Does the first order theory of a finite structure have bounded quantifier rank? Let $\mathfrak{A} $ be any finite structure.
Does its first order theory $ \mathfrak{T} := \mathfrak{TH}(\mathfrak{A}) $ have bounded quantifier rank, in the sense that there is a $ q\in\mathbb{N} $ such that for all $ \varphi\in\mathfrak{T} $ with $ qr(\varphi) &gt; q $ there is a $ \varphi'\in\mathfrak{T} $ with $ qr(\varphi')\leq q $ and $ \varphi'\equiv\varphi $ ?
The theory of any finite structure is model complete. In fact, it is easy to see that any formula is equivalent to an existential formula with one quantifier per each element of the structure, after which all quantifiers of the original formula can be simulated by conjunctions and disjunctions. In particular, the number of quantifiers (hence quantifier rank) is bounded by the size of the structure.