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Has anything that had once been proven true in mathematics actually been shown to be false later?


Short answer—-no: since nothing proven is false.

Long answer—-there have turned out to be models of axioms that seem self-evidently false. And in some cases, those models are not just theoretical curiosities but are of deep practical significance.

Example: There are models in which the parallel axiom (parallels don’t intersect) is false, and one of those models describes actual space.

Example: There are models in which there are infinitesimals (non-null quantities less than any given quantity), and those models are needed to explain actual occurrences (e.g. how it is that a coin can occupy a given place after landing, even though the chances of its doing so are one divided by the number of real numbers, this being smaller than any finite quantity).

Also, there are counterintuitive results, e.g. that a set can have the same number of elements as a proper subset of itself or, to take a similar example, that the points one a line can be mapped onto those of a region, which in turn can be mapped onto those of a plenum, the latter not just being a curiosity, but being necessary to explain the existence of motion.

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