Laver's theorem - Misplaced Pages

Laver's theorem, in order theory, states that order embeddability of countable total orders is a well-quasi-ordering. That is, for every infinite sequence of totally-ordered countable sets, there exists an order embedding from an earlier member of the sequence to a later member. This result was previously known as Fraïssé's conjecture, after Roland Fraïssé, who conjectured it in 1948; Richard Laver proved the conjecture in 1971. More generally, Laver proved the same result for order embeddings of countable unions of scattered orders.

In reverse mathematics, the version of the theorem for countable orders is denoted FRA (for Fraïssé) and the version for countable unions of scattered orders is denoted LAV (for Laver). In terms of the "big five" systems of second-order arithmetic, FRA is known to fall in strength somewhere between the strongest two systems, $\Pi _{1}^{1}$ -CA₀ and ATR₀, and to be weaker than $\Pi _{1}^{1}$ -CA₀. However, it remains open whether it is equivalent to ATR₀ or strictly between these two systems in strength.

References

Fraïssé, Roland (1948), "Sur la comparaison des types d'ordres", Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences (in French), 226: 1330–1331, MR 0028912; see Hypothesis I, p. 1331
Harzheim, Egbert (2005), Ordered Sets, Advances in Mathematics, vol. 7, Springer, Theorem 6.17, p. 201, doi:10.1007/b104891, ISBN 0-387-24219-8
Laver, Richard (1971), "On Fraïssé's order type conjecture", Annals of Mathematics, 93 (1): 89–111, doi:10.2307/1970754, JSTOR 1970754
Hirschfeldt, Denis R. (2014), Slicing the Truth, Lecture Notes Series of the Institute for Mathematical Sciences, National University of Singapore, vol. 28, World Scientific; see Chapter 10
Montalbán, Antonio (2017), "Fraïssé's conjecture in $\Pi _{1}^{1}$ -comprehension", Journal of Mathematical Logic, 17 (2): 1750006, 12, doi:10.1142/S0219061317500064, MR 3730562

v t e Order theory
Topics Glossary Category
Key concepts	Binary relation Boolean algebra Cyclic order Lattice Partial order Preorder Total order Weak ordering
Results	Boolean prime ideal theorem Cantor–Bernstein theorem Cantor's isomorphism theorem Dilworth's theorem Dushnik–Miller theorem Hausdorff maximal principle Knaster–Tarski theorem Kruskal's tree theorem Laver's theorem Mirsky's theorem Szpilrajn extension theorem Zorn's lemma
Properties & Types (list)	Antisymmetric Asymmetric Boolean algebra topics Completeness Connected Covering Dense Directed (Partial) Equivalence Foundational Heyting algebra Homogeneous Idempotent Lattice Bounded Complemented Complete Distributive Join and meet Reflexive Partial order Chain-complete Graded Eulerian Strict Prefix order Preorder Total Semilattice Semiorder Symmetric Total Tolerance Transitive Well-founded Well-quasi-ordering (Better) (Pre) Well-order
Constructions	Composition Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure
Topology & Orders	Alexandrov topology & Specialization preorder Ordered topological vector space Normal cone Order topology Order topology Topological vector lattice Banach Fréchet Locally convex Normed
Related	Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding Isomorphism Order type Ordered field Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz space Partially ordered group Positive cone of a partially ordered group Upper set Young's lattice

Category:

Order theory

See also

References