Counting and Ordering of Numbers

Question

Are there differences between 'counting' and 'ordering'? As such, the whole of rational number is countable, or they order-able too? In what cases counting and ordering are same or not?

Answer 1

"Counting" means finding the cardinality; "ordering" means establishing an order. Certainly, there is a difference: for a simple example, while there is only one possible answer to "How many elements does the set $\{x,y\}$, with $x\neq y$, have?" (namely, "2"), there are at least two different answers to "What is an ordering of the set $\{x,y\}$, with $x\neq y$?" ( either "$x$ smaller than $y$" or "$y$ smaller than $x$"; if you admit partial orders, then there is even a third possible answer, "$x$ and $y$ incomparable").

Now, the rational numbers are already ordered, in their "usual ordering" as a subset of the reals. But that is not what you really mean, I think. Rather, you want to think about a specific "counting" of the rationals (that is, a bijection to the nonnegative integers) as providing a way to "order" the rationals (or any other set that is "countable"; i.e., bijectable with $\mathbb{N}$).

Yes: if you have a set $X$, and you know that $X$ is countable (there exists a bijection $f\colon X\to\mathbb{N}$), then you can use the bijection to endow $X$ with a "total ordering" (in fact, a well ordering): first, pick a bijection $f\colon X\to\mathbb{N}$; then, given $x$ and $y$ in $X$, define $x\preceq y$ if and only if $f(x)\leq f(y)$ (the latter being the "usual" ordering of the natural numbers). But this ordering is "non-canonical": if you choose a different bijection $f$ (and there are many, many bijections if there is at least one) you get a different order.

Answer 2

All infinite countable sets have cardinality $\aleph_0$ (which "counts" them) and can be put into one-to-one correspondence with any countable infinite ordinal, the smallest being $\omega$ (which "orders" them), so there is no difference in this case. If you assume the Axiom of Choice, a more general property is true for all sets: the Well-ordering principle. The rational numbers are countable and can be well-ordered by $\omega$: imagine laying them out in a table (according to each numerator and denominator), and then zig-zagging diagonally starting at the corner.

Cantor's diagonal argument explains why the reals are uncountable. The Well-ordering principle says we can well-order the reals by some uncountable ordinal, but it doesn't say which one. If you assume the Continuum Hypothesis, then we can use the first uncountable ordinal to order them. But even with all of these assumptions, it is impossible to visualize the correspondence in the same way as the case of countable ordinals. Some countable ordinals can be difficult to visualize too, such as $\epsilon_0$.