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In order to work, the definition of the sheaf of rings on page 34 of versions 1 and 2 requires the additional assumption that for any m dividing n both in the trace of L, n is a power of m. I'll leave a complete fix of this bug for some other time, meanwhile I post the shorter version 3, that still contains the solutions to Jónsson's problems.A lattice L is coordinatizable, if it is isomorphic to the lattice L(R) of principal right ideals of some von Neumann regular ring R. This forces L to be complemented modular. All known sufficient conditions for coordinatizability, due first to J. von Neumann, then to B. Jonsson, are first-order. Nevertheless, we prove that coordinatizability of lattices is not first-order, by finding a non-coordinatizable lattice K with a coordinatizable countable elementary extension L. This solves a 1960 problem of B. Jonsson. We also prove that there is no L_{infinity, infinity} statement equivalent to coordinatizability. Furthermore, the class of coordinatizable lattices is not closed under countable directed unions; this solves another problem of B. Jonsson from 1962

VON NEUMANN COORDINATIZATION IS NOT FIRST-ORDER