TITLE: Is 1+1 =2 a theorem?
QUESTION [21 upvotes]: A theorem is defined to be a mathematical statement that is proven to be true. The statement $1+1=2$ has definitely been proven in the history of mankind (Russel and Whitehead had once proven it in the book Principia Mathematica). 
So can it be considered as a theorem? What determines something to be a theorem (besides it being proven to be true)?

REPLY [2 votes]: Depending on your theoretical starting point, $1+1=2$ can be either a definition or a theorem. 
If you start with the addition of natural numbers already defined, then you could simply define $1+1$ to be $2$.
If you start with only a successor function (without addition) as in the modern version of Peano's Axioms, you would define the successor of $1$ to be $2$. Then you would have to construct the add function by selecting an appropriate subset $A$ from the set of ordered triples of natural numbers $N^3$, prove that it actually is a function, that it has the required properties of an add function, and that $(1,1,2)\in A$. Then you would have to define $x+y=z\leftrightarrow (x,y,z)\in A$. Thus $1+1=2$ would be a theorem.