I refer you to the Lowenheim-Skolem Theorem:
Every (countable) first-order theory that has an infinite model, has a model of size k for every infinite cardinal k. You cannot use first-order logic to exclude non-standard numbers unless you want to abandon infinite models altogether.
I refer you to the Lowenheim-Skolem Theorem:
Every (countable) first-order theory that has an infinite model, has a model of size k for every infinite cardinal k. You cannot use first-order logic to exclude non-standard numbers unless you want to abandon infinite models altogether.