(Sorry if I'm a imprecise in my formulations, I am new to math.)
I recently discovered algebraic structures and they seem quite fundamental, I'm wondering whether Peano arithmetic could be expressed as the set {0,1} with the operation +, if yes this would constitute an algebraic structure.
Is there something I'm missing there ? I think this is incorrect because then 1+1 would be outside the set {0,1}. However if we take the set N (natural numbers) with the operation + then we get Peano's arithmetic but we had to define N first using Peano's arithmetic.
Does this mean that sets and operations are first defined using whatever means (logic, other fields of math, ..) and then those "constituents" are used to define algebraic structures ?