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]

• Dushnik–Miller theorem