Suppose there was a predicate, in the language of 1st order $ \mathsf {PA} $, such that it is only true for standard natural numbers i.e. it accepts ALL and ONLY standard natural number, and it rejects any and every non-standard natural number.
Since PA is $ \boldsymbol \omega $-consistent, therefore any such predicate should contradict the Löwenheim-Skolem theorem. But I couldn't find a proof which states such predicates should definitely not exist.
So is there any proof for their non-existence? Or are there any such predicates available, and is there any paper or material that mentions anything about such predicates?
EDIT - It seems that for some reason @spaceisdarkgreen isn't able to respond to my comments. The answer he gave gets 60% of the question, but there's one part to the question that answer doesn't accounts for -
Is it true to say that - " In the intrepretation of any non-standard model No such predicates exist ( in 1st order PA
) which are true for ALL and ONLY standard natural numbers and false for each and every non-standard number " ?
Since in the question I never mentioned about models explicitly, that's because I wanted the answer in general i.e. for all models ( standard and non-standard ) , but in his answer he only refers to standard models, So if anyone can answer this last doubt , so that I can accept the answer.
This is called the overspill principle. If there were such a predicate $\varphi(x)$ for some model $M$, then $M\models\varphi(0)$ and $M \models \forall x( \varphi(x)\to \varphi(x+1))$ (since the successor of a standard element is standard), so by the induction schema, $M\models \forall x\varphi(x).$ So the only model with such a predicate is the standard model.