PERSON
Georg Cantor
The nineteenth-century mathematician who proved that infinity comes in different sizes—and whose diagonal argument, invented to show the real numbers are uncountable, became the master key to the deepest negative results of the twentieth century: Gödel's incompleteness, Turing's undecidability, and the permanent limits of all mechanical computation.
Georg Cantor began with a question his entire discipline had agreed to leave alone—whether infinity is one size or many—and ended by drawing the boundary that all computation still cannot cross. Born in Saint Petersburg in 1845 and confined for most of his career to the University of Halle, Cantor established, in a sequence of papers beginning in 1874, that the counting numbers are a
smaller infinity than the real numbers: you can list the integers in a first, second, third; you cannot list the reals in any order that leaves nothing out. The proof—the diagonal argument of 1891—is a method for manufacturing, from any proposed list, a number that cannot be on it. That method traveled directly into
Kurt Gödel's 1931 incompleteness theorems and into
Alan Turing's 1936 proof that the halting problem is undecidable, and it still marks the outer boundary of what any
language model,