Laver's theorem

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, ${\displaystyle \Pi _{1}^{1}}$-CA₀ and ATR₀, and to be weaker than ${\displaystyle \Pi _{1}^{1}}$-CA₀. However, it remains open whether it is equivalent to ATR₀ or strictly between these two systems in strength.^[5]

See also

• Dushnik–Miller theorem

References

1. 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
2. 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
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, Lecture Notes Series of the Institute for Mathematical Sciences, National University of Singapore, vol. 28, World Scientific; see Chapter 10
5. Montalbán, Antonio (2017), "Fraïssé's conjecture in ${\displaystyle \Pi _{1}^{1}}$-comprehension", Journal of Mathematical Logic, 17 (2): 1750006, 12, doi:10.1142/S0219061317500064, MR 3730562