Are numbers countable?

Let’s take the following set:
N = {0, 1, 2, 3, 4, 5, 6, . . . }

This is a set known as the natural numbers, and it represents all possible values that a set’s cardinality can have.

But what is its cardinality?

Think of the largest number you can, then multiply it by itself and then by itself again. . .

You really can’t determine the last element of this set, because it’s infinite. But the set still needs to have a cardinality associated with it.

And so, the cardinality of the natural numbers is what’s known as an infinite cardinal called aleph-zero or aleph-null.

150px-Aleph0.svg

It’s a handy concept to have access to, since many sets have the same cardinality.

More tomorrow. . .

Let’s talk about cardinality and correspondence

I’ve created the following two sets:
A = {0, 1, 5, 9}
B = {sugar, spice, snips, snails}

Which one is larger?

In terms of the elements it contains, they both have 4 elements, so they are the same size.

The number of elements in a set is called the cardinality of that set, and it’s written so: |A| = |B| = 4

But how could you compare them if you couldn’t count?

What you can do is take one element from A, and one from B, and put them aside. If you run out of elements at the same time, then they have the same cardinality. This is known as correspondence, a one-to-one comparison of elements in each set.

Think of very large finite sets, like all of the people of the world. You could potentially assign each of them a number, and there would be both one number per person and one person per number. In effect, there would be a correspondence between the two.

But what about numbers themselves? Are they countable?

Continued tomorrow. . .