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 (http://planetmath.org/WellOrderingPrincipleImpliesAxiomOfChoice). 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.

单词	CircularReasoning
释义	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 (http://planetmath.org/WellOrderingPrincipleImpliesAxiomOfChoice). 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.
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