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??

"To encode each statement" is not quite right. 
"I had Cheerios for breakfast," for example, cannot be encoded. 
The set of things encoded is a quite carefully described set, and metastatements like yours are not within that limited set. Hence your "contradiction" is not a useful one. 
As a gratuitous aside: this stuff isn't simple. You have to actually understand the details, not just the broad strokes, before you can start building counterexamples. 
