No, not at all. The Incompleteness Theorem is more like, that there will always be things that can't be proven.
Further, it is impossible to find a complete and consistent set of axioms, meaning you can find an incomplete set of axioms, or an inconsistent set of axioms, but not both a complete and consistent set.
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Gödel's incompleteness theorem was a theorem that Kurt Gödel proved about Principia Mathematica, a system for expressing and proving statements of number theory with formal logic. Gödel proved that Principia Mathematica, and any other possible system of that kind, must be either incomplete or inconsistent: that is, either there exist true statements of number theory that cannot be proved using the system, or it is possible to prove contradictory statements in the system.
A quantum theorem does not exist.
Although the mathematical facts of the theorem existed - even before humans did - the theorem itself did not exist until Pythagoras thought of it. In that sense, he did not FIND it because it did not exist until he had thought of it.
no it dose not
It does not.If you consider a right angled triangle with minor sides of length 1 unit each, then the Pythagorean theorem shows the third side (the hypotenuse) is sqrt(2) units in length. So the theorem proves that a side of such a length does exist. However, it does not prove that the answer is irrational. The same applies for some other irrational numbers.