What is an ordered pair actually?

Question

What does $(a,b)$ mean actually? I saw this in the 'formal defintion' of functions, and it tripped me up. We haven't even defined what an ordered pair is, before using it. Is it just a notation of representing two numbers in order?

Answer 1

You probably saw this in a naive rather an axiomatic set theory context. Naively, an ordered pair is understood intuitively, a pair of two things, the first coming from a set specified as the first set, and the second coming from a set specified as the second set.

If you are bothered by the fact that it was left undefined, then rest assured it can be defined. For instance, Kuratowski's definition is that $(a,b)=\{\{a\},\{a,b\}\}$, which, you can convince yourself, does correctly codify what (we all pretend to understand informally in the same way) is the ordered pair.

Answer 2

Let me give motivation for the concept of an ordered pair that is completely free from any preconceptions about sets, since it is rather obvious that any definition of the ordered pair in terms of set theory (ZF) is not quite what we 'intuitively understand' an ordered pair to be.

When we want to describe a sequence of any sort, such as a sentence comprising words, or a procedure comprising steps, or a queue comprising of people, we need to be able to specify the order of the objects in the sequence, and also be able to distinguish between sequences with the same objects but with the objects in different order. For a sequence of exactly 2 objects, that directly corresponds to the following definition of "ordered pair":

For any objects $A,B$:

  Let "$(A,B)$" denote an ordered pair [this is simply a choice of notation]

For any ordered pairs $(A,B),(C,D)$:

  $(A,B) = (C,D)$ if and only if $A = C$ and $B = D$ [this defines when two pairs are identical]

The first part defines the notation in such a way that we can specify the order of the objects simply by writing their placeholders in the order we want. The brackets and the comma are an arbitrary choice. (You may notice that this notation can have many other meanings as well in different contexts, which is unfortunate but we can always state explicitly what it is to disambiguate.)

The second part allows us to identify whether two pairs are identical or different. If $(A,B) = (C,D)$, we of course want each part to be identical according to how we want our notation to correspond to our notion of order. Hence we stipulate that $A = C$ and $B = D$. On the other hand, if $A = C$ and $B = D$, we want it not to matter whether we used $A$ or $C$ as the first object in a pair, and $B$ or $D$ as the second object, so we stipulate that $(A,B) = (C,D)$.

When we see it this way it is clear what is notation (syntax) and what is the meaning (semantics).