Is 1+1 =2 a theorem?

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)?

Solution 1:

The only thing that makes something a theorem (in a particular deduction system) is if a proof of it is known.

Now, as for $1+1=2$, you first must be very precise about what all the symbols mean and what the deduction system is that you allow your proofs to be written in. Once one gets into the details, things get less and less trivial, and far from obvious or straightforward.

So, what do you mean by $1$? what do you mean by $2$? and what do you mean by $+$? and most importantly, what do you mean by "proof"? Different answers to these questions will lead to different answers to the question "is $1+1=2$ a theorem?". You can learn more about these issues by studying logic (model theory) and in particular the set of axioms known as the Peano axioms.

Just to illustrate using two possible interpretations (and avoiding a precise definition of proof, thus relying on some intuitive understanding of what that is). If you define $2$ to be an abbreviation for $1+1$, assuming we know what $+$ is, then $1+1=2$ is certainly a theorem, with a very short proof. However, a more refined possibility is to define the natural numbers as certain sets, and then define the plus operation by induction. Then (commonly) $0=\emptyset$, $1=\{0\}$, and $2=\{0,1\}$. The actual definition of $+$ is a bit more difficult, but then it can, rather easily, be shown that $1+1=2$. I hope this explains things better.

Solution 2:

Theorems are proved from axioms, definitions are axioms.

If you define that the symbols $1+1=2$, then you implicitly wrote an axiom which connects the symbols, and proves that $1+1=2$ is a true sentence.

Often, however, we use the word "theorem" for statements whose proofs are not trivial. In this case, if you define $2$ as $1+1$ then this is not a theorem, this is a definition.

Lastly, as Ittay wrote, you have to be very careful about this. Mathematics require precision, what are the axioms you are assuming? What is the meaning of the symbols, and so on.

For more, see the following:

1. How do I convince someone that $1+1=2$ may not necessarily be true?
2. Prove that 1+1=2
3. How would one be able to prove mathematically that $1+1 = 2$?

Solution 3:

The following are often taken as definitions.

• $1+1=2$
• $1+2=3$
• $1+3=4$

etc. So $1+1=2$ is indeed a definition. However, $2+2=4$ is definitely a theorem.

Solution 4:

The term theorem, and other similar words like lemma; proposition; and corollary, or other related terms like definition and axiom, are often used for pedagogical reasons rather than for technical mathematical reasons.

So, you have to interpret their usage not as something with mathematical content, but because the author is trying to connote something to you through their choice of terms.