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Are there more counting numbers than even numbers? Most people say yes, twice as many. Georg Cantor said no, they are the same size, and then proved that the real numbers are a genuinely bigger infinity. His ideas were resisted for years and are now the foundation of set theory you meet in Class 11. This post explains his method with simple pairings and walks through his most famous argument.
Who was Georg Cantor?
According to his MacTutor biography, Cantor was born in St Petersburg on 3 March 1845 and moved with his family to Germany at eleven. He was an outstanding violinist as well as a mathematician. He studied in Zürich and Berlin, where he attended lectures by Weierstrass, Kummer and Kronecker, completed his doctorate in 1867, and joined the University of Halle in 1869, where he stayed for his whole career. He died in Halle on 6 January 1918.
How do you compare the sizes of infinite sets?
You cannot count an infinite set, but you can pair it with another. Two sets are the same size if every member of one can be matched with exactly one member of the other, with nothing left over on either side. This is a one-to-one correspondence, the same idea as a bijective function in your Class 11 chapter on relations and functions.
Counting numbers and even numbers. Pair 1 with 2, 2 with 4, 3 with 6, and in general n with 2n. Every even number is used exactly once. Same size.
Counting numbers and all integers. List the integers as 0, 1, −1, 2, −2, 3, −3, and so on, then pair them with 1, 2, 3, 4, 5, 6, 7. Same size.
A set that can be paired with the counting numbers is called countable. MacTutor records that in 1873 Cantor proved the rational numbers countable, and also the algebraic numbers, the roots of polynomial equations with integer coefficients.
Why are the real numbers a bigger infinity?
By December 1873 Cantor had proved that the real numbers are not countable: no list, however clever, contains them all. The proof appeared in 1874 in a paper whose title only mentions algebraic numbers, Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. Since the algebraic numbers are countable and the reals are not, it followed that numbers which are not algebraic, the transcendental numbers such as π, must exist, and in a precise sense almost every real number is one of them.
The proof most people learn today is a simpler one, the diagonal argument. The Dictionary of Scientific Biography entry notes that the 1874 proof did not use it; the diagonal method came in a short paper of 1890 to 1891.
Worked example: the diagonal argument. Suppose someone claims to have listed every number between 0 and 1. Their list starts:
0.14159...
0.50000...
0.33333...
0.71828...
Look at the bold digits on the diagonal: 1, 0, 3, 2, and so on down the list. Build a new number by changing each one with a fixed rule: write 5, unless the diagonal digit is 5, in which case write 4. Here the new number begins 0.5555...
It differs from number 1 in the first decimal place (5, not 1).
It differs from number 2 in the second place (5, not 0).
It differs from number 3 in the third place, and from number 4 in the fourth.
It differs from the nth number in the nth place, for every n, so it is not anywhere on the list. Any list misses some number, so the reals cannot be listed. The rule avoids 0 and 9 on purpose, because 0.4999... and 0.5000... are the same number written two ways.
What was the continuum hypothesis?
If the counting numbers are the smallest infinity and the reals are larger, is there anything in between? Cantor believed not, and tried for years to prove it. MacTutor describes how he would think he had proved it false, then find his mistake the next day, then think he had proved it true. This question, the continuum hypothesis, became the first of the problems David Hilbert set out in 1900.
The answer was stranger than either guess. In 1938 Kurt Gödel showed in a short paper that the hypothesis cannot be disproved from the standard axioms of set theory. In 1963 Paul Cohen showed in The independence of the continuum hypothesis that it cannot be proved from them either. Cohen received a Fields Medal in 1966 for this work.
What did the idea cost Cantor?
Not everyone welcomed infinite sets. Kronecker, his former teacher, opposed the work, and a major paper Cantor sent to Crelle's Journal in 1877 was delayed until Dedekind intervened; Cantor never submitted there again. Cantor also lived with depression from 1884 onwards. MacTutor notes that his conflicts and mathematical worries were magnified by his illness but were not its cause.
He also had steady support. His long letters with Dedekind let him test ideas, and when in 1877 he found that the points of a square can be paired with the points of a line segment, he wrote to Dedekind, in MacTutor's translation: "I see it, but I don't believe it!" He helped found the German Mathematical Society and was its first president, and the Royal Society awarded him its Sylvester Medal in 1904.
What can a student take from Cantor's way of working?
Define your terms precisely. "Same size" became a definition you could test, and the paradoxes dissolved.
Trust the proof over the feeling. His own reaction to the square and the line shows that a correct argument can beat intuition.
Find someone to check your thinking. Dedekind's letters were Cantor's sounding board for decades.
If your mood is affecting your work, say so. Talk to someone you trust; in India, Tele MANAS on 14416 is free and confidential.
The pairing ideas here are Class 11 sets and functions. If one-to-one and onto functions are not yet clear, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can revise them before trying the diagonal argument yourself. Our post on sets and relations for Class 11 is a good place to start.
Try it now: write down a pairing between the counting numbers and the odd numbers, and check that nothing is left over.
