TITLE: How is addition defined?
QUESTION [20 upvotes]: I've been reading On Numbers and Games and I noticed that Conway defines addition in his number system in terms of addition. Similarly in the analysis and logic books that I've read (I'm sure that this is not true of all such books) how addition works is assumed. From what I understand the traditional method of building the number system begins with the natural numbers (and zero)
$0:=|\emptyset|$
$1:=|\{\emptyset\}|$
$2:=|\{\emptyset,\{\emptyset\}\}|$
and so forth. In this construction addition could(?) be defined as the disjoint union of the sets associated with the two numbers. Then the integers could be defined as the additive inverse and so forth. Is this the ideal way to do it though, is there a more elegant method?

REPLY [7 votes]: An extremely elegant way of defining numbers I want to mention does not use sets but lambda calculus, i.e. functions and function application. The system used there is called Church encoding.
The idea goes the following: Two, for example, means doing something for two times.
More precisely, when we have some operation (a function) and a value, we apply this function twice on this value. In lambda notation
$$ 2 \equiv \lambda f\,x \mapsto f (f\,x) $$
So generally, any number $n$ is defined as a function that takes another function and returns it's $n$th iterate.
Now we can simply define addition as a function composition.
We first apply the function $n$ times, then $m$ times and thus we got a total of $n+m$ applications.
$$ n + m \equiv \lambda f\,x \mapsto n f (m f x)$$ 
For example, we end up with
$$ 2 + 3 \equiv \lambda f\,x \mapsto f (f (f (f (f\,x)))) \equiv 5 $$