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This page lists notable examples of incomplete or incorrect published mathematical proofs. Most of these were accepted as complete or correct for several years but later discovered to contain gaps or errors. There are both examples where a complete proof was later found, or where the alleged result turned out to be false.
In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction [1] used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. [2]
Oxymoron: a combination of two contradictory terms; Zeugma and Syllepsis: the use of a single phrase in two ways simultaneously; Pun: deliberately mixing two similar-sounding words; Slang: the use of informal words or expressions; Techniques that involve the manipulation of the entire sentence or passage. Dog Latin
If the form of the contradiction is that we can derive a further counterexample D, that is smaller than C in the sense of the working hypothesis of minimality, then this technique is traditionally called proof by infinite descent. In which case, there may be multiple and more complex ways to structure the argument of the proof.
Uncle Joe basically makes the argument that (¬A ⇒ B) and (¬A ⇒ ¬B) are contradictory, saying that the same antecedent cannot result in two different consequents. This purported contradiction is the crux of Joe's "proof". Carroll presents this intuition-defying result as a paradox, hoping that the contemporary ambiguity would be resolved.
The infinitely dense gravitational singularity found as time approaches an initial point in the Big Bang universe is an example of a physical paradox.. A common paradox occurs with mathematical idealizations such as point sources which describe physical phenomena well at distant or global scales but break down at the point itself.
In literature, the paradox is an anomalous juxtaposition of incongruous ideas for the sake of striking exposition or unexpected insight. It functions as a method of literary composition and analysis that involves examining apparently contradictory statements and drawing conclusions either to reconcile them or to explain their presence.
A similar example is the theory of real closed fields, which is essentially equivalent to Tarski's axioms for Euclidean geometry. So Euclidean geometry itself (in Tarski's formulation) is an example of a complete, consistent, effectively axiomatized theory.
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