The Power Set

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We should now have an interesting, if static, perspective on the set of
all things in the real world. All (say) objects in the
``existential Universe'' can be grouped into sets by permutation,
forming with cardinality . These permutations can in turn
be permuted into sets of sets with cardinality .
However, nature *selects* only a *small subset* of -
particular groupings of objects according to certain rules. We simply
don't see any of the vast, the good-friends-with-the-infinite, other
possible set groupings. We therefore for many purposes define *these* objects to be a basic existential set, e.g. the set of all atoms,
and form its power sets instead of including all of the non-observed
sets from the cosmic all.

Every possible grouping of objects into sets, though, is contained in the recursion. Predicate logic and set theory can only be judged to be a ``theory'' or ``valid'' according to whether or not any given predicate, constructed according to any presumed set of rules, successfully identifies objects in the hierarchy. This construction is so far very nearly axiom free. We have really assumed very little about except that it exists, that it has a finite cardinality, and that it contains discrete identifiable (in the formal sense) objects, objects that can each be mentally permuted.

We are ignoring for the moment many questions of interest to
mathematicians - such as what we need to do if the cardinality of
is truly infinite or if is a continuous set. Dealing with infinity
and continuity is *irrelevant* to our *descriptive* process,
because even if is infinite and continuous we can at least imagine a
similar continuous permutive process (which leads instantly to
infinitely infinite infinities) to generate the analog of ,
causing us to throw away infinitely more unrealized possibilities as we
do not *see* either the infinity or the continuity, only the finity
and immediacy of a single slice of the possibly infinite possible. So
to speak.

To speak strictly metaphorically, even though the Universe may live in a
meta-Universe of possible set groupings analogous to the real line,
infinitely divisible and infinitely permutable in every tiny segment no
matter how small, the Universe itself is just *one* of those
groupings. It may well be like unto an irrational number - infinitely
unlikely in a set with uncountably infinite cardinality - but it *is what it is*. Furthermore, we can always renormalize this imagined
real line so that the Universe is the integer number *one*. There
may be lots of other possibilities out there, but if we can't see them
they really don't matter. If we *can* see them, they stop being
``other possibilities''; our Universe and the sets just
turns out to be larger and more complex than we thought but still is
Unitary. This process of conceptually expanding the Universal set
occurs all the time in physics, as we extend into the microcosm.

Accepting as the extremely naive existential set Universe of thought that our brains co-evolved structured wetware and language to cope with, we can at last consider the laws of thought and see what they mean in terms of this fundamental set-theoretic Universe.

The first of the Laws of Thought, the law of identity, states that any
thing that is, is (itself). However, English (and doubtless Greek or
Sanskrit or other languages in which the law is or has been formulated)
is strongly multivalent and thought is an important thing to get right.
We had best proceed extremely carefully and not assume that we actually
understand what this means. We will begin by *defining* a ``thing''
as ``any object in the hierarchy''.

We also have to be careful to define the word ``is'' (and all various
forms of the verb ``to be'' and - in a moment - the concept of ``not
to be'', or negation of being). Among other things we cannot help but
associate different tenses with this verb. We will therefore have to
agree to *mentally* ignore all concepts such as ``was'', ``will
be'', and so on. Our laws of thought are formulated as *static*
statements associated with a *static* description of sets, not with
a *dynamic* conceptualization of predicates that permits us to
convert one set into another. This is actually remarkably consistent
with physics and relativity theory, where time is just another dimension
like space and one can imagine stepping ``outside'' the set of all
space-time events and considering the whole ball of wax to be .

With that carefully established, the law of identity becomes a
beautiful, tautological existential statement. Any ``thing'' is an
object selected from , and as this set hierarchy was itself
imagined (not ``constructed'', as it a priori existed the instant
itself was established) by a process of identification, this law is the
*law of identification*. Our set Universe is precisely that which
can be identified, drawn from the set of *all permutations* of the
existential set that can be identified. If we can identify, that
is, if a statement selects an object from , then that
statement is valid; otherwise it is not.

