Math 2.0

Mathematical logic and foundations Combinatorial set theory Object

Aronszajn tree

Builds on Partial orderWell-orderCountable setUncountable set

Description

A tree, in set theory, is a partially ordered set (T,<)(T, <) in which, for each t∈Tt \in T, the set {s∈T:s<t}\set{s \in T : s < t} is well-ordered. The height of tt is the order type of this set, and level α\alpha of TT is the set of elements of height α\alpha. A branch is a maximal linearly ordered subset of TT. An Aronszajn tree is a tree with these three properties:

  1. Its height is ω1\omega_1, the least uncountable ordinal: the levels α<ω1\alpha < \omega_1 are nonempty, and level ω1\omega_1 is empty.
  2. Each level is countable.
  3. It has no branch of length ω1\omega_1, that is, no uncountable branch.

Remarks

Aronszajn trees exist in ZFC, by a construction of Nachman Aronszajn. One construction uses bounded, strictly increasing sequences of rational numbers, each of countable successor length, ordered by extension. Levels are chosen by recursion on α<ω1\alpha < \omega_1, so that each level is countable and each node has extensions at all higher levels with only slightly larger supremum. An uncountable branch would give a strictly increasing sequence of rationals of length ω1\omega_1, which is impossible since ℚ\mathbb{Q} is countable. This shows that König's lemma, which gives an infinite branch in each infinite finitely branching tree, has no analogue at ω1\omega_1. An Aronszajn tree in which every antichain is countable is a Suslin tree, and its existence is independent of ZFC.

Reading path

To understand this page from the ground up, read these 25 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 relationdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionFunction, definition, no evidence levelFunctiondefinitionPreorder, definition, no evidence levelPreorderdefinitionInjective function, definition, no evidence levelInjective functiondefinitionPartial order, definition, no evidence levelPartial orderdefinitionCountable set, definition, no evidence levelCountable setdefinitionLeast element, definition, no evidence levelLeast elementdefinitionTotal order, definition, no evidence levelTotal orderdefinitionUncountable set, definition, no evidence levelUncountable setdefinitionWell-order, definition, no evidence levelWell-orderdefinitionAronszajn tree, object, no evidence levelAronszajn treeobject
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. Antisymmetric relation definition
  14. Reflexive relation definition
  15. Transitive relation definition
  16. Function definition
  17. Preorder definition
  18. Injective function definition
  19. Partial order definition
  20. Countable set definition
  21. Least element definition
  22. Total order definition
  23. Uncountable set definition
  24. Well-order definition
  25. Aronszajn tree object

Further reading

  1. Web https://en.wikipedia.org/wiki/Aronszajn_tree
About this page
TagKKT4JYJEJJ
URIm2://KKT4JYJEJJ
Kindobject
WritingW0 stub
MSC03E05
UsesPartial order, Well-order, Countable set, Uncountable set
Same as
Changed2 November 2026

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