idea
Gödel numbering: statements become numbers
Gödel did not turn ordinary language into logic. He started with a small formal language (symbols like 0, +, ×, =, “for all”) and gave each symbol a number. A statement is a row of symbols, so it becomes one big number. A proof is a list of statements, so it gets a number the same way.
After that, “this number is a valid proof of that number” is a plain claim about numbers, which arithmetic can express. That is how math ends up talking about itself.