Math 2.0

Mathematical logic and foundations Definition

Infinite set

Builds on Finite set

Definition

A set AA is infinite when it is not finite, that is, when AA is not equinumerous with {1,…,n}\{1, \dots, n\} for any natural number nn.

Remarks

Examples are ℕ\mathbb{N}, ℤ\mathbb{Z}, ℚ\mathbb{Q} and ℝ\mathbb{R}. With the axiom of choice, a set is infinite exactly when it is equinumerous with a proper subset of itself. This is the Dedekind definition. Infinite sets can have different sizes. See Cantor's theorem.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

Show the path as a graph
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 levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionBijective function, definition, no evidence levelBijective functiondefinitionEquinumerous sets, definition, no evidence levelEquinumerous setsdefinitionFinite set, definition, no evidence levelFinite setdefinitionInfinite set, definition, no evidence levelInfinite 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. Codomain definition
  15. Injective function definition
  16. Surjective function definition
  17. Bijective function definition
  18. Equinumerous sets definition
  19. Finite set definition
  20. Infinite set definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Infinite_set, lead section
About this page
Tag01MKW4R283
URIm2://01MKW4R283
Kinddefinition
WritingW1 stated
MSC03E10
DefinesInfinite set
UsesFinite set

Markdown · JSON · Edit on GitHub

What this page needs for W2

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

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by