Is 'order' a primitive notion in the definition of an ordered pair?

Question

I am trying to understand the definition of ordered pairs. The book I am studying is elements of set theory from Enderton.

In the axiomatic approach to set theory, it seems that you start with a primitive notion, followed by axioms, and from there you can use logical deductions to determine properties, theorems etc.

I posted a question Why do definitions need to be 'proved' to work? and the answers were very helpful. I know understand why showing the defined object exists is necessary.

However I am now confused by the notion of 'order' in defining the ordered pair. Both in the book and on wikipedia, sets (and elements) are the only primitive notions that are used to develop the entirety of set theory. In this sense, when we define an ordered pair, it must be a construct from some object that is itself a set, which we have with:

$$\langle x,y\rangle := \{\{x\},\{x,y\}\}$$

Here the ordered pair is defined to be the set on the right hand side. I can follow and accept this. What I am failing to wrap my head around is that there is an assumed sense of order. What I mean by this is we assume that $$\langle a,b\rangle := \{\{a\},\{a,b\}\}$$

while $$\langle b,a\rangle := \{\{b\},\{b,a\}\}$$

My issue here is that in writing out the right hand side of the equations, we looked into the object we defined $\langle x,y \rangle$, used the order in which its members appeared, and then filled in the right hand side.

I'm sure this seems incredibly trivial and silly, but to me I don't understand why there is this assumed sense of 'order' that can be derived from the object. It wasn't a primitive concept, and it was never really defined. So is 'order' simply an intuition? Is it so fundamental that it is silly to consider it further? Or is it simply an implied primitive concept?

Cheers in advance.

Answer 1

“Ordered pair” does not mean “a pair with an order (in the order-theoretic sense)”, it just means “a pair where there the two positions the object can be in are distinguishable”.

Given that we usually write a pair by writing one of the elements first, and the other second, it means that the order you write it in is significant, therefore the name “ordered pair”. But here “ordered” doesn't refer to a mathematical concept (except indirectly, by the mathematical concept of order formalizing the informal concept).

Indeed, in the definition you cite, there's no inherent order; the set $\{\{a\},\{a,b\}\}$ is the same as the set $\{\{b,a\},\{a\}\}$ or the set $\{\{a\},\{b,a\}\}$.

Also, instead of speaking of the “first” and “second” element of a pair, you could also borrow from Lisp, and call the elements of a pair “car” and “cdr”, without changing anything about the mathematical meaning.

Answer 2

I'm sure this seems incredibly trivial and silly, but to me I don't understand why there is this assumed sense of 'order' that can be derived from the object. It wasn't a primitive concept, and it was never really defined.

I think your confusion might lie here. The string of symbols $\langle a, b \rangle$ is a priori undefined. The equation you write for it is a definition of the symbol. Any facts about what the string of symbols $\langle a, b \rangle$ means have to be derived from what it is defined to mean, namely $\{\{a\}, \{a,b\}\}$. With some effort, you can show that $\langle a, b \rangle = \langle b, a \rangle$ -- that is, that $\{\{a\},\{a,b\}\} = \{\{b\},\{a,b\}\}$ -- if and only if $a = b$. Hence this is where the order comes from.

Note that if we had defined $\langle a, b \rangle = \{\{a\}, \{b\}\}$, then in fact you would always have $\langle a, b \rangle = \langle b, a \rangle$.

To be clear, the syntactic expression $\langle a, b \rangle$ does already have an ordering of some sort: namely, the order in which the symbols appear. It is true that this "order" drops sort of out of nowhere, but you wouldn't disagree with it otherwise; for example, you wouldn't complain that $\exists x\forall y (y \notin x)$ means something different from $\forall y \exists x(y \notin x)$, and especially that it means something different from "$x\forall \exists )\notin xy (y$".

Answer 3

The definition of ordered pair is based on set theory. With this skew definition, $$(x,y) := \{\{x\},\{x,y\}\}$$ we can infer that $(x,y) = (u,v)$ (as sets) is logically equivalent to $x=u$ and $y=v$. So order matters here.