Can any statement about natural numbers be written in TNT?

Question

I was trying to understand Godel's theorem from here Godel's First Incompleteness theorem. I still do not completely understand the theorem, but have a broad idea about it. As the link mentions Godel's first theorem addresses statements in Typographical Number Theory ( TNT ) which are true but unprovable using the axioms. The link assumes any statement about natural numbers can be written in TNT, such as "4 is a prime", "2 is the smallest prime number" etc. But it does not prove it in general. I am not even sure what qualifies as a statement about natural numbers. How do I put it mathematically what qualifies as a statement about natural numbers and how do I prove that every such statement can be written using TNT ?
Sorry if my question is vague or lacks correct terminology, I can improve it if suggested.

Answer 1

There are lots of statements which are not expressible in the language of any specific theory, such as TST or Peano arithmetic (which I'm more familiar with). Any statement involving a symbol/word not in the language of the theory can't be directly expressed. Specific examples depend on the theory involved, but for a general kind of example, the sentence "$N$ is (the Godel number of) a sentence in the language $L$ which is true of the natural numbers" is never expressible in the language $L$ - this is Tarski's theorem on the undefinability of truth, and is a direct consequence of the diagonal lemma.

But we're not really interested in non-expressible statements most of the time. The surprise of Godel's theorem is that (for any reasonable system) there are expressible statements which are not decidable.