In standard axiomatic set theory (ZFC) and Peano arithmetic, 1 + 1 is rigorously defined to equal 2.

Modern mathematics establishes the truth of '1 + 1 = 2' through formal axiomatic systems. In Peano arithmetic, which defines natural numbers and their operations, '1' is the successor of '0', and '2' is the successor of '1'. Addition (a + b) is defined recursively: a + 0 = a, and a + S(b) = S(a + b), where S(x) is the successor of x. Thus, 1 + 1 is equivalent to 1 + S(0), which, by definition, becomes S(1 + 0). Since 1 + 0 = 1, then S(1) = 2. This rigorous derivation, famously detailed by Bertrand Russell and Alfred North Whitehead in *Principia Mathematica*, demonstrates that this seemingly simple sum is a logical consequence of foundational definitions.