How would you rigorously prove that 1 + 1 = 2?

Nothing can be "proven"--rigorously or not--without making some assumptions and arbitrarily holding them to be true. Counting systems and simple arithmetic emerged for practical reasons, so that 2 was "the next unit" after unit itself. Of course, for a while this didn't satisfy a number of mathematical logic purists who wanted to believe in some Platonic world where only exact, perfect truth exists, and that's why Alfred North Whitehead and Betrand Russell spent years in drafting their weighty tome--Principia Mathematica--which attempted to answer your very question. It failed. After Russell read Godel's Incompleteness Proof, he wrote that Principia had been a wasted effort. Why? Because while mathematicians from Greek times understood that one needs to begin with definitions and axioms, it had been assumed that there can exist (strong case) an unique set of axioms which would generate all other truths, or (weak case) a set of axioms which will never result in a contradiction of truths derived from them. That is what Whitehead and Russell attempted to do, the weak case. What they failed to prove is that their "really very fundamental sub-Planckian" definitions and axioms which was the foundation of their work would never produce such a contradiction, and Godel's Proof suggested that it may even be impossible to prove it. In other words, it had to be taken on faith that 1+1 = 2. There is still a lot of controversy over this matter, and the larger picture of mathematical rigor in face of Godel's Proof, but for sure the "correct and rigorous" answer that you seek will not fit in here.

As an aside, this problem is similiar to the problem in computer science of proving that a particular software code "will never crash". Whole journals are being devoted to this single issue, because the obvious economic costs involved, but no absolutely failsafe methodology has been developed. This is also related to Turing's famous "Halting Problem"---which states that it is not always possible to prove that a particular computer code--once started--will not run indefinitely.

Addendum 1: I was careful to point out that the weakness of Principia is that they failed to prove self consistency, and that it MIGHT be impossible to prove such consistency. So, the race continues for a "proven" self consistent set of axioms that will finaly demonstrate that 1 + 1 indeed equals 2. However, note that very interesting mathematics (and physics as well) can get gained from differing set of axioms which may or may not be entirely self consistent. The question does need to be asked: Why is it absolutely imperative that mathematics (or physics) be solely based on "perfect" axiomatic systems?

Addendum 2: JCS's erudite answer is exactly what I was referring to about the "controversy". Yes, computer scientists are getting better in working out ways to develop (almost) bug-free software too.

scythian is on the mark here: the positive integers are rigorously constructed using Peano's axioms, and 1+1 is the immediate successor of 1, which is defined as 2.

But I would wholly disagree with scythian's argument that Godel's Incompleteness Theorems are controversial; the only mathematicians that seriously feel this way are fringe logicians and maniacal set theorists. Godel's theorems opened up alot of new venues in mathematics, and some might say "freed" us from the almost certain rigorous demise to which we were headed. Basically, Hilbert in 1900 asked "could somebody please set up a system of axioms which is completely consistent and serves as a basis for all math?", to which Godel responded, several years later, "no, no one can; any axiomatic system describing the integers will have certain unprovable statements, and some which are consistent when treated both as true and as false."

To summarize: 1+1 is 2 because it is defined that way, axiomatically, and thus cannot be proven under the standard system of Peano's axioms.

Steve

EDIT - Above, when I say "unprovable statements", I mean statements treated as true, but not proved as such (not including axioms). When I talk about a statement being "consistent when treated both as true and as false", I mean independent of the current axiomatic framework; this is equivalent to saying the framework cannot prove its own consistency.

"Please don't tell me that's how 1+1 is defined."

It's more a matter of how 2 is defined.

In set theory,

1 is defined as "the common property of all sets which have a single element". Let's call those 1-sets.

2 is defined to be "the common property of all sets which have an element and another element and no more", which we'll call 2-sets.

(It's hard to avoid circular definitions here since these terms are so basic. Defining higher numbers in what is essentially a verbose base 1 notation quickly becomes cumbersome.)

We can create sets out of nothing using the empty set.

Starting with nothing, we can create { }, the empty set.

To avoid an explosion of brackets, let's use the notation Ø for that set. (Ø is the Greek letter phi.)

Then we can put Ø in a set { Ø }, so we have a 1-set ( a set whose lone element is an empty set ).

We can also put THAT set into a set: { { Ø } } ( a set containing a set containing a set containing nothing )

Then the union of { Ø } and { { Ø } } is { Ø, { Ø } }.

The union of a pair of 1-sets is a 2-set.

Then if we slip in (since this is just a Yahoo Answer, not a mathematics course) the idea of cardinality (how many elements a set has), then addition becomes determining the cardinality of the union of sets:

To paraphrase that great mathematician and band leader Lawrence Welk, "and a 1 and a 1 is a 2".

Here's my crack at it:

Assume the integers are closed under addition,

addition is an operation which respects integer ordering,

and there is a least integer greater than 1, and it is defined as 2.

Then since 1+1 is an integer >1 we have by definition of 2 that 2<=1+1. Next, we cannot have 1+1>2 because then there would be an integer b such that 1+1>b>2, implying an integer a such that 1>a>1, which is impossible. Thus 1+1=2.

I will just add a few comments, with little hope that they will be read, given that the list of answers is so very long.

