Set theory definition of addition, negative numbers, and subtraction? Using the definition of natural numbers $0 = \emptyset$ and $S(n) = n \cup \lbrace n \rbrace$ where S is the successor function, what is the definition of addition on natural numbers?
Concerning the definition of negative integers, the wikipedia entry is a bit ambigous: http://en.wikipedia.org/wiki/Integer#Construction
It seems to claim that the integers are defined such that, for example, the natural number 2 is not equal to the integer 2, which is defined as $\lbrace n \in \mathbb{N}^2 | \exists m \in \mathbb{N}:n = (m + 2,m) \rbrace$ or simply $ \lbrace (2,0), (3,1), (4,2), ... \rbrace$, whereas the natural number 2 would be $ \lbrace 0,1  \rbrace = \lbrace \emptyset , \lbrace \emptyset \rbrace \rbrace $. It was to my understanding that the above definition for integers was reserved for the negative integers, for example $ -1 = \lbrace (0,1), (1,2), ... \rbrace $, and the integers would be defined as the union of the sets of negative integers and natural numbers. Is wikipedia wrong and natural 2 = integer 2, or is it the other way around? And in the latter case, are rational 2, real 2 and complex 2 also distinct? (And the ordinal number 2, and the cardinal number 2...)
Assuming the former case, is there an "official" way of defining things next? It seems to me like the easiest way of doing things would be defining first subtraction for $(n,m)$ where $ n \ge m $; that is, $n-m$ is the natural number $ d $ such that $ n = m + d $. Then defining additive inversion, i.e. the unary operation $-$ s.t. $ 0 \mapsto 0$, $ n \mapsto -n$ for nonzero natural $n$ (here $-n$ is the already defined negative integer and not the $-$ operation on $n$), and for negative integers $n$, $ n \mapsto m $ where $m$ is the natural number s.t. $ (0,m) \in n $. From there we can define addition and subtraction more generally for all combinations of natural numbers and negative integers, using additive inversion whenver we get a negative result, for example, $ 3 - (-4) = 3 + 4 $, $ 3 + (-4) = 3 - 4 = - (4-3) $, $ (-3) - 4 = (-3) + (-4) = - (3+4) $.
Tl;dr: what is the definition of addition on naturals, is natural 2 = integer 2 or are they distinct elements, and how do we define addition and subtraction on integers?
 A: There are many different ways of defining the natural numbers, integers, fractions, reals and complex numbers. I for myself do not think there is a canonical way. Thus, Wikipedia is not wrong, and there is not a way to do it *more right".
You certainly do not want to think of all these numbers as their underlying sets. One could start thinking about the intersection of $\frac{2}{3}$ (as fraction) with $\pi$ (as real), but it would make absolutely no sense.
What really matters is the algebraic structure. No matter which definition you give of $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$, you can identify them in a natural way. If $\mathbb{Z}_{1}$ is your first set-theoretic definition of the integers, and $\mathbb{Z}_{2}$ is another one, then there is a canonical function $f \colon \mathbb{Z}_{1} \to \mathbb{Z}_{2}$ such that:


*

*$f(m +_{1} n) = f(m) +_{2} f(n)$

*$f(m \times_{1} n) = f(m) \times_{2} f(n)$

*$f(0_{1}) = 0_{2}$

*$f(1_{1}) = 1_{2}$.


So, what really matters is this algebraic structure of these sets.
A: The set of natural numbers $\mathbb N$, together with its natural addition, is a commutative semigroup. There is a standard way (*) to extend a commutative semigroup $(S,+)$ to a commutative group $(G,+)$: G is the quotient space of $S\times S$ by the relation $(a,b)\sim (a',b')$ if and only if $a'+b=a+b'$, with addition $[a,b]+[a',b']=[a+a',b+b']$. All the formulas with sign are easily shown.
The semi group $S$ is not a subset of $G$ but embeds naturally as a semigroup by the injective map $a\mapsto [a,0]$.  
The construction of ($\mathbb Z$,$+)$ from ($\mathbb N$,$+)$ is exactly the construction above.
(*) but I don't think we can call it "official"
