Orthomodularity is not elementary
Robert Goldblatt · Journal of Symbolic Logic · 1984
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: If is a separable Hilbert space, and is 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 ( , ⊥).