Is there a way to prove that $2+2$ really equals $4$? In elementary school, one learns that $2+2=4$ by experiment (putting two apples next to two other apples), and maybe also from some addition table to be memorized. 
But is there any approach that proves $2+2 = 4$? If so, an example of such a proof would be good.
 A: The usual approach for formally proving that $2+2=4$ is to start from Peano's axioms (which define the set $N$  of natural numbers , $0\in N$ and a successor function on $N$). Using these axioms, along with the rules of logic and set theory, you can formally prove that there exists a unique binary function $+$ such that

$x+0 = x$
$x+(y+1) = (x+y)+1$

where $1$ is the successor of $0$, and $n+1$ is the successor of $n$.
This is a long and tedious process. (Earlier versions of Peano's axioms gave you the above definition to start.)
Then you define 2, 3 and 4 such that

$2=1+1$
$3=2+1$
$4=3+1$

Then you have $2+2=2+(1+1)=(2+1)+1=3+1=4$
A: We assume the Peano axioms.  Specifically:

  
*
  
*Zero is a number. 
  
*If a is a number, the successor of a is a number.
  (We denote the successor of $x$ as $x'$.)
  
*Zero is not the successor of a number. 
  
*Two numbers of which the successors are equal are themselves equal. 
  
*(induction axiom.) If a set S of numbers contains zero and also the successor of every number in S, then every number is in S. 
  

We then define addition recursively as follows:

$$a+0 = a$$
  $$a+b' = (a+b)'$$

Now, we will name some numbers.  We will denote:

$$\begin{align}
0' &= 1 \\
1' &= 2 \\
2' &= 3 \\
3' &= 4
\end{align}$$

We could keep going, but we only need to be able to denote the numbers $0$ through $4$ (inclusive).
Now, showing that $2+2 = 4$ is a simple application of the recursive formula for addition:
$$\begin{align}
2 + 2 &= 2 + 1' \\
&= (2+1)' \\
&= (2+0')' \\
&= ((2+0)')' \\
&= ((2)')' \\
&= 3'\\
&= 4
\end{align}$$
A: There are many ways of defining $\mathbb{N}.$ In the context of this question, the details aren't that important; what matters is that $\mathbb{N}$ ends up being a set equipped with a distinguished function $S : \mathbb{N} \rightarrow \mathbb{N}$ and a distinguished element $0 \in \mathbb{N}$ subject to a theorem that says "definitions by recursion work." This allows us to prove the existence and uniqueness of a binary operation $+$ on $\mathbb{N}$ satisfying the following specifications.
$$n+0 = n, \quad n+S(m) = S(n+m)$$
Now write $4$ as shorthand for $S(S(S(S(0))))$ and write $2$ as shorthand for $S(S(0)).$ Then we have
$$2+2 = 2+S(S(0)) = S(2+S(0)) = S(S(2+0)) = S(S(2)) = S(S(S(S(0)))) = 4$$
Extra Information.
For completeness, here's several ways of defining the naturals.


*

*The algebraic structure $\mathbb{N}$ can be defined as the sole (up to unique isomorphism) model of the Peano Postulates (which are second order). 

*It can also be defined as the free monounary algebra generated by the singleton set $\{0\}$ (I suggest googling this term if you do not know it). 

*Set theorists like defining it as the least set $\omega$ such that firstly, $\emptyset \in \omega,$ and secondly, $x \in \omega$ implies $x \cup \{x\} \in \omega$. The entity $\emptyset$ ends up being our $0$, and the function $x \mapsto x \cup \{x\}$ ends up being our successor function.
A: The proposition "2+2 = 4" is a theorem of the Peano arithmetic (the five Peano's axioms). For example:
$$1 := 0',$$
$$2 := 1' = 0'',$$
$$3 := 2' = (1')' = 0''',$$
and so on.
Yes, it depends on to what meanings we assign "+" and the numerals. 
A: The most commonly used method to define sum in $\mathbb{N}$ is derived from Peano Axioms. 
$0\in\mathbb{N}$ and $s:\mathbb{N}\to \mathbb{N}\setminus \{0\}$ is a given bijection, we can define $1:=s(0)$, $2:=s(1)$ and so on...
We can define a sum: $n+m:=s^{m}(n)$. In that case $2+2=s(s(2))=s(3)=4$. This sum satisfies the properties which we are accustomed.
A: well this kind of proving depands on a set of axiom system of group like Z for example .
and these are axioms that been used to prove that 2+2=4
Axiom 1. Algebraic Properties of Z (Properties of + and .)
Properties of Addition


*

*A1. Associativity. For every x, y, and z in R, ( x + y ) + z = x + (
y + z ).

*A2. Commutativity. For every x and y in R, x + y = y + x.

*A3. Identity. R contains an additive identity, 0, such that for every
x in R,
              x+0=x.

*A4. Additive Inverses. For every x in R, there is an additive
inverse, (-x ),
             in R such that x + (-x ) = 0.
Axiom 2. Order Properties of Z (Properties of <)


*

*(i ) Transitive Property. For every a, b, and c in Z, if a < b and b
< c, then a < c.
(ii ) Trichotomy Property. For every a and b in Z, exactly one of the
following holds: a =b, a < b, or b < a.
(iii ) Additive Property. For every a, b, and c in Z, if a < b, then
a + c < b + c.
(iv) Multiplicative Property. For every a, b, and c in Z, if a < b
and 0 < c, then ac < bc.
(v) Order of Identities. 0 < 1.
Axiom 3. Th e Well-Ordering Principle


*

*For any integer n, there is a next integer n + 1 that comes
immediately after it, with no other integers in between.


with these sets of axioms you can easily prove  now that  2+2=4. just folow these steps
1+1=2
(1+1) + 1 =2+1 =3
(1+1+1)+1=3+1=4
hence
(1+1) +(1+1 )=2 +2 =4 done.
