Showing that it is impossible to prove a statement from some facts.

Question

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Show that the Completeness Axiom can not be proved using the 10 axioms of an ordered field.

The 10 axioms are 5 axioms for ordering properties and, commutativity, associativity, identities, inverses and ditiributivity.

I can not figure out the necessary condition to show something can not be proved by specified facts.

Answer 1

This means that you can find an example of an ordered field which does not satisfy the completness axiom.

Now: $\Bbb Q$ is such a field.

If completeness were a logical consequence of the axioms, it would follow that every ordered field is complete. But $\Bbb Q$ isn't, so completenss is not a logical consequence of the axioms of ordered fields.