# Can we have a theory $T$ that is complete for simple sentences in the language of $T$ that are weaker than $Con(T)$?

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 simple if and only if it is not complex.

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?