Infinity is a quantity that extends beyond limit. The notion of infinity has applications in many areas of mathematics and logic. It forms a key idea in set theory. A set is a collection of elements, such as numbers or objects. The set {1, 2, 3}, for example, has a finite (limited) number of elements—just three. But the set of all natural numbers {1, 2, 3 …} is infinite. It goes on forever.
Ancient Greek philosophers, including Zeno and Aristotle, fiercely debated the idea of infinity. They thought that humans could not comprehend truly infinite quantities. In the early 1600’s, the Italian scientist Galileo noted an apparent paradox (contradiction) with infinite sets. He observed that every natural number {1, 2, 3, 4 …} could be matched with its square {1, 4, 9, 16 …}. The two sets thus appeared to be equally numerous, despite the fact that the set of natural numbers includes all the squares and many numbers that are not squares.
In the late 1800’s, the German mathematician Georg Cantor developed a rigorous mathematical theory of infinity. Embracing Galileo’s paradox, Cantor defined an infinite set as a set that can be matched one-to-one with a subset of itself. Cantor also showed that infinite sets do not necessarily have the same “size,” or cardinality. For example, the real numbers—_the set of points on the number line—have a greater cardinality than the natural numbers. The points on the number line include fractions as well as _irrational numbers, which cannot be written as fractions or as complete decimals.
See also Set theory ; Zeno of Elea .
