Cantor's theorem
Bigger than bijection
Cantor's theorem is the mathematical proof that for any set, the set of all its subsets (its power set) is strictly larger than the original set.
This holds even for infinite sets, establishing that infinities come in different sizes.
The theorem's diagonal argument shows that no set can be put into one-to-one correspondence with its power set.
Not able to enumerate
- countableAble to be tallied
- transfinite numbersInfinite quantities that differ
- continuumUninterrupted, gradual sequence
- real numbersComplete ordered field
Professional in abstract quantities
- nineteenth centuryYears 1801-1900
- GermanyCentral European nation
- mental illnessDisorder affecting cognition or mood
- pioneerEarly explorer or settler
Reducing to diagonal form
- contradictionOpposing claims, incompatible together
- reductio ad absurdumDisproof by absurd consequence
- mathematical proofLogical demonstration of truth
- logical reasoningStructured thought process
lemma · corollary · proof by contradiction · self-reference · function · mapping
Mathematics of abstract collections
- incompletenessState of lacking parts
- consistencyReliable adherence to standards
- formalismPrivileging structure over substance
- PlatonismPhilosophy emphasizing ideal Forms
uncountable
Not able to enumerate
- countableAble to be tallied
- transfinite numbersInfinite quantities that differ
- continuumUninterrupted, gradual sequence
mathematician
Professional in abstract quantities
- nineteenth centuryYears 1801-1900
- GermanyCentral European nation
- mental illnessDisorder affecting cognition or mood
diagonalization
Reducing to diagonal form
- contradictionOpposing claims, incompatible together
- reductio ad absurdumDisproof by absurd consequence
- mathematical proofLogical demonstration of truth
logical reasoning · lemma · corollary · proof by contradiction · self-reference · function · mapping
set theory
Mathematics of abstract collections
- incompletenessState of lacking parts
- consistencyReliable adherence to standards
- formalismPrivileging structure over substance