Math 2.0

Mathematical logic and foundations Theorem

The real numbers are uncountable

Builds on Uncountable set

Statement

The set ℝ\mathbb{R} of real numbers is uncountable.

Remarks

Cantor proved it in his article of 1874. That proof uses nested intervals. A later proof uses Cantor's diagonal argument on decimal or binary expansions. Since the algebraic numbers are countable, the result shows that transcendental numbers exist. Cantor gave this application in the same article.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

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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 setdefinitionThe real numbers are uncountable, statement, no evidence levelThe real numbers are …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  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. The real numbers are uncountable statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Cantor's_first_set_theory_article, lead section
About this page
Tag0518MM6BE1
URIm2://0518MM6BE1
Aliasesuncountability-of-the-reals
Kindstatement, theorem
WritingW0 stub
MSC03E10
StatusThis copy of the site has no build data. Run m2 status 0518MM6BE1.
UsesUncountable set

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