Do the axioms of logic or Peano axioms imply that 1+1=2 ? Could addition be treated differently if the axioms were different such as 1+1=4?

Answer 1

Peano's Postulates prove that 1 + 1 = 2. Substituting the symbol 4 for the symbol 2 does not disrupt the proof.

Value should not be conflated with symbol.