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;[1] Richard Laver proved the conjecture in 1971. More generally, Laver proved the same result for order embeddings of countable unions of scattered orders.[2][3]

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).[4] 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.[5]

See also

References

  1. ^ Fraïssé, Roland (1948), "Sur la comparaison des types d'ordres" (in French), Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. 226: 1330–1331, MR 28912; see Hypothesis I, p. 1331
  2. ^ Harzheim, Egbert (2005), Ordered Sets, Vol. 7, Advances in Mathematics, Springer, Theorem 6.17, p. 201, doi:10.1007/b104891. ISBN 0-387-24219-8
  3. ^ Laver, Richard (1971), "On Fraïssé's order type conjecture", Annals of Mathematics. 93 (1): 89–111, doi:10.2307/1970754. JSTOR 1970754
  4. ^ Hirschfeldt, Denis R. (2014), "Slicing the Truth", Vol. 28, Lecture Notes Series of the Institute for Mathematical Sciences, National University of Singapore, World Scientific; see Chapter 10
  5. ^ 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