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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)?

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Retagged: that's more a question related to mathematical logic. It certainly depends on how you define $1$ and $2$, and $+$ (or even just $+1$). If you use Peano axioms then $1 = S(0)$ and $2 = S(S(0))$. If you further say that $m+1 := S(m)$ for a natural number $m$, then $1+1 = 2$ can be considered as a theorem in such system of axioms, I believe. –  Ilya Apr 2 '13 at 9:08
@Ilya Thank you! –  Archy Wilhes Apr 2 '13 at 9:11
This is the fourth question about $1+1=2$ that I answer. You should read the answers in the links I put in my answer here. –  Asaf Karagila Apr 2 '13 at 9:11
Well, it took Russell and Whitehead over 600 pages to prove this! –  Arthur Fischer Apr 2 '13 at 9:12
The fact that I can describe a system (such as the integers modulo 2) where 1 + 1 = 0, would seem to suggest your statement cannot be a theorem. –  cobaltduck Apr 2 '13 at 14:51

5 Answers 5

up vote 27 down vote accepted

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.

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Thank you for answering my question!! I really appreciate that! –  Archy Wilhes Apr 2 '13 at 9:16
you are welcome :) –  Ittay Weiss Apr 2 '13 at 9:47
Ittay Weiss .. nice answer !! –  Grijesh Chauhan Apr 2 '13 at 16:47

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 $1+1 = 2$?
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Thanks a lot for the answer! It is very useful!! –  Archy Wilhes Apr 2 '13 at 9:17

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.

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Thanks. I've gained a greater insight into the matter! +1 –  Archy Wilhes Apr 2 '13 at 9:30
Thanks. I like your username. –  goblin Apr 2 '13 at 9:36
The definitions above are common in abstract algebra (e.g. ring theory). In the context of Peano Arithmetic, $1$ is defined as $S(0)$, $2$ is defined as $S(S(0))$, and addition is a different operation than successor, so $1+1 = 2$ is a theorem in that context as well. –  Carl Mummert Apr 2 '13 at 12:03
@CarlMummert, yes good point. –  goblin Apr 2 '13 at 12:11
I would write those equations the other way around. –  Elements in Space Apr 2 '13 at 12:19

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.

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Thank you very much for answering my question! –  Archy Wilhes Apr 2 '13 at 9:13

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.

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Thanks!! This is awesome! –  Archy Wilhes Apr 8 '13 at 17:44

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