An Ordering of the set of Sentences of Peano Arithmetic

Aleksandar Ignjatović

Publications de l'Institut Mathématique, _N_S_38 (1985) no. 52, p. 13 .

Résumé

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 $$, 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 $T$.

Détail

Classification : 03H15