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Well, I can easily not have a 2 in a system of my own definition. The OP put no limit on what system they wanted by ending the equation with a question mark. Can't I have a Field that only contains the set (0,1) would like set symbols here instead of parenthesis. Then, 0+0=0, 0+1=1, 1+0=1, and 1+1=0...

As long as addition has been properly defined, it's 2. The proof can be found here: https://blog.plover.com/math/PM-translation.html This. With the important part being how addition is defined. Must say, when I took Abstract Algebra, this concept that you can have a system where 1 + 1 does not equa...