Let’s denote a sentence $ P$ as “** weak Godel sentence of theory $ T$ **“, if and only if $ $ [\neg (T \vdash P) \wedge \neg (T\vdash \neg P)] \wedge [Con(T)=Con(T+P) \wedge Con(T)=Con(T+ \neg P)] $ $

In English this is: $ P$ is independent of $ T$ and the addtion of $ P$ or $ \neg P$ to $ T$ doesn’t prove the consistency of $ T$ .

Let’s denote a sentence as ** complex** if it has a proper subformula of it that is a sentence, or when de-prenexed it results in a sentence that has a proper subformula of it that is a sentence. A sentence is

**if and only if it is not complex.**

*simple*Let’s fix the language of $ T$ to a classical first order logic language that doesn’t contain any constants in its signature. By sentence it is meant the usual meaning of a fully quantified formula (i.e. has no free variables).

Definition: $ $ T \text{ is complete for simple sentences below } Con(T) \iff \forall P (P \text { is a weak Godel sentence of }T \to P \text { is complex})$ $

*In other words:* all sentences if the addition of them or their negation to $ T$ doesn’t result in a theory that can prove the consistency of $ T$ , that are simple, then those sentences are decidable by $ T$ .

Question: is it possible to have a theory that meets Godel’s incompleteness criteria and yet is complete for simple sentences below its consistency level?