How To Actually Count To Infinity? -  A deep-thinking stick figure is asking the following question in its head: "How do I count beyond infinity?" Below this bubble is seen the following word highlighted inside a square block: Infinity. Beside this block is a seal that says 'For Dummies - Approved' on it.

Welcome to infinity for dummies: a crash course on how to count to infinity. But before we even begin, isn’t infinity something uncountable by definition? If that is the case, how are we supposed to count to infinity?

Well, the answer to that question lies at the core of understanding the fundamental principles behind the notion of infinity. Before we count to infinity, we need to start with the concept of counting first.

So, this is the first challenge that I will be tackling in this essay. Once we get a grasp of how counting works, I will dive into basic set theory. This will be of immense help to us in extending what we know about counting from the realm of finite quantities to that of infinite quantities.

By the end of this essay, you would be able appreciate the beauty and complexity of infinity even if you are a non-technical person. Without any further ado, let us begin.

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The Magic of Counting

Most recently, I learnt that dogs can count. While that’s a cute fact, it joins a wealth of research that indicates that many life forms can count to a limited extent. So, we could argue that counting exists in nature and is not a mathematical invention.

Human beings might be significantly better than dogs at counting, but we are still not infinitely better (see what I did there?). Nonetheless, counting to infinity is not out of our reach. Well, it IS out of our reach if we approached the task in the traditional sense.

When we talk about counting, we typically think of listing positive whole numbers. A good example of this is when young children with closed fists start opening one finger at a time to “count” candies. This is not the approach we will be using to count to infinity (at least, not entirely).

There is another age-old counting method that is significantly less demanding in terms of cognitive load. Imagine a hypothetical situation where a music band is about to play in a dome. There are 100 seats in the dome.

The lead singer never spends any cognitive effort counting the audience as the people enter. But after a while, she notices that all the seats are filled. She instantly announces that there are 100 people in the crowd. How did she know that?

Well, she knew that there were 100 seats in the dome (someone counted them using the classic method). Using this information, she arrived at the total count of her audience without actually counting the people one by one.

To be precise, this method defers the cognitive load of counting to comparing. We do the counting of one set of quantities once (chairs) and leverage it to count another set of quantities using comparison (people).

Just to drive home the point, here is another example. A chef wishes to boil exactly one litre of water. He holds a glass jug under a running tap and closes the tap as soon as the water level in the jug reaches the marking that reads “1 L”.

Someone counted (technically, measured) and marked the jug for 1 litre of fluid. Using this knowledge, the chef achieves his goal of measuring/counting via visual comparison alone; he need not count water molecules.

This method of counting might seem trivial for finite numbers. But it becomes powerful when we shift to the realm of infinity. Now that we have established how counting works via comparison, let us proceed to the next piece of the puzzle.

The Magic of Set Theory

In the previous section, I mentioned that the counting of one set of quantities is done once and is leveraged to count another set of quantities via comparison. The key feature here is that we are talking about two sets!

In mathematics, a set is a collection of elements that follows the following simple rules:

1. Each element is unique (no duplicates).

2. The order of elements is irrelevant.

Examples of sets are the collection of all whole numbers, the collection of all living birds, etc. In mathematics, we compare two sets using a property known as cardinality, which refers to the total number of elements in any given set.

We say that two sets have the same cardinality if they have the same number of elements. In a more strict sense, two sets have the same cardinality if each element of one set can be uniquely mapped to an element of the other set.

Before things get too abstract, let us look at an example. Consider two rows of cards with the following numbers:

How To Actually Count To Infinity? — An illustration showing two sets of cards: A (on top) and B (on bottom) with the following numbers: Row A: 1, 2, 3, 4; and row B: 1.2, 2.2, 3.2, 4.2
Comparing two sets of cards — Illustration created by the author

Each of the elements in the upper row can be uniquely paired with one of the elements in the lower row. Here is one such pairing:

How To Actually Count To Infinity? — An illustration showing two sets of cards: A (on top) and B (on bottom) with the following numbers: Row A: 1, 2, 3, 4; and row B: 1.2, 2.2, 3.2, 4.2. Each element in the lower row is obtained by adding 0.2 to the corresponding element in the upper row.
Pairing upper-row elements with lower-row elements — Illustration created by the author

You can also observe that there are four pairs of mappings. This means that both these sets have the same cardinality of 4. In mathematics, this is known as one-to-one correspondence.

