circular reasoning

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. The step where circular reasoning is used is surrounded by brackets [ ].

Let $C$ be a collection of nonempty sets. By the well-ordering principle, each $S \in C$ is well-ordered. [For each $S \in C$ , let $<_S$ denote the well-ordering of $S$ .] Let $m_S$ denote the least member of each $S \in C$ with respect to $<_S$ . Then a choice function $\displaystyle f \colon C \to \bigcup_{S \in C} S$ can be defined by $f(S)=m_S$ .

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 $S \in C$ , an ordering is chosen. This cannot be done in general without appealing to the axiom of choice.