Über eine elementare Frage der Mannigfaltigkeitslehre

Cantor, G. (1891). Über eine elementare Frage der Mannigfaltigkeitslehre. Jahresbericht der Deutschen Mathematiker-Vereinigung, 1, 75-78.

Type

Journal article

Intellectual base

Primary research

Year

1891

Link

https://eudml.org/doc/144383

Cited from

computer_science, mathematics, philosophy

Cited by

6 citations across 6 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Cardinality
  • … B$; by the Schröder-Bernstein theorem (proved by Cantor 1887, Dedekind c. 1887, Schröder 1898, Bernstein 1898), if $|A| \leq |B|$ and $|B| \leq |A|$ then $|A| = |B|$; the theorem is purely combinatorial — it requires no choice principles — and supplies the antisymmetry of $\leq$ on cardinals; Cantor's theorem
• Computability
  • Three further structural features travel with the computability role. Self-reference and diagonalization: uncomputability proofs almost universally run the move "consider a procedure that would solve the problem, then apply it to its own description," the same diagonal that Cantor, Gödel, Turing, and Tarski each deploy at different levels

    This sourceIntroduces the diagonal argument, the self-reference engine reused by Gödel, Turing, and Tarski.

• Diagonal Impossibility
  • In a system rich enough to encode descriptions of its own analyzers — truth-predicates, halting-checkers, set-membership deciders, provability-checkers — the assumption that a single total analyzer exists for some property of these descriptions can be turned against itself.

    This sourceThe original diagonal argument proving the reals uncountable; also the structural template behind Russell's set-membership flip.

• Infinity
  • and his (1891) diagonal argument

    This sourceCantor diagonal argument formal treatment.

• Paradox
  • as early diagnostic tool. Set-theoretic and mathematical paradoxes

    This sourceCantor diagonal argument formal treatment.

• Set and Membership
  • Cantor's original 1874 paper established that the real numbers are uncountable (by a nested-intervals / bisection argument), which was the origin of modern set theory as a mathematical subject; his later 1891 paper

    This sourceCantor diagonal argument formal treatment.

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• https://www.digizeitschriften.de/id/37721857X_0001%7Clog14 ×1

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