[FOM] RE: FOM Real Numbers

[Matt Insall]

Slater:
Indeed, if the number 5 were the same as the set {p|p<5} then not ...

Insall:
In the approach to which I referred, 5 is not so identified.  It is
identified with a set (actually an equivalence class) of cauchy sequences.
Its complement is the calss of all sets which are not members of that
particular equivalence class, and is a proper class.  (It is not especially
useful to deal with such a complementation operation, but if you are wanting
to be pedantic in requiring that real numbers have complements, like the
rationals do, then you are in a sense stuck with weird things like this.)
The number 5 also has a relative complement in the class of equivalence
classes of convergent cauchy sequences, and this complement has as its
members convergent cauchy sequences - it is the union of all the real
numbers other than the number 5.