I'm so sorry but confusions abound in your article. First of all the comparison between there being two square roots and the law of non-contradiction is unfounded. That all positive numbers have two square roots does not entail a contradiction.
Second, you misunderstand Gödel's incompleteness theorems, which pertains to proof, not truth. Truth cannot be established within the theorem. As for application to physics, that's gobbledygook as the proof is purely formal. You may want to look at Turing computability to see the implications that extend to idealized machines, which subsume possible ones