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Does Gödel's Incompleteness Theorems means 2+2=5?

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"I am a layman trying to understand above theorems. This could be a stupid question."

There is no such thing as a stupid question!

"Does these theorems imply that we actually cannot prove that 2+2 = 4???"

Gosh, I may have to reconsider!

No, Godel's theorem say that, given any set of axioms large enough to encompass the properties of the non-negative integers there must exist some theorem that can neither be prove nor disproved.  It does not say that a specific theorem cannot be proved.  In fact, if we were able to identify a specific theorem that can not be proved nor disproved, we can always extend the axioms, perhaps by adding that theorem itself as an axiom, so that theorem can be proved.   Of course, there would then be still another theorem that cannot be proved nor disproved.

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