Independence and consistency proofs in quadratic form theory

Abstract

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.