(1) The best answer, by far, is by MathManTG. He just forgot to add that what he did, identifying the naturals with the initial ordinal sets, is possible because the axioms of Formal Arithmetic systems (First-Order Peano is the most widely known, but there are others, weaker ones, used by logicians to study these system's capabilities) are interpretable within Zermelo-Fraenkel Set Theory; so, in this sense, Set Theory contains First-order Arithmetic.

(2) I must strongly disagree with the persons that assert that the (true) sentence 1+1=2 is a merely a definition, or otherwise not provable. When you set up an axiomatic system, you have undefined terms (for example, the points and lines of euclidian geometry), axioms and rules of inference. Inthe case of Arithmetic, the usual undefined terms are 0 (a constant), s, +, ×, etc.; the other symbols are the usual logical ones, the rules of inference are the ones used in classical systems of logic, and the axioms depend on the particular fragment of Arithmetic you are studying.

Now, there are special terms, called numerals (NOT numbers), that are of the form s(s(...s(0))) that, under a suitable interpretation, will reference the usual natural numbers, but not necessarily.

The important point is that for any practical axiom system for arithmetic the sentence s(0) + s(0) = s(s(0)) will be PROVABLE from the axioms, using the accepted rules of inference, and that the usual interpretation of this sentence, with the naturals and common arithmetic operations as the model, simply means 1 + 1 = 2.

Now, if your axioms are true in this model, and your inference rules are sound, then this interpreted sentence will be true.

But it is not correct to say that this is not provable because is implicitly contained in the axioms; the (interpreted) sentence 123651423512734 + 46237864729387= 169889 288242121 is also true, and is formal counterpart (with the s), is also provable from the axioms, but I doubt that anyone will say that it's a definition and not provable.

(3) A final comment on Russell, Hilbert, Gödel and a few others. Frege and Russell were the initiators of the Logicist Programme (LP): their aim was to show that Mathematics was reducible to Logic; they failed, but not because of Gödel's theorems; in fact, LP was, for all practical purposes, dead when Gödel published his results and he chose to present them using Principia Mathematica's logical system because it was the best formal logical system available (in fact, everybody focus on the first incompletness theorem, which is just a result that shows that Truth and Provability are different concepts, and forgets that he also proved that first-order logic is complete, that is, any logically valid sentence can be mechanically proved). What caused the demise of the LP was not Gödel's theorems, but the circularity paradoxes that started with Frege (who founded modern mathematical logic) and the attempts to escape them. Russell tried to do this by introducing what is nowadays known as a typed system, which only allows the formation of sets using elements from lower types. This blocks the circularity paradox but, unfortunately, it also blocks the existence of infinite sets (Russell was so desperate that he introduced an additional axiom, the Axiom of Reducibility, to bypass this: the result was again the paradoxes; Hermann Weyl, upon reading this, is said to comment "he just killed himself").

What was deeply affected by Gödel's theorems on incompletness was the Formalist Programme of Hilbert and his students. I will not write about this now, but I should add that something called "Relativized Hilbert's Programmes" are an active topic of research today, so overblown comments about Gödel are not warranted. In fact, many logicians today are coming to the conclusion that these theorems are not Gödel's greatest work, and they are simply part of the landscape and not really surprising; his later writings, about Set Theory and the need for new axioms, the Dialectica interpretation, etc. are revealing themselves to be much more interesting.

trigo is the key here.

consider a right triangle with sides 3-4-5,such that 5 is the hypotenuse.

now,sin^2theeta + cos^2theeta=1

sin^2theeta+sin^2theeta+cos^2theeta+cos^2theeta=2*(16/25)+2*(9/25) [taking in original form where p/h=sintheeta]

=32/25+18/25=50/25= which is 2.

proving the result 1+1=2 (sin^2theeta+cos^2theeta=1)

addition can also be proved by the principle of mathematical induction

I thot of this question so many yrs ago and then eventually gave up when i eventually got the live example of me tagging along with one more friend and we were two people, but when i went out with 4 individuals we were 4 people i can't say either of those 4 individuals i went with were non-existant thus zero wasn't there in that calculation. No ghost was accompanying me who were visible so the no. of invisible ghosts were me + my friend + my Friend + my friend + Space

Thus due to this i accept it as true coz the only other option remaing was to bang my head so hard that it'd split into 2 equal parts but right now i just want a single whole head which is just 1.

Surely it follows immediately from the definition. n! = 1*2*3 . . . *(n - 1)*n = n*{(n - 1)*(n - 2)* . . . 3*2*1} because multiplication is commutative. = n*(n - 1)! P.S. Anyway. I always thought that n! was defined in the order n*(n - 1)*(n - 2) . . . 3*2*1

I am not so sure that it can be proven. It is possible to construct mathematical systems in which 1 + 1 ≠ 2. Moreover, some of these mathematical systems have practical uses. One such system is that used in parity and cyclic redundancy code checkers in computers. In this system of arithmetic, the numbers are represented in base 2 but addition is done without carries. In other words, 1 + 1 = 0 in this system of arithmetic. It also turns out that addition and subtraction are the same thing in this system of arithmetic.

My knowledge of mathematics doesn't extend this far into the theoretical, but I recall taking a set theory class where the professor proved that 1 + 1 = 2 by building sets upon sets.

So it can be done with some manipulation of set theory. But that's about all I know.

Maybe this will give you a kick-start.

-John