Set theory
The mathematics of collections treated as objects in their own right, and the theory of the infinite made exact. Georg Cantor proved that infinity is not one thing: some infinite sets are strictly larger than others, and there is no largest.
Das tiefe Archiv ist vorerst auf Englisch verfasst — geprüfte Übersetzungen stehen auf der Roadmap. Die Übersetzungsfunktion Ihres Browsers funktioniert auf dieser Seite gut.
✦ Moment, wirklich?
Cantor proved you cannot even list the real numbers. Suppose you had a complete numbered list of every decimal between 0 and 1; build a new number whose first digit differs from the first listed number's first digit, whose second differs from the second's second, and so on down the diagonal. It differs from every entry somewhere, so it was never on the list — no list can be complete. The fractions, by contrast, can be listed. There are strictly more real numbers than fractions, though both are endless.
Was es ist
A set is any collection considered as a single thing. Cantor's radical move, in an 1874 paper in Crelle's *Journal*, was to compare the sizes of *infinite* sets by pairing: two sets have the same size if their members can be matched one-to-one, with none left over. Match the counting numbers 1, 2, 3, … against a candidate set, and if the matching can be completed the set is "countable" — the smallest infinity, which Cantor later named ℵ₀ (aleph-null). The whole numbers are countable; so, less obviously, are the fractions, and even the algebraic numbers. But the real numbers are not. No matter how you try to list them, the diagonal construction always builds one you missed. Infinity, it turns out, comes in sizes.
Warum es zählte
For two millennia the infinite had been a source of paradox to be handled at arm's length — Aristotle allowed only a "potential" infinity, never a completed one; Galileo noticed that there seem to be as many squares as whole numbers and backed away puzzled. Cantor made the actual infinite a precise object with an arithmetic and, astonishingly, a hierarchy. His theorem shows that the set of all subsets of any set is strictly larger than the set itself — so from any infinity you can always build a bigger one, without end. Set theory then became the common bedrock on which nearly all of mathematics could be rebuilt: numbers, functions, and spaces alike defined as sets. The work was savaged — Kronecker, once his teacher, worked to block his career — but Hilbert defended it in 1926 with a line that stuck: "No one shall expel us from the paradise that Cantor created."
Was es erschloss
Set theory became the foundation of modern mathematics, its axioms (the ZFC system) the base layer on which the rest is formally constructed. It gave measure theory and hence rigorous probability, general topology, and the modern definition of function, number, and space. Through the paradoxes it exposed — most famously Russell's — it forced the birth of mathematical logic, and through the diagonal method it seeded the theory of computation. Almost every careful answer to "what *is* a number, really?" given since is written in Cantor's language.
Dieser Eintrag wartet noch auf seine vollständige Darstellung — die Kartografen sind am Werk. Sein Platz im Graphen ist bereits verifiziert.
Setzt voraus
Stimmt etwas auf dieser Seite nicht? Jede Behauptung hier soll dem Widerspruch standhalten. Eine Korrektur vorschlagen →