Consider a familiar logical system, such as language, which is capable of reasoning through statements such as “All men are mortal. Socrates is a man” to conclude that “Socrates is mortal.” It’s easy to see, and easy to formally prove, that the last statement follows logically from the first two, given a simple set of algebraic rules (if A = B and B = C then A = C). But Gödel showed that any logical system complex enough to prove mathematical theorems has a fundamental limitation: application of their rules can generate statements that are true, but these statements cannot be proved with the
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