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Independence and consistency proofs in quadratic form theory

Published online by Cambridge University Press:  12 March 2014

James E. Baumgartner  and
Otmar Spinas
James E. Baumgartner
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
Otmar Spinas
Affiliation:
Mathematisches Institut, Universität Zürich, CH-8001 Zürich, Switzerland Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel
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Extract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 4 , December 1991 , pp. 1195 - 1211
Copyright
Copyright © Association for Symbolic Logic 1991

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References

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