We consider the following properties of uncountable-dimensional quadratic spaces (E, Φ):
(*) For all subspaces U ⊆ E of infinite dimension: dim U˔ < dim E.
(**) For all subspaces U ⊆ E of infinite dimension: dim U˔ < ℵ0.
Spaces of countable dimension are the orthogonal sum of straight lines and planes, so they cannot have (*), but (**) is trivially satisfied.
These properties have been considered first in [G/O] in the process of investigating the orthogonal group of quadratic spaces. It has been shown there (in ZFC) that over arbitrary uncountable fields (**)-spaces of uncountable dimension exist.
In [B/G], (**)-spaces of dimension ℵ1 (so (*) = (**)) have been constructed over arbitrary finite or countable fields. But this could be done only under the assumption that the continuum hypothesis (CH) holds in the underlying set theory.