Method of the Auxiliary Hypothesis

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The method rests on the Criterion of Deduction:

Let $A$ be a relation in $I$, and $I*$ be the the theory obtained by adjoining $A$ to the axioms of $I$. If $B$ is a theorem in $I*$, then $A \implies B$ is a theorem in $I$.

The proof is on the page 30 in the book Theory of Sets by Bourbaki.


Suppose a field, where the inverse of the zero-element and its existence are not defined in axioms.

The auxiliary hypothesis: adjoin the inconsistent assumption (not yet proved) to the axioms that the zero-element has an inverse:

${0}^{-1}*0 = 1$

According to the axioms of the field:

$0^{-1}*0 = 0*0^{-1}= 0$

Contradiction, since an axiom of the field states the uniqueness of the product. So the zero-element has no inverse by the Method of the auxiliary hypothesis. Otherwise, the set of axioms would be inconsistent.

A common mistake is to assume the non-existence of the inverse for the zero-element from the axioms of the field. It is a consequence, not a definition per se.


Similarity to a proof method from Propositional Logic

If $a \land \neg b$ leads to contradiction, then the proposition $(a \implies b)$ is true.

$(a \implies b) \iff (\neg a \lor b) \iff \neg (a \land \neg b)$


Theory of Sets by Bourbaki