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In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. [1] For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are ...
One example is the parallel postulate, which is neither provable nor refutable from the remaining axioms of Euclidean geometry. Mathematicians have shown there are many statements that are neither provable nor disprovable in Zermelo–Fraenkel set theory with the axiom of choice (ZFC), the standard system of set theory in mathematics (assuming ...
Shianghao Wang found a counterexample in 1948 and published a corrected version of the theorem in 1950. In 1934 Severi claimed that the space of rational equivalence classes of cycles on an algebraic surface is finite-dimensional, but Mumford (1968) showed that this is false for surfaces of positive geometric genus.
This is now known to be false. The first counterexample was constructed by Atiyah & Hirzebruch (1961). Using K-theory, they constructed an example of a torsion cohomology class—that is, a cohomology class α such that nα = 0 for some positive integer n —which is not the class of an algebraic cycle. Such a class is necessarily a Hodge class.
Models And Counter-Examples (Mace) is a model finder. [1] Most automated theorem provers try to perform a proof by refutation on the clause normal form of the proof problem, by showing that the combination of axioms and negated conjecture can never be simultaneously true, i.e. does not have a model.
This page will attempt to list examples in mathematics. To qualify for inclusion, an article should be about a mathematical object with a fair amount of concreteness. Usually a definition of an abstract concept, a theorem, or a proof would not be an "example" as the term should be understood here (an elegant proof of an isolated but particularly striking fact, as opposed to a proof of a ...
For example, a particular statement may be shown to imply the law of the excluded middle. An example of a Brouwerian counterexample of this type is Diaconescu's theorem, which shows that the full axiom of choice is non-constructive in systems of constructive set theory, since the axiom of choice implies the law of excluded middle in such systems.
Many natural problems in geometric graph theory, especially problems of recognizing geometric intersection graphs and straightening the edges of graph drawings with crossings, belong to , and are complete for this class. Here, completeness means that there exists a translation in the reverse direction, from an arbitrary sentence over the reals ...