Infinity is infinity. There is only one infinite quantity, and all infinities are the same size.

Before Georg Cantor published his work on infinite sets in the 1870s and 1880s, mathematicians treated infinity as a single concept. The word meant 'without end,' and the question of whether one infinite collection might be larger than another was not considered a meaningful mathematical question. Infinity appeared in calculus as the result of unbounded processes, and in philosophy as the attribute of the divine, but comparing two infinities and asking which was bigger was not a problem that mathematics had the tools to address.

Cantor's first key insight appeared in 1873. He showed that the rational numbers, meaning all fractions, can be put into a one-to-one correspondence with the counting numbers: 1, 2, 3, and so on. This meant that in a precise set-theoretic sense, there are exactly as many rational numbers as counting numbers, even though the rationals appear far denser on the number line. He demonstrated this by constructing a diagonal enumeration that listed all fractions in a regular sequence. He then showed that the real numbers cannot be put into any such correspondence.

The proof, now called the diagonal argument, appeared in 1891. Cantor showed that given any proposed complete list of real numbers between 0 and 1, he could construct a new real number guaranteed not to be on the list: take the first digit of the first number, the second digit of the second, and so on, and change each digit by one. The resulting number differs from every entry on the list at at least one decimal place. Since any proposed list must be incomplete, the real numbers form a strictly larger infinite set than the counting numbers. Two distinct infinities had been established, and the second was provably larger than the first.

The response from Cantor's contemporaries was hostile in a way unusual even for mathematical controversy. Leopold Kronecker, one of the dominant figures in German mathematics, called Cantor a corrupter of youth and worked to block his publications and appointments. Henri Poincare described set theory as a disease from which mathematics would eventually recover. Many mathematicians held that completed infinite sets were philosophically incoherent, that infinity was a process rather than a thing. Cantor suffered severe depression and spent extended periods in a sanatorium. He died there in 1918.

His rehabilitation came through David Hilbert, among the most influential mathematicians of the early 20th century, who described Cantor's theory as 'the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.' Hilbert placed transfinite mathematics at the center of his program for the foundations of the discipline. By the 1920s, Cantor's framework was standard in mathematical research.

School curricula absorbed set theory slowly and selectively. The 'new math' curriculum of the 1960s introduced elementary set notation into American K-12 schools, and some teachers encountered Cantor's ideas in that context. The hierarchy of infinite cardinals, with its distinction between countable and uncountable infinities, was typically treated as advanced mathematics rather than a fundamental corrective to basic intuition. Students who graduated through the 1950s, 1960s, and 1970s almost universally left school having heard 'infinity is infinity' as a closed statement, without the tools to ask whether the infinities in calculus and the infinities in set theory were the same kind of thing.