This is a question from Stefan Bilaniuk's very good free online book [A Problem Course in Mathematical Logic](http://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=olbp12123):


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?