I Thought a Cardinal Was a Bird: Exploring the Interactions Between Cardinal and Ordinal Numbers

Shaafi Flener

In class, we discussed what cardinality of a set was: a measure of how large or small that set is. But that begins to raise the question, how do we quantify a set that has infinite elements, or, honestly, any set beyond just assigning a number to it based on the number of elements? In this blog post, I will be showing the difference between cardinal and ordinal numbers, and how they might interact.

Cardinal numbers

First, we look at cardinal numbers. These represent the size or quantity of elements in a set. They answer the question, “How many?” We already know that cardinality defines the size of a set, for both finite and infinite sets.. For finite sets, cardinal numbers are standard natural numbers (0, 1, 2, 3, …). The cardinality of a set $A$ is denoted as $|A|$ .

But what about infinite sets? As we showed through Cantor’s Theorem, we know that infinite sets can have different sizes. The smallest infinite set, the set of natural numbers ( $\mathbb{N}$ ), has an infinite cardinality denoted as $\aleph_0$ (aleph-null). Other infinite sets, such as the set of real numbers ( $\mathbb{R}$ ), have a greater cardinality, represented as $\aleph_1$ (aleph-one).

Ordinal numbers

While the idea of a cardinal number is a basic one, the ordinal numbers take the notion of a number a step further. A set might have $|A| = 3$ , but what does that mean in terms of the elements contained within?

Ordinal numbers establish the position or order of elements within a set. While cardinal numbers answer the question “how many?” ordinal numbers ask the question, “in what order?” Ordinality characterizes the arrangement of elements and is denoted by an ordinal notation, such as 1st, 2nd, 3rd, etc. Unlike cardinal numbers, ordinal numbers reflect a hierarchical structure.

In set theory, ordinal numbers are constructed based on the idea of well-ordering. A well-ordered set is one where every non-empty subset has a least element. The smallest well-ordered set is the empty set, denoted as $\varnothing$ . The first non-empty well-ordered set is $\{ \varnothing \}$ , the second is $\{ \varnothing, \{ \varnothing \}\}$ , the third is $\{ \varnothing, \{ \varnothing \}, \{ \varnothing, \{ \varnothing \}\}$ , and so on. As a general rule, the set corresponding to the ordinal number $n + 1$ is equivalent to the set $A \cup \{A\}$ where the set $A$ corresponds to the ordinal number n.

Ordinal numbers continue beyond finite sets, extending into the realm of “transfinite” numbers, which each denote some sort of infinite set. The smallest transfinite ordinal is $\omega$ (omega), representing the order type of the natural numbers. Beyond that, we have $\omega+1$ , $\omega+2$ , and so on, reflecting the order types of the natural numbers followed by one, two, and so forth. This is a little more complex to think about, but the general gist comes from the idea that the set corresponding to the ordinal number $\omega+1$ is composed of all values within ω and then an additional element where it is not contained within the set of $\omega$ . Other notable ordinal numbers include $\epsilon_0$ (epsilon-null), the order type of the positive integers, and $\omega_1$ (omega-one), representing the order type of the countable ordinals. These different symbols are pretty arbitrary, but the idea remains, that they represent different orderings.

Distinction between cardinal and ordinal numbers

Now that we have developed a basic idea of what cardinal and ordinal numbers are, I introduce a lemma:

Lemma: (Well-Ordering Principle). Let $A$ be a set. There is a binary relation $\leq_{A}$ on $A$ such that $(A,\le{_A})$ is well-0rdered.

Now, that is a lot of math jargon. To put it in more plain terms, this is equivalent to the Axiom of Choice that we saw in class, stating that for all sets, there must be some function that roams the entirety of it. However, this takes it a step further, and expresses that that set must also have a least element! Based on this lemma, we have a much more interesting proposition, where we first fix a set $A$ , and then have from the lemma that the following set is non-empty:

$O(A) = \{\alpha \in \text{ordinal numbers} |\alpha \cong (A,\leq_{A}) \text{ for some well-ordering} \leq_{A} \text{of } A\}$ .

From here, we define the cardinality of a set $A$ as:

$\mathrm{Card}(A) = \text{the least element of}~O(A)$ .

This is just a way of saying that the cardinality of every set can be expressed as an ordinal which represents the shortest possible well-ordering of the set.

Based on these two ideas and the formerly shown lemma – which we will not prove right here due to how it is more complex to prove – we can construct a new lemma relating ordinals and cardinals:

Lemma: For any sets $A,$ and $B$ , $|A| = |B|$ if and only if $\mathrm{Card}(A) = \mathrm{Card}(B)$ . Furthermore, $|A| \leq |B|$ if and only if $\mathrm{Card}(A) \leq \mathrm{Card}(B)$ .

This lemma allows for some amazing takeaways including the fact that for every cardinal number, there exists an ordinal number that is the same! On its own this is a unique concept which can be extended further to looking at well-0rdering in finite sets, but going another direction, we can create values for the cardinality of infinite numbers. We let $\aleph_0$ (a cardinal number) be equal to $\omega$ which we have already described as the least infinite cardinal. Then from the ordinal definition we had earlier, we can continue to define larger and larger infinite sets, allowing us to tackle a whole new realm of mathematics.

As you can see, this one lemma based on the interactions between cardinal and ordinal numbers can open up the realm to a ton of other different theorems and ideas. As I have introduced these ideas to you all, I hope it inspires to explore more the ideas around ordinals and cardinality, and go off on your math journeys!

Sources

Solomon, Reed. Notes on ordinals and cardinals

I Thought a Cardinal Was a Bird: Exploring the Interactions Between Cardinal and Ordinal Numbers

Cardinal numbers

Ordinal numbers

Distinction between cardinal and ordinal numbers

Sources

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