC), you can build one. In a universe obtained by a large cardinal collapse, you provably cannot.
This is not about esoteric axioms making esoteric objects appear or disappear. Almost disjoint families are combinatorial — they concern how subspaces can be packed together while maintaining bounded overlap. The packing question has different answers depending on the ambient axioms, even though the subspaces themselves are the same.
The philosophical pressure is real. “Does this mathematical object exist?” feels like it should have an absolute answer. The paper shows it doesn't — existence is relative to the universe, and both universes are consistent. Mathematical existence is not discovered but negotiated with the axioms you inhabit. The structure is the same; only the context of inquiry determines whether it can be assembled.