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Orthomodularity is not elementary1

Published online by Cambridge University Press:  12 March 2014

Robert Goldblatt
Robert Goldblatt*
Affiliation:
Victoria University Wellington, New Zealand
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Extract

In this note it is shown that the property of orthomodularity of the lattice of orthoclosed subspaces of a pre-Hilbert space is not determined by any first-order properties of the relation ⊥ of orthogonality between vectors in . Implications for the study of quantum logic are discussed at the end of the paper.

The key to this result is the following:

Ifis a separable Hilbert space, andis an infinite-dimensional pre-Hilbert subspace of, then (, ⊥) and (, ⊥) are elementarily equivalent in the first-order languageL2of a single binary relation.

Choosing to be a pre-Hilbert space whose lattice of orthoclosed subspaces is not orthomodular, we obtain our desired conclusion. In this regard we may note the demonstration by Amemiya and Araki [1] that orthomodularity of the lattice of orthoclosed subspaces is necessary and sufficient for a pre-Hilbert space to be metrically complete, and hence be a Hilbert space. Metric completeness being a notoriously nonelementary property, our result is only to be expected (note also the parallel with the elementary L2-equivalence of the natural order (Q, <) of the rationals and its metric completion to the reals (R, <)).

To derive (1), something stronger is proved, viz. that (, ⊥) is an elementary substructure of (, ⊥).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 2 , June 1984 , pp. 401 - 404
Copyright
Copyright © Association for Symbolic Logic 1984

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Footnotes

1

The work reported in this paper was done in late 1975. At the time it presented itself not so much as a contribution to quantum logic as a counterexample to a possible line of enquiry. Recently I explained the results to Bas van Fraassen, who encouraged me to make them available for publication.

References

REFERENCES

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