# Variant of the Lévy hierarchy on formulas

Consider the following variant of the Lévy hierarchy on formulas : let $\Phi$ be the set of all meaningful formulas on the alphabet $\in,=,\vee,\wedge,(,),\neg,\exists,\forall$ and a countable set of variables $x_1,x_2, \ldots$. Let $\Sigma_0=\Pi_0$ consists of all formulas in $\Phi$ without quantifiers at all (in the usual Lévy hierarchy, we would replace this with bounded quantifiers), and by induction, let $\Sigma_n$ be the set of formulas of the form $\exists t_1 \exists t_2 \ldots \exists t_r \psi$ where $\psi$ is in $\Pi_{n-1}$, and let $\Pi_n$ be the set of formulas of the form $\forall t_1 \forall t_2 \ldots \forall t_r \psi$ where $\psi$ is in $\Sigma_{n-1}$.

Now let $\Delta$ be the set of sentences that are equivalent to some $\Sigma_1$ formula and equivalent also to some $\Pi_1$ formula at the same time (here "equivalent" means provably equivalent : I say that $\phi$ is equivalent to $\psi$ when $\phi \Leftrightarrow \psi$ is a theorem of ZFC). For example, if $\phi$ is an universally false sentence (i.e. $\neg \phi$ is a theorem of ZFC), we have $\phi \Leftrightarrow \exists x\ (x \neq x)$ and $\phi \Leftrightarrow \forall x\ (x \neq x)$, so that $\phi$ is in $\Delta_1$. Are there other sentences in $\Delta_1$ besides the universally true or universally false ones?

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In order to make the question precise, you'll need to explain what you mean by "equivalent" and what you mean by "universally true or universally false". – Carl Mummert Sep 24 '11 at 23:39
@Carl : Indeed. I updated the post accordingly. – Ewan Delanoy Sep 28 '11 at 19:01

If "equivalent" just means "logically equivalent" then the answer is that any $\Delta_1$ sentence $\phi$ is either true in every structure or false in every structure. From elementary model theory, we know that because $\phi$ is logically equivalent to an existential formula, it is preserved under taking superstructures. Because it is logically equivalent to a universal formula, it is preserved under taking substructures.
Let $A$, $B$ be any two structures for your language. Then the "disjoint union" $C$ of $A$ and $B$ is a superstructure of both, where you rename the elements of $B$ if necessary so that $|A| \cap |B| = \emptyset$ and then you let $\in^C$ be $\in^A \cup \in^B$. Therefore $\phi$ has the same truth value in $A$ and $B$ because $B$ is a substructure of a superstructure of $A$.