There are two problems with this.
First, writing something down in English is very different from formalizing it. Natural language interacts with human brains in all kinds of complicated ways that we do not fully understand, so just because we can make a compelling-sounding argument in English doesn't mean that argument doesn't have holes somewhere.
Second, the incompleteness theorems apply only to given formal systems, not to formality in general. Given a system, you can produce a Godel sentence for that system. But any Godel sentence for a given system has a proof in some other more powerful system. This is trivially true because you can always add the Godel sentence for a system as an axiom.
This is a very common misconception about math even among mathematicians. Math produces only conditional truths, not absolute ones. All formal reasoning has to start with a set of axioms and deduction rules. Some sets of axioms and rules turn out to be more useful and interesting than others, but none of them are True in a Platonic sense, not even the Peano axioms. Natural numbers just happen to be a good model of certain physical phenomena (and less-good models of other physical phenomena). Irrational numbers and complex numbers and quaternions etc. etc. turn out to be good models of other physical phenomena, and other mathematical constructs turn out not to be good models of anything physical but rather just exhibit interesting and useful behavior in their own right (elliptic curves come to mind). None of these things are True. At best they are Useful or Interesting. But all of it is formalizable.