Uniquely undefinable elements ∗ Greg

There exists a model in a countable language having a unique element which is not definable in Lω1,ω. We answer problem 10.12 from the updated version of Arnie Miller’s problem list, [3]. Definition For a structure M and a ∈ M, we say that a is definable over Lω1,ω if there is a formul ψ ∈ Lω1,ω such that for all b ∈ M a = b ⇔ M |= ψ(b). Theorem 0.1 There is a countable language L and a model M for that language such that there is a unique a ∈ M which is not definable over Lω1,ω. 1 Ranks for formulas For the entirety of this section, L is a countable language. Definition For φ ∈ L we define the rank of the formula by induction. If φ is an atomic formula, then it has rank 0. If φ has rank α, then ∃xφ and ∀xφ both have rank α+ 1 and ¬φ has rank α. If (φn)n∈N is a sequence of formulas, each φn having rank αn, then