Difference between cardinal and ordinal

Question

I think this question has already been asked, but I do not find any precise trace of it. I do not get the difference between cardinal and ordinal numbers: why aren't they both matching the equivalence classes under the relation "being in bijection with"?

Answer 1

This is not easy to understand, because in finite sets there is not much of a difference. The idea is that for ordinal numbers you also have a relation between the numbers, so you can put them in line. Let's take as an example the numbers $$ 0,\frac{1}{2},\frac{3}{4},\frac{7}{8},...,\frac{2^n-1}{2^n},...,1. $$ Now this set has the same cardinality as $\mathbb{N}$ (because there is a bijection) but if you wanted to find a bijection $\Phi$ to the natural numbers preserving the order of numbers, this would not be possible. Preserving order means $ x < y \Longrightarrow \Phi(x) < \Phi(y)$. It's not possible, as you have the biggest element 1, which would need to be bigger than all natural numbers (but it's not obvious). In a way the ordinal numbers are a finer tool to describe how big a set is. For example $ 2 \omega + 5$ has the same ordinal number as $$ 0,\frac{1}{2},\frac{3}{4},\frac{7}{8},...,\frac{2^n-1}{2^n},...,1,1+\frac{1}{2},...,1+\frac{2^n-1}{2^n},...,2,3,4,5,6 $$ has the same ordinal number as $$ \frac{7}{8},...,\frac{2^n-1}{2^n},...,1,1+\frac{1}{2},...,1+\frac{2^n-1}{2^n},...,2,3,4,5,6 $$ because it has to "limit-dents" and afterwards 5 elements.

Answer 2

If I need numbers to count "how many items do I have" then I use cardinal numbers. If I need to make a list, or a enumeration, I use ordinal.

For finite sets there is no difference. If you have cardinality $3$, you can have the list: $[a,b,c]$ or $[a,c,b]$ or $[b,c,a]$ or ... but they have the same structure, so there is only one ordinal corresponding to the cardinal $3$.

If you have the set $\mathbb{N}=\{0,1,2...\}$ and you add one item $x$, the amount of item is the same as before, because is again $\aleph_0$. However, if you are working with list things are different.

You can decide that $x$ is between $2$ and $3$, in this case the new list $0,1,2,x,3,4,...$ is essentially the same as the original $\mathbb{N}$. You can decide that $x$ is bigger than any previous number, in this case the list $0,1,2,3,...,x$ is different from $\mathbb{N}$. For example, the new list has a maximum, $\mathbb{N}$ is unbounded.

So there are essentially different way to put an order over a infinite set, so you can't collapse concept of ordinal and cardinal.