Changes

|}

|}

Finally, the reflectively extended naming function <math>\operatorname{Nom}' : O' \to S'\!</math> is defined as <math>\operatorname{Nom}' = \operatorname{Nom} \cup Nom_1.\!</math>

Finally, the reflectively extended naming function Nom' : O' > S' is defined as Nom' = Nom U Nom'.

A few remarks are necessary to see how this way of defining a CRE can be regarded as legitimate.

A few remarks are necessary to see how this way of defining a CRE can be regarded as legitimate.

In the present context an application of the arch notation "<x>" is read on analogy with the use of any other functional notation "f(x)", where "f" is the name of a function f, "f( )" is the context of its application, "x" is the name of an argument x, and where the functional abstraction "x > f(x)" is just another name for the function f.

In the present context an application of the arch notation, for example, <math>{}^{\langle} x {}^{\rangle},\!</math> is read on analogy with the use of any other functional notation, for example, <math>f(x),\!</math> where <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> is the name of a function <math>f,\!</math> <math>{}^{\backprime\backprime} f(~) {}^{\prime\prime}\!</math> is the context of its application, <math>{}^{\backprime\backprime} x {}^{\prime\prime}\!</math> is the name of an argument <math>x,\!</math> and where the functional abstraction <math>{}^{\backprime\backprime} x \mapsto f(x) {}^{\prime\prime}\!</math> is just another name for the function <math>f.\!</math>

It is clear that some form of functional abstraction is being invoked in the definition of Nom1, above.  Otherwise, the expression "x  > <x>" would indicate a constant function, one that maps every x in its domain to the same sign or code for the letter "x".  But if this is allowed, then it seems either to invoke a more powerful concept, lambda abstraction, than the one being defined or else to attempt an improper definition of the naming function in terms of itself.

It is clear that some form of functional abstraction is being invoked in the definition of Nom1, above.  Otherwise, the expression "x  > <x>" would indicate a constant function, one that maps every x in its domain to the same sign or code for the letter "x".  But if this is allowed, then it seems either to invoke a more powerful concept, lambda abstraction, than the one being defined or else to attempt an improper definition of the naming function in terms of itself.