Ueber eine elementare Frage der Mannigfaltigkeitslehre¶

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

Type: Journal article
Intellectual base: Primary research
Year: 1891
Volume: 1
Pages: 75-78
Link: https://eudml.org/doc/144383
Cited from: computer_science, mathematics, philosophy

Cited by¶

7 citations across 7 artifacts.

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

Primes¶

Cardinality
- Mapped back to the six-component structural signature: the carrier collections are \(\mathbb{Q}\), \(\mathbb{R}\), \(\mathcal{P}(\mathbb{N})\) in turn; the equinumerosity bijections are the diagonal-traversal \(\mathbb{N} \leftrightarrow \mathbb{Q}\) and the binary-expansion \(\mathcal{P}(\mathbb{N}) \leftrightarrow \{0,1\}^{\mathbb{N}} \leftrightarrow \mathbb{R}\); the cardinal numbers as equivalence classes are \(\aleph_0\) and \(\mathfrak{c} = 2^{\aleph_0}\); cardinal arithmetic gives \(|\mathcal{P}(\mathbb{N})| = 2^{\aleph_0}\) from the function-space construction; ordering by injection plus Cantor's theorem yields \(\aleph_0 < \mathfrak{c}\) strictly; and the use is the foundational classification of analysis (continuum-many reals; countably-many computable reals; transcendentals exist by cardinality alone; CH cannot be settled from ZFC — guiding the choice of forcing or inner-model extensions for further work).
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.

Mechanisms¶

Diagonalization Impossibility Proof
- The discipline is to pin the exact class and model the certificate covers — Cantor's diagonal argument
  This sourcePresents Cantor's 1891 diagonal construction, the canonical model for diagonal arguments.

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