  We should comment on the ≈ notation here.  We are concerned with modeling various types of ordinary mathematical objects in set theory, and one kind of object that we want to model will be that of a system of equations.  This is an unusual thing to do.  In anticipation of things to come, we use the ≈ sign for equations we would like to solve.  So in our discussion of x = ½x + 1 above, we would prefer to write x ≈ ½x + 1.  The point is that ‘x’ here is a symbol, but whatever we take symbols to be, it will almost never be the case that the symbol x is identical to the expression ‘½x + 1’ or to anything related to it.  For the solution to an equation or a system of them, we will use a “dagger” to refer to the solution.  Thus for this equation, x† = 2; the reason that 2 satisfies the equation is that 2 = ½(2)+ 1 (and here we use = rather than ≈).