nLab counterexample

Idea

In mathematics, a counterexample is an example for the negation of a statement.

A counterexample to a conjecture is a way to satisfy the hypotheses of the conjecture such that the conclusion is a contradiction.

In constructive mathematics one considers also “weak counterexamples”. See there for more.

Examples

• counterexamples in algebra

• counterexamples in category theory

• Cauchy's mistake

Related concepts

mathematical statements

• theory

• judgement, assertion

• hypothesis, consequence

• axiom

• assignment

• definition (inductive, coinductive)

• lemma

• proposition/type (propositions as types)

• theorem

• proof/program (proofs as programs)

• example, counterexample

• conjecture, folklore

• exercise

• analogy

• tautology

• paradox

• no-go theorem

References

See also:

• Wikipedia, Counterexample

On counterexamples in topology:

• Lynn Arthur Steen, J. Arthur Seebach, Counterexamples in Topology, Springer (1970/1978) [doi:10.1007/978-1-4612-6290-9]

• $\pi$-Base: Topology – a community database of topological counterexamples [topology.pi-base.org]