understanding ordinals in set Theory

Question

This is a question about understanding ordinals. I’m very confused about what an ordinal is. In my notes it says an ordinal is a well ordered set where every element is its initial segment which is a bit confusing. I thought a set being well-ordered meant every subset had a minimum element and so by using this property we can sort of arrange the elements of the set in order from smallest to biggest by taking different subsets and finding the minimum element each time. If this is the case then what is an ordinal and how does its extra property make it differ to a well-ordered set? Is an ordinal a set in which every element has a determined position?

Once we know what an ordinal is, what do we do with them? What is their significance?

Answer 1

An important thing to understand is that there is no deep conceptual difference between ordinals and well-ordered sets. Ordinals are special examples of well-ordered sets, but any well-ordered set corresponds to (is isomorphic to) a unique ordinal, in a unique way.

In an ordinal $\alpha$, the comparison relation $x < y$ for $x,y\in \alpha$ is just the set element relation $x\in y$. In other words, if $x$ and $y$ are elements of an ordinal $\alpha$, they are themselves ordinals, and either $x\in y$ or $y\in x$, and that describes the order between ordinals. You can also say that an ordinal is always the set of all ordinals that come before.

But this is just a model of ordinals. It turns out that it is a convenient model, and everyone uses it nowadays, but you could maybe come up with another construction of a certain collection of well-ordered sets such that any well-ordered set is isomorphic to exactly one your collection. It does not matter very much, which is why everyone is happy to use the classical definition I discussed above.

The crucial property of ordinals is the following:

For any set of ordinals $S$, there is a "next" ordinal $\alpha_S$, which is strictly bigger than all the ordinals in $S$, and is the smallest with this property.

Any statement about ordinals can be deduced from that one. For example, start with $S=\emptyset$ (when you start, you haven't constructed any ordinal yet). Then there is a next ordinal, which we call $0$, and by definition it is the smallest ordinal. Now we can take $S=\{0\}$, which gives us a next ordinal, which we call $1$. Once we defined recursively all finite ordinals this way, we can take the set $S=\{0,1,2,\dots\}$ of all finite ordinals, and there is a next one, which we call $\omega$. Then comes $\omega+1$, etc.

Their significance is (among other things, this is just my personal take on ordinals, I am not a logician) that they encode all recursive behaviors. I think this is illustrated by the fundamental property I highlighted above: they naturally come up when you have some kind of "process" such that whatever steps you did, there is a "next step".

Answer 2

A set $x$ is said to be transitive iff

$$\forall y \forall z (z \in y \in x \rightarrow z \in x)$$

That is, every element of every element is an element.

A set $x$ is said to be hereditarily $\varphi$ iff $x$ is $\varphi$ and every element of $x$ is hereditarily $\varphi$.

An ordinal is a hereditarily well-founded, hereditarily transitive set.

We use ordinals to describe or represent the order-type of a well-ordered set. Thus ordinal operations have an interpretation in terms of order-type.

For example, $\beta \times \alpha$ is the order-type that consists of $\alpha$ copies of $\beta$: That is, the order-type of the Cartesian product $A \times B$ of a set $A$ of order-type $\alpha$ and a set $B$ of order-type $\beta$ under the lexicographic order

$$(a_1, b_1) \leq (a_2, b_2) \longleftrightarrow a_1 < a_2 \lor (a_1 = a_2 \land b_1 \leq b_2)$$

for all $a_1, a_2 \in A$ and $b_1, b_2 \in B$. (Notice the convention for ordinal multiplication is "backwards" from what one might expect. Just an annoying detail.)

Likewise, $\alpha + \beta$ is the order-type that consists of $\alpha$ followed by $\beta$: That is, the order-type of the disjoint union $A + B = (\{0\} \times A) \cup (\{1\} \times B)$ under the lexicographic order (as defined above).