The notion of cardinality is so powerful that we can define counting as the act of finding the largest number in the equivalent cardinality set of consecutive whole numbers starting with 1.

If we find at least one element in one of the sets that doesn’t have a match in the other, we may say that the sets feature different cardinalities.

Now that we have established the fundamentals of set theory, we are ready to shift to the realm of infinity.

How to Count to Infinity?

This is the part where you would need to muster up some focus; it can get tricky. Do you remember how we established that two sets share the same cardinality? Let us now do the same exercise, but involving one infinite set.

Imagine two rows of cards, one of which (the upper row) holds infinite cards and another (the lower row) holds a billion cards:

How To Actually Count To Infinity? — An illustration showing two sets of cards: A (on top) and B (on bottom) with the following numbers: Row A: 1, 2, 3, 4, …upto infinity; Row B: 1, 2, 3, …. upto 1 billion.
Comparing an infinite set with a finite set — Illustration created by the author

When we try and map the elements from the upper row to the lower row, we will be able to match them uniquely up to one billion. All cards beyond the billionth card in the lower row will be left without matches.

On the one hand, this shows us that these sets have different cardinalities. On the other hand, this also shows us that infinite sets have different cardinalities as compared to finite sets.

Now, let us switch things a bit and imagine two infinite rows of cards with the following numbers:

How To Actually Count To Infinity? — An illustration showing two sets of cards: A (on top) and B (on bottom) with the following numbers: Row A: 1, 2, 3, 4, …up to infinity; Row B: 2, 3, 4, 5 …. up to infinity.
Comparing infinite sets — Illustration created by the author

Do you think these two sets have the same cardinalities? Typical intuition might say “no”, but the answer is “yes”. How can that be? We are missing a card with ‘1’ in the lower row after all. Does this not mean that the lower row will always have a cardinality of ‘1’ lesser than the upper row?

The reason that they have the same cardinality is that each of the upper-row elements can be uniquely mapped to a lower-row element. If ’n’ represents an upper-row element, the corresponding lower-row element is (n + 1).

Similarly, if ’n’ represents a lower row element, then the corresponding upper row element is (n − 1). To drive home this process, let us compare two more rows of infinite cards holding the following patterns of numbers:

How To Actually Count To Infinity? — An illustration showing two sets of cards: A (on top) and B (on bottom) with the following numbers: Row A: 1, 2, 3, 4, …up to infinity; Row B: 1/2, 1/4, 1/8, 1/16 …. up to infinity.
Comparing infinite sets (example 2)— Illustration created by the author

Let ‘x’ represent an upper row element and ‘y’ represent a lower row element. Then, the following relationship holds:

How To Actually Count To Infinity? — y = 1/(2^x) and x = log_base2(1/y)
Mapping the relationship between sets — Math illustrated by the author

This relationship ensures a one-to-one correspondence between the two sets. Therefore, we can be certain that they share the same cardinality.

The key point to note here is that even though we are not counting the elements of these infinite sets by listing them, we can compare them using the concept of cardinality.

At this point, you might think that all infinite sets share the same cardinality. But the really perplexing thing about infinity is that this is not the case (I’ll save this discussion for a future essay)!

Summary — How to Count to Infinity?

Here is a list of points summarising what we covered in this essay:

1. We can count not only by listing whole numbers, but also by comparing sets.

2. We can determine if two sets contain the same number of elements by matching the elements in one set with the elements in the other set.

3. This matching technique might be trivial for finite sets. But for infinite sets, it is a powerful concept.

4. When all the elements of one set match uniquely with the elements of another set, we say that they share the same cardinality.

5. The set theoretic concept of cardinality lets us compare infinite sets, even though we cannot count their elements by listing.

6. Not all infinite sets share the same cardinality.

The reason why not all infinite sets share the same cardinality is quite complex; it would be beyond the scope of this essay. So, as I had mentioned in the previous section, I plan to cover it separately in a future essay.

For now, I hope that you enjoyed reading this one!

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