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.
- 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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