# Is every theory meeting Gödels incompleteness, also incomplete below its consistency level?

Is it always the case that for any theory $$T$$ that meets Godel's criteria for incompleteness, there is a sentence $$P$$ such that neither $$T \vdash P$$, nor $$T\vdash \neg P$$; and such that $$T+P$$ is equi-consistent with $$T$$, and $$T+ \neg P$$ is equi-consistent with $$T$$?

• math.stackexchange.com/q/1975706/462 – Andrés E. Caicedo Dec 1 '18 at 23:58
• Do you want to also require that $T+\neg P$ be equiconsistent with $T$? If not, then $\neg Con(T)$ is an example of such a $P$ (as long as the base theory with respect to which we're establishing equiconsistency is strong enough to prove Godel's theorem). – Noah Schweber Dec 2 '18 at 18:03
• Yes of course!, I'll add it. – Zuhair Dec 2 '18 at 18:37