yes jdvelasc it's a little hard to figure out what your objection actually is here if this sentence is not a formalization of what it's asserted to be a formalization of then there just is no such thing as a formalization in first order logic no first order sentence can fix its interpretation your original claim is that x can't be a natural number due to the soundness theorem but that's not so the soundness theorem says only that there is really no such natural number x it does not refute the possibility that the theory t happens to prove that there is such a natural number t may prove false things and if it's omega inconsistent must prove false things the soundness theorem does not refute that because the soundness theorem applies only if all the axioms are true and some of the axioms of t may in fact in this case must be false