What Is Infinity? And Why Are There Different Sizes of Infinity?

There are exactly as many even numbers as there are whole numbers, even though the whole numbers include all the evens plus all the odds. And yet there are more numbers between 0 and 1 than there are whole numbers in total.

Both statements are proven mathematics, not wordplay. They come from Georg Cantor, a German mathematician who, in the 1870s, did what philosophers had considered impossible: he learned how to compare the sizes of infinite sets.

"Part" Can Be as Big as "Whole"

In everyday life, a part is always smaller than the whole. Take away some apples from a basket and you have fewer apples.

Infinite sets break that rule. In 1638, Galileo Galilei noticed that every whole number has a square:

1 → 1
2 → 4
3 → 9
4 → 16
…

The squares are a small part of the whole numbers, and they get rarer as you go. Yet the list pairs every number with exactly one square, with nothing left over on either side. Galileo concluded that "equal," "greater" and "less" simply don't apply to infinite quantities. This puzzle is now called Galileo's paradox. Browse the list of square numbers to see how sparse they become.

Two Kinds of Infinity in Philosophy

For over two thousand years, most thinkers followed Aristotle, who distinguished:

• Potential infinity: a process that never ends, like counting 1, 2, 3, … forever
• Actual infinity: a completed infinite collection, like "the set of all whole numbers" as a single object

Aristotle accepted the first and rejected the second. Calculus used potential infinity for centuries through limits. Cantor's radical move was to treat actual infinite sets as real mathematical objects, with sizes you could compare.

Cantor's Idea: Measure by Pairing

How do you know two finite groups are the same size without counting? Pair them up. If every chair has exactly one person and every person has a chair, there are as many people as chairs.

Cantor applied this to infinite sets. Two sets have the same cardinality (size) if there's a one-to-one correspondence between them: a pairing that uses every element of both sets exactly once.

By that definition, the whole numbers and the even numbers are the same size:

1 ↔ 2
2 ↔ 4
3 ↔ 6
n ↔ 2n

Any set that can be paired with the whole numbers is called countably infinite. Its size is written ℵ₀ ("aleph-null"), using the first letter of the Hebrew alphabet.

Surprisingly Countable Sets

Many sets that look bigger turn out to be countable too:

• All integers: list them as 0, 1, −1, 2, −2, 3, −3, …
• All fractions: arrange them in a grid with numerator on one axis and denominator on the other, then walk through the grid diagonally, skipping duplicates

Even though there are infinitely many fractions between any two whole numbers, there are no more fractions than whole numbers. At this point you might suspect every infinite set is the same size. Cantor proved otherwise.

The Diagonal Argument: A Bigger Infinity

In 1891, Cantor published his famous diagonal argument, showing the real numbers between 0 and 1 can't be listed.

Suppose someone claims to have a complete list:

1st: 0.5 1 3 8 2 …
2nd: 0.1 4 1 5 9 …
3rd: 0.7 1 8 2 8 …
4th: 0.3 3 3 3 3 …
5th: 0.9 0 0 0 1 …

Build a new number by going down the diagonal and changing each digit. Take the 1st digit of the 1st number, the 2nd digit of the 2nd number, and so on, and replace each one (say, with 5, or with 4 if it's already 5):

Diagonal digits:  5, 4, 8, 3, 1
New number:       0.4 5 5 5 5 …

The new number differs from the 1st number in its 1st digit, from the 2nd number in its 2nd digit, and from the nth number in its nth digit. So it's not on the list anywhere. (Using only digits 4 and 5 also avoids the 0.999... = 1 issue of numbers with two decimal forms.)

No matter what list you try, there's always a real number missing. The real numbers are uncountable, a strictly larger infinity than ℵ₀. Its size is called the cardinality of the continuum.

An Insider Reference: The Fight Over Cantor's Work

Cantor's ideas were bitterly opposed. Leopold Kronecker, one of the most influential mathematicians in Berlin, reportedly called Cantor a "corrupter of youth" and worked to block publication of his papers. Henri Poincaré was also skeptical of Cantor's set theory. Cantor struggled with severe depression and spent time in sanatoriums in his later years.

But the work endured. In 1926, David Hilbert wrote the line that became its defense: "No one shall expel us from the paradise that Cantor has created." Set theory is now the standard foundation of modern mathematics.

Two Concepts Worth Knowing

Cardinality

Cardinality is the size of a set. For finite sets it's the number of elements. For infinite sets it's defined by one-to-one correspondence, which is why ℵ₀ and the continuum are different sizes.

Bijection

A bijection is a pairing that's one-to-one (no element used twice) and onto (no element left out). It's the precise mathematical version of "pairing up."

Quick Answer: Are There Different Sizes of Infinity?

Yes. Two infinite sets have the same size if their elements can be paired one-to-one. The whole numbers, even numbers, integers and fractions are all the same size, called countable infinity. Cantor's diagonal argument proves the real numbers can't be paired this way, so they form a larger, uncountable infinity.

Try Them Yourself

• Square Numbers List: Galileo's paradox in a table
• Even Number Checker: pair each number n with 2n
• Counting Tables: the whole numbers, one by one
• Square Roots List: irrational numbers the list can't catch
• Number Theory Formulas: properties of the integers
• Unlocking the Secrets of Set Operations: the language Cantor built

Write down five decimals, then build a sixth that differs from each along the diagonal. You'll have reproduced the argument that changed mathematics.