Godel proved incompleteness by building a sentence that says of itself 'this is not provable'

In his 1931 paper, Kurt Godel proved that any consistent formal system rich enough for arithmetic must contain statements that are true but cannot be proved within it. He did so by encoding logical formulas as numbers, a technique now called Godel numbering, and then constructing a sentence that in effect says of itself “this statement is not provable.” If the system could prove it, the system would be inconsistent; if it cannot, the statement is true and yet unprovable. His second theorem went further: no such system can prove its own consistency.