I believe the simple explanation for Godel's incompleteness theorems is that there cannot be a set of all sets including itself, with the exception perhaps of God. Einstein and Godel had offices at Princeton. I believe Einstein said he didn't understand Godel's work.
There is an explanation of the first incompleteness theorem.
"The first incompleteness theorem shows that the Gödel sentence GF of an appropriate formal theory F is unprovable in F. Because, when interpreted as a statement about arithmetic, this unprovability is exactly what the sentence (indirectly) asserts, the Gödel sentence is, in fact, true (Smoryński 1977"
There are practical consequences for the demonstration of the incompleteness or limitations of formal systems. Philosophers at least, as well as computer scientists and mathematicians got more to think about.
The Godel criterion seems reflected in reality at the quantum level in some respects. Even a fundamental particle or wave *in the beginning, requires a precursor outside or before the system starts. Incompleteness exists apparently in the infinities that arise as math physics regress toward the singularity so the model is incomplete.
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