Total Order Definition Motivation

Question

A Relation $R$ is a strict total order on $A$ if the following are true for all $x,y \in A$:

$(i) \hspace{.75in}\neg xRx \hspace{.75in} \text{(irreflexive)}\\ (ii) \hspace{.30in} xRy \wedge yRz \rightarrow xRz \hspace{.25in} \text{(transitive)}\\ (iii) \hspace{.25in} xRy \vee yRx \vee x=y \hspace{.25in} \text{(connex).}$

A set $A$ strictly totally ordered by $R$ is sometimes said to be a linearly ordered set, or $R$ is sometimes said to be a linear order relation. My question is one that has to do with intuition for the definition.

How do we know that these requirements $(i)$-$(iii)$ are sufficient to describe a set that can be ordered linearly? (This can mean intuitively, that the points can be placed into a line in which points to the right of $x$ satisfy $xRy$ and points to the left of $x$ satisfy $yRx$. However within $ZFC$, this line is not an object, only an intuitive tool to help motive our definition)

I'm able to show graphically, for a finite set of points $T$, that given a total order relation $R$ on $T$, the points can be ordered into a line. However, I don't know how to extend this reasoning to infinite sets. How do we know that the total order definition is sufficient to intuitively define a linear order for infinite sets?

Answer 1

It's a definition abstracted from linear orders we know from "real life", like the order on the integers, or the rationals, or the reals (and their subsets). We never need any property of an order that is not one of these axioms. These are the essential ones we use in all proofs that work in all linearly ordered sets )this is just based on experience of the people doing those proofs).

You could add other axioms. But you'd have to argue that there is some intuitively obvious property of all linear orders we can construct, that we cannot prove without this newly proposed axiom.

What usually happens is that some mathematician suggests axioms for some structure, shows that they hold in the "canonical examples" of such a structure, and derives all sorts of usual facts from these axioms alone. If these proofs are intuitive and easy to follow, the axioms are intuitive etc. they will get accepted by the community.

A linear order should "order" any pair so given $x \neq y$ you you should know whether $xRy$ or $yRx$ holds (which one is "smaller"). This is the third axiom. In this instance we want a strict order, so no point can be smaller than itself (axiom 1) and if $x$ is smaller than $y$ and $y$ is smaller than $z$ then $x$ should also be smaller than $z$ (a partial order and an equivalence relation have to obey the same condition). A sort of consistency. You cannot reasonably ask for a maximum element (though some orders have one, all finite ones e.g.) as the integers (which we want to be a linearly ordered set) does not have one. It would make the theory of linear orders too specific or narrow. Etc. etc.

Answer 2

How do we know that the total order definition is sufficient to intuitively define a linear order for infinite sets?

This question doesn't really make sense. An "intuition" is inherently an imprecise idea. Your intuition for something might be different from mine. The entire point of a mathematical definition is to be a precise idea where everyone can agree on what it means. Mathematical definitions are often inspired by intuitive ideas, but this is just inspiration, nothing more.

So, does the mathematical definition of a total order match your intuitive idea of a "linear order"? Maybe, maybe not. For instance, if your idea of a "line" is connected to the geometric idea of $\mathbb{R}$, you might think the definition is inadequate because not every totally ordered set can embed in $\mathbb{R}$. But the purpose of the definition of a total order is not to exactly capture your intuitive idea of a linear order. The purpose is to be a useful, precise definition that can be used to discover, formulate, and prove useful theorems. If, for some purposes, you find this definition inadequate at capturing your intuition, that is totally fine and not in any way a problem with the definition. It just means you may want to use some different definition for your purposes (and you should call it something other than "total order", because everyone else has agreed on what "total order" means and you don't want to confuse them).