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## Axiom of integers

Because we treat the integers we have axiom for integers instead of the axiom of the empty set. of the integers.

This may not seem satisfactory since it assumes the integers already exist. However we are assuming the framework of C ++ and they do exist in that framework. All of these axioms are informal. If you want to do a fully formal system you need to start with first order arithmetic or some other system strong enough to embed a universal TM and then define the constructs in C ++ that we use in that system. The code for this is trivial. You only need to have an integer object as a member of class Functional and return the value of that object for member function int_value. To write the code for this we need to define the class Functional. The full class definition is in Figure .

Figure: Functional base class

Many of the member functions have already been described. We will describe here some additional functions that are generic. Others will member functions and the code for member functions that are not a part of the class definition will be included where and if they are needed. Some functions are for housekeeping or other purposes and will not be described here. They included because this is compiled C ++ code which is intended to be used and has been to a limited degree tested. The source code is available at TO BE DETERMINED.

We use virtual C ++ functions to make this classes open ended. A Functional maps Functional's to other Functional's through a virtual function operator(). By deriving classes from base class Functional and by redefining the virtual member functions, such as operator(), we have a completely open structure with flexibility similar to that of a `set' in set theory. Because every instance of a Functional must be implemented as an effective procedure, we need to know if a given parameter is valid. This is determined by another virtual function valid_parameter  . This is the base class for all functionals no matter what structure they represent. As new levels of the hierarchy are defined the virtual functions in Functional must be defined in derived classes to work with all previously defined code.

If an integer is being represented then the_limit is the integer being represented otherwise it is -1. Member union_depth  is used to take the union of an infinite set. Virtual member function clone_base  is used to create a copy of this object including all the derived classes that this object may also be an instance of. This function is never called directly but only by function clone  .

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