What Is Infinity, Actually? (beyond the pop-sci headlines)
For over 2,000 years, humanity believed infinity was only potential, meaning you could always add one more but never truly arrive at a complete infinity. This view was held by thinkers like Aristotle and Gauss. However, Cantor introduced a revolutionary concept, treating infinities as completed objects that could be manipulated, leading to ongoing controversy. Cantor's "heresy" was to insist that actual infinities exist as concrete objects with which mathematical operations could be performed.
A potential infinity is a continuous process, like counting 1, 2, 3, 4, without ever stopping or claiming a complete collection. An actual infinity, in contrast, posits that such a complete collection exists as a single object, allowing its properties to be examined and compared to other collections. Cantor's first peculiar discovery was demonstrating that there are as many even numbers as natural numbers. Mathematicians define the "size" of sets using cardinality, where two sets have the same size if their elements can be paired exactly in a one-to-one correspondence, or a bijection. This concept applies not only to even numbers and natural numbers but also to integers and rational numbers, despite the latter being dense on the real line. Cantor proved a bijection between the natural numbers (N) and the rational numbers (Q). He called these "countably infinite sets" and assigned them the symbol "aleph null" (ℵ₀), which represents their shared size.
