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(Mathematical Logic:)
A theory is consistent iff it is free of contradictions: For any proposition P, it's possible to prove at most one of P, ~P from the axioms of the theory.

One may prove that a theory is consistent iff it has a model:

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If it's consistent, we need the axiom of choice to get a model. The proof is extremely technical, but in a sense trivial. We work in a universe which contains just those objects that the theory's language can mention. We arrange for all provable propositions (theorems) P to hold, and then Choose for any unprovable pair (P,~P) which holds, in a "consistent" manner.
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If it has a model, then for any proposition P either P holds in the model or ~P holds in the model. If P holds then ~P cannot be proven (or it would be true in the model, since the model satisfies the axioms), and if ~P holds then P cannot be proven. So a contradiction can never be proved.

This direction is free of the axiom of choice.

This is usually known (for first order logic) as Gödel's completeness theorem (NOT the same as his incompleteness theorem!)

Con*sist"ent (?), a. [L. consistens, p.pr.: cf. F. consistant.]

1.

Possessing firmness or fixedness; firm; hard; solid.

The humoral and consistent parts of the body. Harvey.

2.

Having agreement with itself or with something else; having harmony among its parts; possesing unity; accordant; harmonious; congruous; compatible; uniform; not contradictory.

Show me one that has it in his power To act consistent with himself an hour. Pope.

With reference to such a lord, to serve and to be free are terms not consistent only, but equivalent. South.

3.

Living or acting in conformity with one's belief or professions.

It was utterly to be at once a consistent Quaker and a conspirator. Macaulay.

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