It is perhaps worthwhile to formalize this, to define and extend
traditional naive set theory algebraically just a bit to encompass the
null set, the ``set of all things that cannot be put into sets'', where
we insist that all subsets in any set theory *already* de facto
include the empty set, so that this ``set'' isn't one and is necessarily
distinct from the empty set.

First, like good algebracians let us give the null set the symbol
suggested above: . This will help us differentiate it from the
empty set . To simplify the algebra and show cleanly that the
empty set is inside of it, we will introduce at the beginning an ``empty
object'' which is in our existential set Universe. Rather than
introduce an extra ``empty object placeholder'' in a list of objects
(which would work just fine) we will treat the *brackets
themselves*, the set boundary, as the empty object.

Then given a Universal set of objects with their
identity subsets
(recognizable as permutations of all the group's objects one at a time
and the *implicit* empty identity subset (the
permutation of all the group objects zero objects at a time), they can
be grouped into subsets in many ways via the union process
e.g. :
where in particular
. Each of these subgroups represents a
unique (unordered) permutation of the set objects. Note well that *all sets* include the brackets and hence the empty set.

In spite of the apparently discrete index on the set objects, do not be
fooled - this index is discrete only in the sense of indicating
uniqueness and should not be taken to mean that we can actually
algebraically *specify* all the identity subsets for any given space
in the sense of creating a mapping between some set of symbols and the
set objects. In this I am being no sloppier, really, than any set
theorist is when discussing a set that might have infinite
cardinality (uncountably infinitely many members) such as any interval
of the real number line.

In English, these existential set theoretic statements say that all things that exist (set objects) can be placed in identity subsets, the union of all things that exist is all things that exist and that all non-empty subsets of all things that exist can be built out of unions of identity subsets (all of which seems pretty obviously true, given a Universal set of ``things that exist'' and a union and permutation process capable of handling continuum manifolds if that is what the Universal set happens to be).

Given this, the following three statements (plus the notion that any given subgroup can be formed by - or better yet selected out of the permutations of - the unions of identity groups) fully specify the notion of the Law of Identity:

(3.4) |

(3.5) |

(3.6) |

In this approach we *do not require any special treatment of the
empty set* in the algebra. It is just the ``zero'' of the algebra and
lies within it just as in arithmetic so all numbers
``contain zero'', and can be (the empty object) as
easily as a nonempty member.

Now, however, we *add* the following `black hole'' relations:

(3.7) |

(3.8) |

These are *very different from the properties of the empty set!* Set
operations involving (the undefined or null set) are without
exception themselves *undefined or null*. One *cannot* in any
sensible way take the union of ``undefined'' (which is neither an object
nor the absence of an object) with a list of objects and end up with a
list of objects, not even an empty one. Nor can one take the
intersection. isn't, really, a set and doesn't live ``in'' the
Universe .