The idea that one infinity can be larger than another seems to defy common sense, yet it is a foundational concept in modern mathematics. For finite collections, comparing size is straightforward. However, when dealing with infinite sets, the traditional method of counting elements breaks down, requiring a more abstract definition of “size.” This mathematical framework, developed in the late 19th century, reveals a surprising hierarchy where not all infinities are equal. The difference lies in how densely the elements of a set are packed together, leading to distinct levels of infinity.
How Mathematicians Compare Infinite Sets
Mathematicians use the concept of cardinality to define the size of any set, whether finite or infinite. Comparing cardinality relies on establishing a one-to-one correspondence, also known as a bijection. This process involves pairing every element in one set with exactly one element in the other, ensuring no element is left over in either collection. If such a perfect pairing can be made, the two sets have the same cardinality, meaning they are the same size.
For example, one can compare a set of chairs to a set of people by having each person sit in a chair. If every person is seated and every chair is occupied, the two sets are the same size. This pairing method extends to infinite sets, where it often yields counter-intuitive results. An infinite set is considered “larger” than another if a one-to-one correspondence can be established between the smaller set and a subset of the larger set, but a complete pairing between the two entire sets is impossible.
The Countable Infinity
The smallest type of infinity is known as countable infinity, which is the size of the set of Natural Numbers ($\mathbb{N}$), the familiar counting numbers $\{1, 2, 3, 4, \dots\}$. Any set that can be put into a one-to-one correspondence with the Natural Numbers is called a countably infinite set. This means its elements can theoretically be listed in an ordered sequence, like an endless line of items. The cardinality of this set is denoted by the symbol $\aleph_0$ (aleph-null).
Surprisingly, sets that appear much larger than the Natural Numbers also share this size. The set of all Integers ($\mathbb{Z}$), which includes negative numbers and zero, is countably infinite. A pairing can be created by alternating between positive and negative numbers: $1 \to 0$, $2 \to 1$, $3 \to -1$, $4 \to 2$, $5 \to -2$, and so on. Even the set of Rational Numbers ($\mathbb{Q}$), which are all numbers that can be expressed as a fraction, is also countably infinite. This is demonstrated by a method of listing all possible fractions, proving that the Rational Numbers can be counted despite their density on the number line.
The Uncountable Infinity of Real Numbers
The existence of a larger infinity is demonstrated by the set of Real Numbers ($\mathbb{R}$). This set includes all numbers that can be represented on a continuous number line, encompassing both rational and irrational numbers (like $\sqrt{2}$ and $\pi$). Irrational numbers have non-repeating, non-terminating decimal expansions. The Real Numbers are proven to be a strictly larger infinity because they cannot be put into a one-to-one correspondence with the Natural Numbers. This larger size is established through Cantor’s Diagonal Argument.
The argument begins by assuming, for contradiction, that a complete list of all Real Numbers between 0 and 1 is possible. Each number on this hypothetical list is written as an infinite decimal. A new real number is then constructed by looking at the digits along the diagonal of this list.
The first digit of the new number is chosen to be different from the first digit of the first number on the list. The second digit is different from the second digit of the second number, and so forth. This newly constructed number is guaranteed to be a Real Number between 0 and 1, but it cannot be on the original list.
Since the new number differs from every number on the assumed complete list, the initial assumption that all Real Numbers could be listed must be false. This proves that the set of Real Numbers is uncountable, possessing a cardinality strictly greater than $\aleph_0$.
The Infinite Ladder of Infinities
The discovery that the Real Numbers represent a larger infinity than the Natural Numbers is not the end of the hierarchy. Mathematicians have shown that this process of finding larger infinities can be repeated indefinitely. This is formalized by Cantor’s Theorem, which states that for any set, the collection of all its possible subsets, called the Power Set, is always strictly larger than the original set. Applying this theorem repeatedly generates increasingly larger infinities.
This leads to an endless sequence of ever-increasing infinite sizes, denoted by the Aleph Numbers: $\aleph_0$, $\aleph_1$, $\aleph_2$, and so on. The cardinality of the Real Numbers is known to be equal to the cardinality of the power set of the Natural Numbers. The question of whether the cardinality of the Real Numbers is the next size of infinity, $\aleph_1$, is known as the Continuum Hypothesis. This hypothesis posits that there is no set with a size strictly between the countable infinity of the Natural Numbers and the uncountable infinity of the Real Numbers. This question was proven to be independent of the standard axioms of set theory, meaning it can neither be proven true nor false within that system.