What Do Gödel's Incompleteness Theorems Truly Mean?
Arthur T Knackerbracket writes:
In 1931, by turning logic on itself, Kurt Godel proved a pair of theorems that transformed the landscape of knowledge and truth. These "incompleteness theorems" established that no formal system of mathematics - no finite set of rules, or axioms, from which everything is supposed to follow - can ever be complete. There will always be true mathematical statements that don't logically follow from those axioms.
I spent the early weeks of the Covid pandemic learning how the 25-year-old Austrian logician and mathematician did such a thing, and then writing a rundown of his proof in fewer than 2,000 words. (My wife, when I reminded her of this period: "Oh yeah, that time you almost went crazy?" A slight exaggeration.)
But even after grasping the steps of Godel's proof, I was unsure what to make of his theorems, which are commonly understood as ruling out the possibility of a mathematical "theory of everything." It's not just me. In Godel's Proof (a classic 1958 book that I heavily relied upon for my account), philosopher Ernest Nagel and mathematician James R. Newman wrote that the meaning of Godel's theorems "has not been fully fathomed."
Maybe not, but six decades have passed since then. Where are we with these ideas today? Recently, I asked logicians, mathematicians, philosophers, and one physicist to discuss the meaning of incompleteness. They had plenty to say about the implications of Godel's strange intellectual achievement and how it changed the course of humanity's unending search for truth.
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It is easy to lose one's sense of wonder at the fact that such a blindingly obvious set of axioms - the Peano axioms for arithmetic (the set of rules about the natural numbers 0, 1, 2, 3 ... closely related to the system that Godel used in his proof, such as the rule, "Every number has a successor") - is essentially incomplete and undecidable, meaning that all axiomatizable consistent extensions are incomplete and undecidable. Hold on to that wonder! The incompleteness theorems teach us that when it comes to our attempt to master the conceptual order, whether it be in mathematics or, for that matter, in any other domain, we will always fail - and indeed, in this case more than any other, we should be glad to have failed, for failure was clearly the more interesting, the more profound, outcome.
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