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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

REFERENCES

[B/G]Baur, W. and Gross, H., Strange inner product spaces, Commentarii Mathematici Helvetici, vol. 52 (1977), pp. 491–495.CrossRefGoogle Scholar
[vD]van Dòuwen, E. K., The integers and topology, Handbook of set-theoretic topology (Kunen, K. and Vaughan, J. E., editors), North-Holland, Amsterdam, 1984, pp. 111–167.CrossRefGoogle Scholar
[F]Fremlin, D. H., Consequences of Martin's axiom, Cambridge University Press, Cambridge, 1984.CrossRefGoogle Scholar
[G1]Gross, H., Quadratic forms in infinite dimensional vector spaces, Birkhäuser, Basel, 1979.Google Scholar
[G2]Gross, H., Handwritten notes.Google Scholar
[G/O]Gross, H. and Ogg, E., Quadratic spaces with few isometries, Commentarii Mathematici Helvetici, vol. 48 (1973), pp. 511–519.CrossRefGoogle Scholar
[Ku]Kunen, K., Set theory: an introduction to independence proofs, North-Holland, Amsterdam, 1980.Google Scholar
[Ma]Magidor, M., Letter to Otmar Spinas, dated 10 30, 1988.Google Scholar
[S1]Spinas, O., Konsistenz- und Unabhängigkeitsresultate in der Theorie der quadratischen Formen, Ph.D. thesis, University of Zürich, Zürich, 1989.Google Scholar
[S2]Spinas, O., Iterated forcing in quadratic form theory (in preparation).Google Scholar
[S3]Spinas, O., An undecidability result in lattice theory, Abstracts of Papers Presented to the American Mathematical Society, vol. 11 (1990), p. 161.Google Scholar
[T]Todorčević, S., Partitioning pairs of countable ordinals, Acta Mathematica, vol. 159 (1987), pp. 261–294.CrossRefGoogle Scholar