Orthomodularity is not elementary

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Abstract
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 (, ⊥).