It would take a book-length response to adequately reply to your question - more about that later. But in short here's an answer: a theorem is the conclusion of a valid argument (i.e. a "proof") based on certain assumptions ("hypotheses" or "axioms"). And Godel's result is such a theorem. I won't attempt to define a paradox, but certainly it seems his theorem might also be regarded (and has been) as a paradox as well. But a better answer might be found in the book "Godel's Theorem: an incomplete guide to its use and abuse", by Torkel Franz\'en - I hugely recommend it, if you haven't already done so. -= rags =- PS: Godel's theorem isn't based on the liar paradox - that's Tarski's theorem - but rather on a closely related paradox about provability - which isn't really a paradox after all ... PPS - you don't really think theorems "close development or debate", now, do you?! Experience suggests otherwise I think. On Sun, 8 Jan 2017, Patrik Eklund wrote:
Since the Incompleteness Theorem uses the Liar Paradox, why is it called the Incompleteness Theorem and not the Incompleteness Paradox?
A Theorem closes a development or debate, and calls for admiration (because the inventor did something supposedly good), whereas a Paradox opens up development and debate (since the detector has pointed at something being wrong), and delays the call for admiration of the disruptively innovative solution until it is really deserved.
Best,
Patrik
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