How is infinity used in mathematics?

In mathematics, infinity—represented by the symbol $\infty$—is a concept describing something endless or without bound. It is utilized in several fundamental ways:12

Calculus and Infinitesimals

Infinity is central to the development of calculus. Early mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz used infinitesimals—quantities smaller than any finite value but not zero—to define derivatives and slopes. While initially controversial, these concepts were later placed on a rigorous foundation through the study of limits and, in the 1960s, nonstandard analysis.34

Set Theory and Transfinite Numbers

Modern mathematics uses infinity to compare the sizes of infinite sets. Georg Cantor, the founder of set theory, proved that not all infinities are equal. He introduced transfinite numbers to classify these sizes, such as:53
* Aleph-null (${\aleph }_0$): The cardinality of the set of counting numbers.36
* Continuum ($c$): The larger cardinality of the real numbers.36

Cantor’s work demonstrated that infinite sets can be larger than others, leading to an endless hierarchy of cardinalities. The continuum hypothesis remains a famous, undecidable problem regarding the relationship between these specific infinite sizes.53