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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.   "