Why is Godel's incompleteness theorem difficult to prove: the point is in the formulations and not only in the essence

Roughly speaking, Gödel’s incompleteness theorem asserts that there are true mathematical statements that cannot be proved. When I was in the 11th grade, the three of us together with the geometry teacher Mr. Olsen and my friend Uma Roy spent five weeks reading the original proof of Gödel. Why so long? Partly because we were still schoolchildren. Partly because the 24-year-old Gödel was not the most talented writer. But mainly because the proof is actually quite difficult.

This may seem surprising, because all the evidence in fact can fit in one paragraph. Godel begins by building a mathematical statement, essentially equivalent to a sentence,

This statement is impossible to prove.

Godel then considers what happens if this statement is false. That is, if this statement can be proved. But any statement that can be proved must be true — here is a contradiction. From this, Godel concludes that the statement must be true. But since the statement is true, it follows that the statement cannot be proved. Please note that this final statement is not a contradiction. On the contrary, this is the proof of Gödel's theorem.

So why is real evidence so complex? The trick is that what may sound like a valid mathematical statement in English is often not (especially when the sentence refers to itself). For example, consider the following sentence:

This sentence is false.

A sentence is meaningless: it cannot be false (because it would make it true) and it cannot be true (because it would make it false). And, of course, it cannot be written in the form of a formal mathematical statement.
')
Here is another example (known as the Berry paradox):

Define {x} as the smallest positive integer that cannot be described in less than 100 words.

This may look like a valid mathematical definition. But again, it does not make sense. And, which is important for the sanity of mathematics, no similar statement can be written formally, that is, mathematically.

Even statements in the language of mathematics may be meaningless:

$$

$$S = \ {A \ mid A \ not \ in A \}$$

(i.e $$$S$- this is a set of sets $$$A$which are not elements of themselves).

This is again a nonsensical definition (known as the Russell Paradox). In particular, as soon as we have identified $$$S$we can ask the question whether $$$S$yourself? If so, then $$$S$cannot be a member $$$S$- a contradiction; and if not, then $$$S$will be a member $$$S$- again a contradiction.

The point of these three examples is that if you want to prove theorems about mathematical statements, then you should be very careful about the fact that you really use mathematical statements. Indeed, from the 46 definitions at the beginning to the surprisingly dense evidence at the end, the original article by Gödel is nothing more than a massive exercise in caution.