Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen.¶
Cantor, G. (1874). Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. Journal für die reine und angewandte Mathematik, 258-262.
Cited by¶
4 citations across 4 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Cardinality
- … a set's size such that: (1) two sets $A$ and $B$ have the same cardinality ($|A| = |B|$) iff there exists a bijection between them — a one-to-one correspondence matching each element of $A$ to exactly one of $B$ and vice versa; this bijection-based equinumerosity is the foundational definition introduced by Cantor
This sourceFirst proof of the uncountability of the reals, using a nested-intervals (bisection) argument — NOT the diagonal argument — and establishing that the algebraic numbers are countable while the reals are not (forcing the existence of transcendentals by cardinality alone).
- … a set's size such that: (1) two sets $A$ and $B$ have the same cardinality ($|A| = |B|$) iff there exists a bijection between them — a one-to-one correspondence matching each element of $A$ to exactly one of $B$ and vice versa; this bijection-based equinumerosity is the foundational definition introduced by Cantor
- Dense Set
- The two corollaries are vivid here. Size-independence: \(\mathbb{Q}\) is countable while \(\mathbb{R}\) is uncountable
This sourceEstablishes the uncountability of the reals against the countability of the rationals/algebraic numbers — the size-independence corollary of density.
- The two corollaries are vivid here. Size-independence: \(\mathbb{Q}\) is countable while \(\mathbb{R}\) is uncountable
- Infinity
- and matured decisively with Cantor's (1874) proof that the real numbers are uncountable
This sourceFirst proof of the uncountability of the reals, using a nested-intervals (bisection) argument — NOT the diagonal argument.
Supported in partVerified against the publisher's abstract
Cantor's 1874 Crelle paper (77: 258-262) is confirmed as the cited work, but no text of it was obtained, so the uncountability proof itself is not shown here.
“Article Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen. was published on January 1, 1874 in the journal Journal für die reine und angewandte Mathematik (volume 1874, issue 77).”
- and matured decisively with Cantor's (1874) proof that the real numbers are uncountable
- Set and Membership
- Cantor's
This sourceFirst proof of the uncountability of the reals, using a nested-intervals (bisection) argument — NOT the diagonal argument.
- Cantor's
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