In 1874 Georg Cantor published a brief paper that, while titled as a note on algebraic numbers, actually contained his diagonal proof that the real numbers cannot be listed in a one‑to‑one correspondence with the natural numbers—showing that infinity comes in different sizes. Cantor’s argument relied on suggestions from his correspondent Richard Dedekind, who had earlier proved that the algebraic numbers are countable and advised Cantor to examine the interval [0,1]. By presenting Dedekind’s result as a decoy, Cantor smuggled his revolutionary conclusion past the conservative editor Leopold Kronecker. Recently uncovered letters reveal that Dedekind’s input was far more substantial than previously acknowledged, challenging the traditional story of Cantor as a lone genius and underscoring the collaborative, ethical dimensions of the discovery that laid the foundations of modern set theory.
1. In 1874, 28‑year‑old mathematician Georg Cantor published a paper about sets of numbers.
2. The paper presented the idea that infinity comes in different sizes.
3. By 1872, German mathematicians Richard Dedekind and Georg Cantor were independently working on constructing the number line.
4. Dedekind used a method now called a Dedekind cut to construct a complete and continuous number line.
5. Cantor also constructed a complete and continuous number line independently.
6. Their work showed that infinity exists in every interval of the number line, not only at its far end.
7. In summer 1872, Cantor and Dedekind met while vacationing in a Swiss village.
8. In November 1873, Cantor wrote to Dedekind, beginning a mathematical correspondence.
9. Cantor’s letter asked whether the real numbers can be placed in one‑to‑one correspondence with the natural numbers.
10. Dedekind replied that he did not know how to prove that, but had proved that the algebraic numbers are countable.
11. Dedekind suggested Cantor examine only the interval between 0 and 1 to test the countability of the real numbers.
12. Cantor proved that for any countable list of real numbers between 0 and 1, a new real number can be constructed that is not on the list (diagonal argument).
13. Cantor sent his proof to Dedekind, who later noted that he had improved the argument significantly.
14. The conclusion was that the real numbers are uncountable while the natural numbers are countable, implying different sizes of infinity.
15. The paper was published in 1874 in a journal edited by Leopold Kronecker, who opposed the use of infinite concepts in mathematics.
16. Cantor presented Dedekind’s proof of the countability of algebraic numbers nearly word for word, placing his own argument about real numbers elsewhere in the paper.
17. The published paper is about four pages long and titled “On a property of algebraic numbers”.
18. Over the following decades, Cantor’s ideas formed the basis of set theory, which became a unifying framework for mathematics.
19. In 2024, researcher Demian Goose discovered missing letters from Dedekind to Cantor in the archive of the University of Halle.
20. The letters contain Dedekind’s proof that the algebraic numbers are countable, which Cantor later used without explicit attribution.
21. Cantor’s reply to Dedekind acknowledged that Dedekind’s remarks were of great assistance to him.
22. The discovery shows that Cantor’s famous paper incorporated Dedekind’s result without credit.