One of the most underappreciated concepts of mathematics, in my view, this quite easily applies to a number of things, from quantitative finance and the economic bailout to psychology and behavior to amorality and the substitution of science for religion by many educated people. This is easily one of the five most important things I've learned about as an undergraduate, in any of the multiple majors I considered.
The link has about 8 different explanations of how this works. I posted an extremely simplistic summary one.
http://www.miskatonic.org/godel.html
Gödel showed that within [any] rigidly logical system... propositions can be formulated that are undecidable or undemonstrable within the axioms of the system. That is, within the system, there exist certain clear-cut statements that can neither be proved or disproved. Hence one cannot, using the usual methods, be certain that the axioms of arithmetic will not lead to contradictions ... It appears to foredoom hope of mathematical certitude through use of the obvious methods. Perhaps doomed also, as a result, is the ideal of science - to devise a set of axioms from which all phenomena of the external world can be deduced.
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