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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 $T$?

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    $\begingroup$ math.stackexchange.com/q/1975706/462 $\endgroup$ – Andrés E. Caicedo Dec 1 '18 at 23:58
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    $\begingroup$ 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). $\endgroup$ – Noah Schweber Dec 2 '18 at 18:03
  • $\begingroup$ Yes of course!, I'll add it. $\endgroup$ – Zuhair Dec 2 '18 at 18:37

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