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 is denoted as
.
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 (), has an infinite cardinality denoted as
(aleph-null). Other infinite sets, such as the set of real numbers (
), have a greater cardinality, represented as
(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.… Read the rest