Math 2.0

Mathematical logic and foundations Definition

Ordinal number

Builds on Well-orderElement

Definition

A set XX is transitive when each element of an element of XX is an element of XX. An ordinal number, in the definition of von Neumann, is a transitive set α\alpha that is well-ordered by the relation ∈\in, with β<γ\beta < \gamma when β∈γ\beta \in \gamma.

Remarks

The first ordinals are 0=∅0 = \varnothing, 1={0}1 = \{0\}, 2={0,1}2 = \{0, 1\}, and in general n+1=n∪{n}n + 1 = n \cup \{n\}. The set ω={0,1,2,… }\omega = \{0, 1, 2, \dots\} of all finite ordinals is the first infinite ordinal. Each well-ordered set is order isomorphic to exactly one ordinal, its order type. Each ordinal α\alpha is the set of all ordinals smaller than α\alpha.

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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 pairdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionPreorder, definition, no evidence levelPreorderdefinitionPartial order, definition, no evidence levelPartial orderdefinitionLeast element, definition, no evidence levelLeast elementdefinitionTotal order, definition, no evidence levelTotal orderdefinitionWell-order, definition, no evidence levelWell-orderdefinitionOrdinal number, definition, no evidence levelOrdinal numberdefinition
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  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Implication definition
  5. Predicate definition
  6. Ordered pair definition
  7. Universal quantifier definition
  8. Cartesian product definition
  9. Subset definition
  10. Binary relation definition
  11. Antisymmetric relation definition
  12. Reflexive relation definition
  13. Transitive relation definition
  14. Preorder definition
  15. Partial order definition
  16. Least element definition
  17. Total order definition
  18. Well-order definition
  19. Ordinal number definition

Where this leads

37 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Ordinal_number, lead section
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