Georg Cantor

Two sets have the same size if their members can be paired off one to one. For finite sets that is counting without numbers. Cantor used it as the definition for infinite sets, and then proved that the definition does not collapse: the natural numbers and the real numbers cannot be paired. His theorem is more general.

Theorem (Cantor). For every set X there is no function from X onto its power set P(X). Hence |X| < |P(X)|.

Proof. Suppose f maps X onto P(X). Let D be the set of those x in X that are not members of f(x). Because f is onto, D = f(y) for some y. If y is in D, then y is not in f(y) = D. If y is not in D, then y is in f(y) = D. Both cases are contradictions, so no such f exists. ■

The same argument, applied to the integers or to digit lists, is the “diagonal”. Applied again and again to N, it gives P(N), P(P(N)) and so on without end. The first step of that ladder, from N to P(N), is the step from the countable to the continuum. Whether any size lies between the two is the Continuum Hypothesis, which Gödel (1940) and Cohen (1963) showed ZFC can neither refute nor prove; Chapter 8 says this and builds a beth ladder for exactly that reason.

Counted with tools/corpus_count.py (entity-aware, anchored) at HEAD, not counting this page, its verify script, the Book 7 index, any generated index, the audit log or the generated link boxes (which name this page’s title on other pages); .html and .md files only (the counter does not read .lean). “Chapters” leaves out tools, audit notes, archives, scaffolding and generated indexes.

Where the theorem itself is used: Whitehead and Russell block [5] runs the diagonal over every one of the 65,536 maps from a four-element set to its power set; Chapter 8, Nested Infinities tells the argument and builds the cardinality staircase; the Ocio chapter finds that a displayed “ceiling formula” is Cantor’s theorem under another name; and Orthogenesis/NestedInfinities/EightChecks.lean states it as a < 2 ^ a for any cardinal, taken from Mathlib’s Cardinal.cantor. This page does not repeat any of them. The Lean file was not compiled for this page.

The ratio is the finding. The man with the most citations as an opponent is in 5 files; his correspondent is in 2; Hilbert, who defended the result, is in 43. The corpus tells Cantor’s story through his theorem and his enemies, and barely at all through the people he wrote to.

He wrote his set theory in German. From about the age of eight he attended a German Lutheran school in St Petersburg whose timetable listed Russian, German and French, with English and Latin optional; he did not take Latin. A biographer quotes Grattan-Guinness that he seemingly never wrote in Russian although he must have known it. French versions of his papers appeared in Acta Mathematica in 1883, translated by others, among them Poincaré; Jourdain made the English edition of 1915. Not found: the language spoken at home; the language of the 1867 dissertation text (only its Latin title was seen); Marotte’s 1899 French translation. The fuller table, with what is only plausible, is in Cantor and Crossing Infinities (Book 3).