We consider a partial ordering of the set of sentences of Peano arithmetic $P$ induced by a theory $T$ extending $P$, which orders sentences according to the complexity of their ``proofs". Using some properties of the ordering induced by the theory $P+\neg\text{Con}_p$ we prove that $P$ doesn't have the Joint Embedding Property. We also describe models for $P$ which do not enrich the ordering induced by $P$, i.e., models satisfying $<_{\text{Th}({\frak M})} =<_P$, and we prove that for every consistent theory $T$, $T\supset P$, there is a theory $T'\supset P$ such that the ordering induced by the theory $T'$ is a linear extension of the ordering induced by the theory $T$.