Orthomodularity is not elementary

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 I of orthogonality between vectors in b. Implications for the study of quantum logic are discussed at the end of the paper. The key to this result is the following: (1) If Xk is a separable Hilbert space, and ?? is an infinite-dimensional pre-Hilbert subspace of AX, then (CA, I) and (Xk, I) are elementarily equivalent in thefirst-order language L2 of 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 (CA, I) is an elementary substructure of ( I, I). This is done by showing that any element of Xk can be moved inside 9 by an automorphism of Xk2 that leaves fixed a prescribed finite subset of b. Familiarity is assumed with the basic theory of Hilbert spaces, and for this purpose the very accessible exposition of Berberian [3] has been followed. THEOREM 1. Let Xk be a separable Hilbert space, and 2" an infinite-dimensional linear subspace of Xk2. Then if a,,... , an E ?? and b E X{, there exists an isomorphism T: Xk -+ Xk such that T(a1) = ai for 1 < i < n, and T(b) E '. PROOF. (By an isomorphism is meant a bijective linear transformation that preserves inner products and hence leaves the orthogonality relation invariant.) Suppose that the ai's are ordered so that for some k < n, {al,.. , akj is a linearly