Unprovability of circuit upper bounds in Cook's theory PV
  We establish unconditionally that for every integer $k \geq 1$ there is a language $L \in \mbox{P}$ such that it is consistent with Cook's theory PV that $L \notin Size(n^k)$. Our argument is non-constructive and does not provide an explicit description of this language. 
Jan Krajicek and Igor C. Oliveira
math.LO cs.CC