How to Understand Infinity in Mathematics?

“Infinity” is not only a concept in mathematics but also widely used in physics and philosophy. If we divide the understanding of infinity by time, taking the German mathematician Georg Cantor (born in Russia but a German national) and his introduction of “set theory” as a milestone, there are two ways to understand “infinity”:

Infinity Before the Emergence of Set Theory:

Mathematicians and philosophers have had various interpretations of “infinity” throughout history. Different eras and thinkers offered diverse views—for example, Aristotle saw infinity as a never-ending process, while Plato regarded it as an ideal existence. Medieval mathematicians considered infinity to be a vague and difficult-to-handle concept. Even after the development of calculus and the introduction of infinitesimals and infinite quantities, infinity was treated mainly as a type of limit.

Infinity After the Emergence of Set Theory:

With Cantor’s introduction of set theory, the understanding of infinity underwent a fundamental transformation. Mathematicians began to clearly classify the structures of infinity, recognizing that infinity can come in different types and levels.

1. The Two Most Common Categories of Infinity: Countable and Uncountable

Countable Infinity

A set is countably infinite if its elements can be listed in a sequence according to a certain rule. For example, the set of integers is countably infinite.

Uncountable Infinity

A set is uncountably infinite if its elements cannot be listed in any fixed sequence. For instance, the set of real numbers is uncountably infinite.

2. Infinite Sets with Different Cardinalities: Infinities Can Be Larger or Smaller

Cantor introduced the concept of cardinality to measure the “size” of infinite sets. He denoted the smallest infinite cardinality by “aleph-null” (ℵ₀), which corresponds to the size of any countable infinite set. The larger the cardinality of a set, the “larger” the infinity. (This may sound counterintuitive—how can infinities differ in size?)

3. Infinities of Different Dimensions:

Examples include infinitesimals in quantum mechanics, point sets in Euclidean spaces, and fractal sets.

Since Cantor, “infinity” has become a fundamental mathematical tool playing an essential role across many disciplines.