Math 2.0

Mathematical logic and foundations Concept

Cantor's diagonal argument

Builds on Uncountable setCountable set

Summary

Cantor's diagonal argument shows that some infinite sets are uncountable. Given a list s0,s1,s2,…s_0, s_1, s_2, \dots of infinite sequences of 00 and 11, define a sequence tt by t(n)=1−sn(n)t(n) = 1 - s_n(n). Then tt differs from each sns_n at position nn, so tt is not in the list. So the set of all such sequences is not countable.

Remarks

Georg Cantor published the argument in 1891. His first proof that ℝ\mathbb{R} is uncountable, from 1874, used another method. The same idea proves Cantor's theorem for every set. Diagonal arguments also appear in the proofs of the halting problem and of Gödel's incompleteness theorems.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionInjective function, definition, no evidence levelInjective functiondefinitionCountable set, definition, no evidence levelCountable setdefinitionUncountable set, definition, no evidence levelUncountable setdefinitionCantor's diagonal argument, concept, no evidence levelCantor's diagonal arg…concept
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  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Implication definition
  5. Predicate definition
  6. Ordered pair definition
  7. Existential quantifier definition
  8. Universal quantifier definition
  9. Cartesian product definition
  10. Subset definition
  11. Uniqueness quantifier definition
  12. Binary relation definition
  13. Function definition
  14. Injective function definition
  15. Countable set definition
  16. Uncountable set definition
  17. Cantor's diagonal argument concept

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Further reading

  1. Web https://en.wikipedia.org/wiki/Cantor's_diagonal_argument, lead section
About this page
Tag0XCH8QVDCQ
URIm2://0XCH8QVDCQ
Aliasesdiagonal-argument, diagonalization
Kindconcept
WritingW1 stated
MSC03E10
Part ofProof methods
UsesUncountable set, Countable set
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