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**Example text**

3 Hilbert spaces As noted in the introduction, Hilbert space structures have the form shown in Eq. 4 as H¯ x = { Hx , + x , − x , · x , −, − x , ψx }. (31) Complex numbers are included implicitly in that Hilbert spaces are closed under multiplication of vectors by complex numbers. Also scalar products are bilinear maps with complex values. As was the case for numbers, parallel transformation of H¯ y to x maps H¯ y onto H¯ x . If scaling of the numbers is included, then the local representation of H¯ y , C¯ y on H¯ x , C¯ x is given by H¯ xr , C¯ xr .

The universes are all equivalent in that any mathematical system present in one universe is present in another. It follows that y contains systems for the different types of numbers as ¯ y , I¯y , Ray , R¯ y , C¯ y . It also contains H¯ y , Op , etc. Universe equivalence means here that for any N y system type, S, S¯y is the same system in y as S¯ x is in x . For the purposes of this work, it is useful to have a speciﬁc deﬁnition of mathematical systems. Here the mathematical logical deﬁnition of a system of a given type as a structure (Barwise, 1977; Keisler, 1977) is used.

Here ay is the same (or Fy,x -same) number value in C¯ y as a x is in C¯ x . Opy is the same operation in C¯ y as Op x is in C¯ x . Op denotes any one of the operations, +, −, ×, ÷. Note that Fy,x is independent of paths between x and y. This follows from the requirement that for a path P from x to y and a path Q from y to z, Q∗ P Q P Fz,x = Fz,y Fy,x . (17) Here Q ∗ P is the concatenation of Q to P. If z = x then the path is cyclic and the ﬁnal structure is identical to the initial one. This gives the result that Q∗ P Fx,x = 1.