Intuition of Ordinal Number

Question

I am learning the concept of ordinal numbers.

In the book of Set Theory and Metric Spaces by I. Kaplansky (Sec. 3.2, pg. 55), the author states

We attach to every well-ordered set an ordinal number; two well-ordered sets are awarded the same ordinal number if and only if they are order-isomorphic.

In Wiki, it states that

A natural number (which, in this context, includes the number 0) can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. When restricted to finite sets, these two concepts coincide. When dealing with infinite sets, however, one has to distinguish between the notion of size, which leads to cardinal numbers, and the notion of position, which leads to the ordinal numbers described here.

My first question is then: according to Wiki, an ordinal number should describe the position of an element in a set. Then why the ordinal number is attached to a set? Shouldn't it be attached to an element of a set?

Moreover, I am also confused by the meaning of "set A has a larger ordinal than set B". I mean, for cardinal numbers, I understand that set A has a larger cardinal than B generally means A has a larger size than B. For example, $A=\{0,1,2\}$ and $B=\{0\}$. But how can we say the "position" or the "ordering" of A is larger than that of B?

It will be appreciated if you can explain with some concrete examples. I have checked the answers here and here, but found no help.

Answer 1

There are a few results and definitions which might be helpful to know about, when trying to understand Ordinals.

I won't be proving these results, but they are usually proven in any Introduction to Axiomatic Set Theory Textbook. ( Enderton,Halmos etc..)

Theorem:

Every Well-Ordering (X,R) is Isomorphic to a unique ordinal. We call such an ordinal the type(X,R).

Definition (Predecessor)

Let (X,R) be a well-order. Let x$\in$X, then

pred(x,X,R) = {y$\in$X| (y,x)$\in$R}

Definition: (Initial Segment of a well-order)

An initial segment of a well-order (X,R) is

(pred(x,X,R),R) for some x$\in$X

If pred(x,X,R) $\neq$ X, such an initial segment is called a proper initial segment.

Theorem: No Well-Order is isomorphic to any of its proper initial segments.

We also have that (pred(x,X,R),R) is a well-order, so by our earlier theorem is is isomorphic to some ordinal. Even more, it is known that the ordinal it is isomorphic to is an element of the ordinal that (X,R) is isomorphic to.

That is the sense in which Ordinals "measure" the length of well-orderings.

We can think of well-orders as lines, and ordinals measure the length of the line, initial segments of the line, have a length measured by an ordinal which is an element of the ordinal which measured the whole line.

Your second question is about why do we associate a well-order with a type if ordinals are supposed to measure the order of the sets elements?

To answer that we need to define what it means for a well-order to have type(X,R) = $\alpha$.

Such an $\alpha$ is the unique ordinal, s.t. there is an isomorphism f:(X,R)≅($\alpha$,$\in$)

By definition, we have that f is a bijection between X and $\alpha$ and we have that for any a,b$\in$X(aRb $\iff$ f(a)$\in$f(b))

It is in this sense, that ordinals give an "order" to the well-ordering.

This property means that the R-least element is associated with $\emptyset$, the second R-least element is associated with 1, and so on.

Therefore, we can associated each element in X with some ordinal which denotes its "order/position" in the well-ordering.