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Proof theory is a major branch [1] of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical ... For example, it ...
A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of probability theory. Probabilistic proof, like proof by construction, is one of many ways to prove existence theorems. In the probabilistic method, one seeks an object having a given property, starting with a large set of candidates.
Kőnig's theorem (set theory) Kőnig's theorem (graph theory) Lagrange's theorem (group theory) Lagrange's theorem (number theory) Liouville's theorem (complex analysis) Markov's inequality (proof of a generalization) Mean value theorem; Multivariate normal distribution (to do) Holomorphic functions are analytic; Pythagorean theorem; Quadratic ...
Indeed, the above proof that the law of excluded middle implies proof by contradiction can be repurposed to show that a decidable proposition is ¬¬-stable. A typical example of a decidable proposition is a statement that can be checked by direct computation, such as " n {\displaystyle n} is prime" or " a {\displaystyle a} divides b ...
A complete proof for this case was published by Aschbacher and Smith in 2004. In 1986, Spencer Bloch published the paper "Algebraic Cycles and Higher K-theory" which introduced a higher Chow group, a precursor to motivic cohomology. The paper used an incorrect moving lemma; the lemma was later replaced by 30 pages of complex arguments that ...
The example mapping f happens to correspond to the example enumeration s in the picture above. A generalized form of the diagonal argument was used by Cantor to prove Cantor's theorem: for every set S, the power set of S—that is, the set of all subsets of S (here written as P(S))—cannot be in bijection with S itself. This proof proceeds as ...
Ne’Kiya Jackson and Calcea Johnson have published a paper on a new way to prove the 2000-year-old Pythagorean theorem. Their work began in a high school math contest.
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]