How Can Infinity Come in Many Sizes?

Intuition breaks down once we’re dealing with the endless. To begin with: Some infinities are bigger than others.

Mark Belan
Jordana Cepelewicz
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Published February 23, 2026

I

nfinity invites resistance. Aristotle rejected the existence of the infinite entirely; to him, infinity was simply a limit that could never be reached, not a true mathematical entity. In the early 17th century, Galileo wrote that typical ways of thinking about sets and numbers held no meaning in the realm of the infinite, and that mathematicians would only find paradoxes if they tried to apply their usual tool kit to it. And when, 200 years later, Georg Cantor formalized the idea that infinity comes in many sizes, he was met with anger and fear. His colleagues dismissed his work as that of a madman.

But in time, Cantor’s work on sets and infinity would form the bedrock of modern mathematics. As David Hilbert, another mathematical great, later wrote: “No one shall expel us from the paradise that Cantor has created for us.”

So how can infinity have different sizes?

Welcome to Cantor’s paradise.

Part 1

What Does It Mean To Count?

Mathematicians count using sets.

A set is any collection of objects, and the number of objects in a set is its size, or “cardinality.”

When we count objects in a set, we’re actually matching the natural numbers (1, 2, 3, and so on) to each object.

When mathematicians apply this method of counting to infinite sets, things get interesting.

Part 2

Counting to Infinity

The set of natural numbers …

… appears to be twice the size of the set of even numbers.

After all, the set of natural numbers contains both all the evens and all the odds.

But here’s why this intuition is wrong.

Start writing down the numbers in each set.

You can match each natural number to each even number.

Conclusion: This complete matching means that both sets are the same size.

Any set that matches one-to-one with the natural numbers is “countably” infinite. These sets are the smallest kind of infinity.

Part 3

A More Complicated Count

The set of natural numbers …

… seems much smaller than the set of rational numbers, or fractions.

After all, there are infinitely many rationals just between the numbers zero and 1.

Let’s compare them.

First, arrange the set of rational numbers in a grid. The top row is just every natural number, written as a fraction: 1/1, 2/1, 3/1, and so on.

In the next row, add 1 to every denominator, giving you 1/2, 2/2, 3/2, and so on. Repeat this step to get infinitely many rows, each infinitely long.

Let’s try to match this set to the natural numbers. If you match each natural number with a number in the first row, you’ll never get to the second row.

But there is a way of drawing a line through the grid that hits every number. Just move through the numbers along a snaking path.

Now pair each natural number with a rational number as it appears along this path. When you hit a number you’ve already seen (such as 2/2, which is equivalent to 1/1), skip it. In this way, you’ll match every natural number to every rational number.

Conclusion: Once again, the sets are the same size.

Yet Cantor showed that there are bigger infinities — “uncountably” infinite sets that can’t be matched to the natural numbers.

Part 4

A Bigger Infinity

The set of natural numbers …

... seems much smaller than the set of real numbers, which includes all fractions as well as numbers like √2 and π.

Let’s see what happens when we compare them.

Let’s assume that as with the previous examples, you can match every real number to a natural number, with none left over.

We’re going to prove that this will never work.

Examine your list of all real numbers. Now use it to build a new number. Take the first digit of the first number in the list and add 1 to it. This will be the first digit of your new number. To get the second digit, add 1 to the second digit of the list’s second number. Keep doing this down the list, forever.

The new number you’ve generated can’t be in your original list. Its first digit is different from that of the first number, its second digit is different from that of the second number, and so on. Our original assumption — that we could write down a list of all real numbers and match them to the natural numbers — was false.

Conclusion: The set of real numbers must be larger than the set of natural numbers.

The real numbers are uncountably infinite.

Part 5

Real Connections

The set of real numbers between zero and 1 …

… seems as if it should be half as large as the set of real numbers between zero and 2.

This feels like our first example, where there seemed to be twice as many natural numbers as even numbers, but both sets ended up having the same cardinality.

Is the same true here?

Choose any number in your first set — say, 0.6. Match it to the number that’s twice its size (in this case, 1.2) in the second set.

By doing this for every number in the first set, you’ll get a one-to-one match between the sets.

Conclusion: The sets are the same size.

In fact, the set of all real numbers between zero and 1 has the same size as the set of all real numbers, period. Any portion of the real number line has the same size.

Epilogue: Cantor’s Paradise

These are just a few of the ways that infinite sets can defy our intuition. Infinite sets that seem to be different sizes might actually have the same cardinality — or one infinite set might really be larger than another. And we only showed two possible sizes of infinity. There are infinitely many.

By proving this, Cantor called into question what mathematicians thought they knew. This led to new areas of study, and to a reckoning in the field — about what math was capable of, and what math even was.

Today, mathematicians continue to explore Cantor’s paradise to test math’s limits. They’re no longer so afraid of the inhabitants of this wild, paradoxical kingdom.