This is a question from Stefan Bilaniuk's very good free online book A Problem Course in Mathematical Logic:

Problem 18.6. Suppose $\Sigma$ is a recursive set of sentences of LN. Find a sentence of LN, which we'll denote by $Con(\Sigma)$, such that $\Sigma$ is consistent if and only if $A \vdash Con(\Sigma)$.

By Godel's second incompleteness theorem isn't this impossible?