Axiom of Empty Set

We have arrived in Halmos' book at chapter III. However, because we are lacking a 'Specification Axiom' (a la Halmos chapter II), it is necessary to start with at least one set which comes out of the blue sky. Fortunately though, the Axiom of empty set may be one of the axioms of Zermelo Fraenkel set theory. The empty set is a set without elements. The axiom of Extension says that all such sets are equal. Thus there is only one empty set. The usual symbol for the empty set is: $$ \emptyset $$ But because it's is a set with no elements (between its curly braces) another useful notation for this "empty box" is: $$ \{\} $$ The only property of the empty set to be mentioned in this section is the fact that it is a subset of any set $A$: $$ \emptyset \subseteq A $$ How is the empty set implemented in our set theory? As an empty array:

SetLength(empty,0);

And indeed, this is true in theory as well as in practice: the empty array is a subset of any array.
Another implementation of the empty set in our set theory is a bitmap of zeroes; the bitmapped implementation is any with no bits up:

    .. 0000000000

However, exactly the same bit sequence can be interpreted as a number. What number does it represent then? Answer: the number zero. Therefore we conclude that the empty set is equivalent with zero. Or rather that it is zero. Not quite perhaps. Let's stay cautious for the moment being and write the zero with an overline, to indicate that it is a set, not (yet) a number: $$ \{\} = \overline{0} $$ A little philosophy may be in place here. It is noted that the content of the empty set is emptiness itself. But the empty set is equivalent with zero. Thus zero is not emptiness itself. It contains emptiness. Emptiness itself is represented in programming languages by something even more nothing than zero, commonly called NIL or NULL. End of philosophical note.