Are all models of peano arithmetics descibed using first order logic non standard?

Question

It is known that there are non-standard models of Peano Arithmetics when it is described using first order logic. My question is if there is standard model (one which does not contains non-standard elements) of PA described in FOL. What is example of such canonical model?

Answer 1

It isn't clear what "described in FOL means".

The first-order theory of Peano Arithmetic certainly has a standard model, which consists of the standard natural numbers with their usual arithmetical operations. Peano Arithmetic also has many nonstandard models.

The thing that cannot be done in first-order logic is to make a theory $T$ such that the only model of $T$ is the standard model of arithmetic. This has nothing to do about arithmetic per se; a first order theory that has an infinite model has infinitely many other infinite models, regardless of the subject matter of the first order theory.

Answer 2

By Gödel's Incompleteness Theorems any construction of a model for PA must transcend PA. Using a bit of set theory (some small fragment of ZF) one can show that $\omega$ (the first transfinite von Neumann ordinal) together with appropriate definable arithmetic operations forms a model of PA and has no non-standard elements.