While often conceived as a single, boundless concept, infinity in mathematics presents a more nuanced reality. Mathematicians have discovered that infinity itself comes in different “sizes,” revealing a complex landscape beyond a singular, all-encompassing notion. Understanding these varying magnitudes changes our perception of what it means for something to be infinite.
Understanding Infinite Sets and Cardinality
To grasp the different sizes of infinity, it is helpful to first understand mathematical “sets” and “cardinality.” A set is a collection of distinct objects, such as numbers, letters, or even ideas. Cardinality is the way to measure the “size” of a set, indicating the number of elements it contains. For finite sets, cardinality is simply the count of elements.
Comparing the sizes of infinite sets requires a different approach than simple counting. Mathematicians use one-to-one correspondence, also known as bijection. This involves pairing each element from one set with exactly one element from another. If such a pairing is possible, the two sets have the same cardinality, meaning they are the same “size.” This principle extends to infinite sets, allowing for their comparison.
Countable Infinities
The smallest type of infinity is known as “countable infinity.” A set is countably infinite if its elements can be put into a one-to-one correspondence with the set of natural numbers (1, 2, 3, and so on). This means every element in the set can be assigned a unique natural number. The set of natural numbers itself is a primary example.
Other sets, seemingly larger, also possess this same “size” of infinity. The set of all integers (positive, negative, and zero) is countably infinite. This can be shown by devising a systematic way to list them, such as 0, 1, -1, 2, -2, and so forth, effectively pairing each integer with a natural number. Similarly, the set of rational numbers (numbers that can be expressed as a fraction) is also countably infinite. A method exists to list them without missing any, confirming their one-to-one correspondence with the natural numbers.
The Realm of Uncountable Infinities
Beyond countable infinities lies a category of “larger” infinities known as uncountable infinities. These are sets whose elements cannot be put into a one-to-one correspondence with the natural numbers. This implies that no matter how one tries to list the elements of an uncountable set, elements will always be left out, showing they are larger than countable sets.
The most prominent example of an uncountably infinite set is the set of real numbers. Real numbers encompass all numbers on the number line, including integers, rational numbers, and irrational numbers like pi (π) and the square root of 2. While rational numbers are countable, the addition of irrational numbers makes the entire set of real numbers uncountable. The density and continuous nature of the real number line prevent any complete listing by pairing them with natural numbers.
Proving “Bigger”: Cantor’s Revolutionary Insight
The insight that some infinities are larger than others came from mathematician Georg Cantor in the late 19th century. His most famous method for demonstrating this difference is Cantor’s diagonal argument. This argument provides a proof by contradiction, showing that assuming countability for certain infinite sets leads to an impossibility. The diagonal argument illustrates that the set of real numbers, even just those between 0 and 1, is uncountable.
To understand the argument, imagine trying to list all real numbers between 0 and 1 in an infinite sequence, assigning a natural number to each. Each real number in this list would have an infinite decimal expansion, like 0.12345… or 0.98765…. Cantor’s argument constructs a new real number that is guaranteed not to be on this list. This is done by looking at the first digit of the first number, the second digit of the second, and so on, forming a “diagonal” of digits.
For each digit on this diagonal, a new digit is chosen for the new number that is different from the original (e.g., if the diagonal digit is 5, choose 6; if it’s not 5, choose 5). This process creates a new real number whose first decimal digit differs from the first number on the list, its second differs from the second, and so forth.
Because this new number differs from every number on the list in at least one decimal place, it cannot be found anywhere on the original list. This contradiction proves that the initial assumption—that all real numbers between 0 and 1 could be listed—must be false. Therefore, the set of real numbers is uncountable, meaning it is a larger infinity than the set of natural numbers.
The Hierarchy of Infinities
The existence of uncountable infinities reveals a hierarchy of increasingly larger infinities. Mathematicians use “aleph numbers” to categorize these different sizes of infinite sets. The smallest infinite cardinality, that of the natural numbers (and all countable sets), is denoted as aleph-null (ℵ₀). The cardinality of the real numbers, a larger infinity, is often referred to as the cardinality of the continuum, and is represented by 2^ℵ₀.
While the exact relationship between 2^ℵ₀ and the next aleph number, aleph-one (ℵ₁), is complex and subject to ongoing mathematical research (known as the continuum hypothesis), the principle remains that larger infinities exist.
For any given infinite set, it is always possible to construct an even larger infinite set by considering its “power set.” The power set of a set is the set of all its possible subsets. For example, if a set has ‘n’ elements, its power set has 2^n elements, which is always larger than ‘n’. This concept extends to infinite sets, meaning the power set of an infinite set will always have a strictly larger cardinality than the original set. This ability to continually generate larger infinities demonstrates that the question “what is bigger than infinity?” has an endless answer.