The Rationals have an AZ‐Enumeration

Let M be an ω‐categorical structure (that is, M is countable and Th( M ) is ω‐categorical). A nice enumeration of M is a total ordering ≺ of M having order‐type ω and satisfying the following. Whenever a i , i <ω, is a sequence of elements from M , there exist some i < j <ω and an automorphism σ of M such that σ( a i ) = a j and whenever b ≺ a i , then σ( b )≺ a j . Such enumerations were introduced by Ahlbrandt and Ziegler in [ 1 ] where they showed that any Grassmannian of an infinite‐dimensional projective space over a finite field (or of a disintegrated set) admits a nice enumeration; this combinatorial property played an essential role in their proof that almost strongly minimal totally categorical structures are quasi‐finitely axiomatisable. Recall that if M is ω‐categorical and ā is a k ‐tuple of distinct elements from M (with tp (ā) non‐algebraic), then the Grassmannian Gr( M ; ā) is defined as follows. The domain of Gr( M ; ā) is the set of realisations of tp (ā) in M k , modulo the equivalence relation xEy if x and y are equal as sets. This is a 0‐definable subset of M eq , and now the relations on Gr( M ; ā) are by definition precisely those which are 0‐definable in the structure M eq . (In particular, Gr( M ; ā) is also ω‐categorical.) Notice that it is by no means clear that if M admits a nice enumeration, then so do Grassmannians of M . However, there is a strengthening of the notion of nice enumeration for which this is the case.