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The choice of axioms for any formal system is to some degree a matter of aesthetics, as it is commonly the case that many different selections of formal rules will serve as axioms to derive all the rest as theorems.  As it happens, the example of an algebraic law that we noticed first, <math>a(~) = (~),\!</math> as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms.

The choice of axioms for any formal system is to some degree a matter of aesthetics, as it is commonly the case that many different selections of formal rules will serve as axioms to derive all the rest as theorems.  As it happens, the example of an algebraic law that we noticed first, <math>a \texttt{( )} = \texttt{( )},~\!</math> as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms.

We might also notice at this point a subtle difference between the primary arithmetic and the primary algebra with respect to the grounds of justification that we have naturally if tacitly adopted for their respective sets of axioms.

We might also notice at this point a subtle difference between the primary arithmetic and the primary algebra with respect to the grounds of justification that we have naturally if tacitly adopted for their respective sets of axioms.