What does Godël’s Incompleteness Theorem actually mean and imply? I just saw Ted-Ed’s video on this topic and didn’t fully understand what it means or what the implications of this are.
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It basically just says that, for certain types of mathematical systems (I can never remember the actual definition, but it ends up being most systems we care about), there will always be theorems that are _independent_ of the axioms of the system and/or that the system itself is _inconsistent_ (i.e. implies a contradiction). An independent statement is a statement that cannot be definitely proven true or false _within_ that system. Or, more precisely, there will be _models_ of the system where the statement is true and others where its false.
The analogy that I typically use is that, for a given work of fiction, there will almost certainly be questions that have no canonical answer because the author simply hasn’t provided sufficient information within the canon to answer. For example, in the Harry Potter universe, there’s no canonical answer to the question of how _exactly_ a horcrux is made because J.K. Rowling never specified. Hence, there can be fan fictions that answer the question one way and others that answer it a different way, and, all else equal, each one would be equally valid, so long as they don’t contradict anything from the original work. Of course, this is only an analogy and certainly an imperfect one, but it really helped me get comfortable with the idea of something being “true in one model but false in another.”
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