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One of the widely used types of impossibility proof is proof by contradiction.In this type of proof, it is shown that if a proposition, such as a solution to a particular class of equations, is assumed to hold, then via deduction two mutually contradictory things can be shown to hold, such as a number being both even and odd or both negative and positive.
In mathematics there is the concept of proof of impossibility referring to problems impossible to solve. The difference between this impossibility and that of the no-go theorems is that a proof of impossibility states a category of logical proposition that may never be true; a no-go theorem instead presents a sequence of events that may never occur.
Proving a negative or negative proof may refer to: Proving a negative, in the philosophic burden of proof; Evidence of absence in general, such as evidence that there is no milk in a certain bowl; Modus tollens, a logical proof; Proof of impossibility, mathematics; Russell's teapot, an analogy: inability to disprove does not prove
This does not follow from Abel's statement of the theorem, but is a corollary of his proof, as his proof is based on the fact that some polynomials in the coefficients of the equation are not the zero polynomial. This improved statement follows directly from Galois theory § A non-solvable quintic example. Galois theory implies also that
Impossibility of performance, in contract law an excuse for non-performance of a contract; Impossibility defense, a criminal defense for a crime that was legally impossible to commit; Proof of impossibility, in mathematics a proof that demonstrates that a particular problem cannot be solved
Baron Munchausen pulls himself out of a mire by his own hair.. In epistemology, the Münchhausen trilemma is a thought experiment intended to demonstrate the theoretical impossibility of proving any truth, even in the fields of logic and mathematics, without appealing to accepted assumptions.
Such a proof is again a refutation by contradiction. A typical example is the proof of the proposition "there is no smallest positive rational number": assume there is a smallest positive rational number q and derive a contradiction by observing that q / 2 is even smaller than q and still positive.
Bertrand's postulate and a proof; Estimation of covariance matrices; Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational