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.
Self-defining sets have something of a tataulogous nature. Godel wrote;”
“Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory.”
Google’s AI wrote; “An example of the incompleteness theorem in basic arithmetic is a statement like “There exists a natural number that is not the Gödel number of any provable statement within Peano Arithmetic,” which, if Peano Arithmetic is consistent, is true but cannot be proven within Peano Arithmetic itself; essentially, it describes a true statement about natural numbers that the system cannot demonstrate as true using its own rules.”
Piano Arithmetic https://youtu.be/PkqZU8Crc6k?feature=shared
Chris Weare If one exists within an incomplete system theories made from within would experience a paradigm comparable to the incompleteness theorems. Pragmatic value…Quine-Putnam’s indispensability argument.line in support of the necessity of math for physics would probably remain within the parameters of incompleteness.
If there are multiple cosmological accounts that are complete and work, it is probably that only one could be true, and that one incomplete even so, like Newton’s theory of gravity and as Einstein’s is likely to be, as well as the next though modern cosmologists claim a ‘theory of everything’s is just around the corner.
Garrison Gibson: books and blog 