Math 2.0

Mathematical logic and foundations Definition

Uncountable set

Builds on Countable set

Definition

A set AA is uncountable when it is not countable, that is, when there is no injective function from AA to ℕ\mathbb{N}.

Remarks

The real numbers and the power set of ℕ\mathbb{N} are uncountable. With the axiom of choice, a set is uncountable exactly when its cardinality is at least ℵ1\aleph_1, the least cardinal larger than ℵ0\aleph_0. A superset of an uncountable set is uncountable.

Reading path

To understand this page from the ground up, read these 16 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 setdefinition
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

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Uncountable_set, lead section
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MSC03E10
DefinesUncountable set
UsesCountable set

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  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

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