Question about Gödel's Incompleteness Theorem Gödel used prime factorization to encode each statement with a unique number (which is Gödel numbering).
But I wonder if this statement can be encoded:

"If this statement can be encoded, then this statement is false."

If it can be encoded, then it will be false and the truth will become "This statement can be encoded and this statement is true." And that leads to a contradiction, so it cannot be encoded.
Are there any mistakes? If not, did I just prove that Gödel's numbering is wrong??

Godel proved his result by first setting up a system of mathematical logic that can do the basics of arithmetic.  It includes the symbols $\Rightarrow$ "if then", $\wedge$ "and", etc.  It also includes the natural numbers and some of their operations.  In order for your statement to work, you would need to encode "If this statement can be encoded, then this statement is false." into his system.  Then you would need to show that this encoded statement is not itself undecidable.  One or both of these things is not possible.
