I am trying to understand Gödel's First Incompleteness Theorem. Since Imaginary numbers make mathematics closed, using Gödel's First Incompleteness Theorem it would mean there is an inconsistency. I would like to know if this thinking is correct.
$$\sqrt{x}\space\cdot\space\sqrt{x} = \sqrt{x^2} \rightarrow \lvert x \rvert $$ $$\therefore \sqrt{-1} \space \cdot \space \sqrt{-1} = 1$$ $$but \space i \space \cdot \space i = -1$$
Would this be considered an inconsistency in mathematics due to Gödel's Incompleteness Theorem?