User blog comment:Rgetar/Number distinction/@comment-31837178-20171209160848

I had thought of something very similar to this, which I called "indeterminate numbers". An indeterminate number can be expressed as the set of its possible values, for example {1,2,3,5}. You can use this to state that the solution of a polynomial equation has just one indeterminate root, for example the solution of x^3-x^2-12x=0 is {-3,0,4}, or the square root of 25 is {-5,5}.

You can also establish an identity element (I called it "@") equal to {0,1} and describe an indeterminate number as a linear combination of @'s. For example, 8@+@+@ = {0,8}+{0,1}+{0,1} = {0,1,2,8,9,10}. A lot of the work I did on this seemed to duplicate work I'd seen on sumsets, so I never really pursued this concept further, but it's a really interesting concept on its own merit.