Is there a first order formula that is satisfied by $\mathbb{N}$, but not by any other models of Peano axioms? For example, is there a formula that expresses "there is an element that is greater than any natural number"? My first try was $$\exists v_0 (0<v_0 \wedge \forall (v_1\; v_1<v_0 \Rightarrow Sv_1 <v_0)). $$ But I see that this formula is not true in non-standard models either, because every non-zero element has a predecessor.
Is there a first order formula that is satisfied by $\mathbb{N}$, but not by any other models of Peano axioms
first-order-logiclogicnonstandard-modelspeano-axiomspredicate-logic
Best Answer
You can appeal to Gödel's incompleteness theorem: clearly the Peano axioms plus your one axiom are still recursively enumerable, and clearly they are strong enough to encode arithmetic (since they contain the Peano axioms). Thus the theory must be incomplete, and in particular have more than one model by Gödel's completeness theorem.