Circular reasoning is an attempted proof of a statement that uses at least one of the following two things:
the statement that is to be proven
a fact that relies on the statement that is to be proven
Such proofs are not valid.
As an example, below is a faulty proof that the well-ordering principle implies the axiom of choice (http://planetmath.org/WellOrderingPrincipleImpliesAxiomOfChoice). The step where circular reasoning is used is surrounded by brackets [ ].
Let be a collection of nonempty sets. By the well-ordering principle, each is well-ordered. [For each , let denote the well-ordering of .] Let denote the least member of each with respect to . Then a choice function can be defined by .
The step surrounded by brackets is faulty because it actually uses the axiom of choice, which is what is to be proven. In the step, for each , an ordering is chosen. This cannot be done in general without appealing to the axiom of choice.
|Date of creation||2013-03-22 16:06:32|
|Last modified on||2013-03-22 16:06:32|
|Last modified by||Wkbj79 (1863)|