Mathematicians and dreamers may object that this definition is cold and
heartless - it excludes all sorts of reasoning about *non*-existential Universes, things we might imagine, things we might
dream up. Basically all abstract thought. Not so - it merely
acknowledges that those subjects contain an *infinity of traps* for
the unwary mind that will require *axioms* to deal with, as it is
absolutely trivial to conceive of imaginary universes in which six
impossible things happen before breakfast^{3.16}. Abstract thought will turn
out to be a simply *lovely* game and all sorts of fun, but we need
to remember that it is a game where we can easily twist the rules back
onto themselves into impossibilities, inconsistencies, paradoxes, and
worse. Not so with . It is the very definition of mundane.

What now of the *difficult* laws, the ones involving *nothing*
and *non-being*, the *negation* of the two ideas that we had to
work so hard to clearly and unambiguously define above so that the law
of identity could be viewed (literally) as a Universal Truth?

Note well that *negation* is a *very subtle and difficult
concept*, so much so that positive set theory^{3.17} *excludes* it and manages to get along amazingly well without it.

Nevertheless, in the English statement of the laws of thought (and in
Aristotle's and Parmenides' Greek statements as well) negation is very
much present, and of course negation is a key part of *logic*, which
either proceeds from the laws of thought or the laws of thought proceed
from logic (depending on who you happen to be speaking to at the time)
so we have to at least figure out what we are going to do with it in our
set theoretic expression of those Laws. Let us start with the law of
contradictions (as I wrote it a couple of sections ago: No thing can
both be and not be. This was a somewhat clumsy way of writing it, but
now that clumsiness will serve us well as we have at last defined what a
``thing'' is and what ``being'' means, which gives us at least a chance
at defining what ``no thing'' and ``non being'' are.

Even so, we will discover that there are *many distinct linguistic
meanings* of negation of ``thingness'' and ``being'' with regard to the
existential Universe. Let us list a few of them. A
``thing'' in the existential set is an object in its associated
Universe, so ``no thing'' might be:

- An object in
*another*, disjoint Universe. For example, if is the set of natural numbers (without worrying in the least about whether or not this set closes at infinity - we will adopt the point of view that infinity is something that we reach by a limiting process and that existential truth is derived from truths associated with finite sets of arbitrary (variable) size), and we introduce an irrational number, that number is ``no thing'' in - it can be identified with no nonnegative integer or set of sets of sets of nonnegative integers. If we consider the set of all apples, it might be an orange. - An
*imaginary*(or if you prefer,*hypothetical*object. This is really a special class of the first case, because we can define an existential set Universe associated with the imagination and awareness, self and otherwise, itself. It is an*important*special class because for each one of us, it is the*only set Universe that we directly experience*. It is also where e.g. all abstract sets such as those of mathematics live, as it is otherwise remarkably difficult to find a piece of in your refrigerator. - This is the really tough one - no object at all, in any set
theory. The negation of all set theories. The utter lack of an
existential set Universe of any sort. If this sounds scary, it should.
The utter negation of thingness of any sort is
*almost inconceivable*to a conscious mind, and this is the source of many religious beliefs that basically assert that this kind of negation cannot exist.

This is the easy part. What about ``being''? We defined being in the
law of identity as ``being identifiable'', where being identifiable
itself basically meant being a set in the hierarchy. Not
being is then pretty straightforward. It means not being in the
existential set Universe, period. Of course this is now an *existential tautology* and *every thing* is *in* the existential
set Universe, unless we somehow embed that Universe in a larger one as
one might embed the natural numbers in the complex plane. Which is
cheating in *so* many ways, especially if the Universe one is trying
to embed is the actual existential physical Universe in which we
live^{3.18} *No thing* is not in this Universe.

However, what about the inheritance from the *other* two forms for a
``thing''? And more important, what about predicates? People tend to
*use* the laws of thought to decide *propositions* or the
set equality of *predicate expressions*. Let us consider these
separately. Being is now well defined for our existential set Universe
and indeed is a tautological extension of the law of identity for that
Universe. We don't really need a law of contradiction in this approach,
only a criterion for establishing identity, which is doubtless the
observation that led to the development of positive set theory.

We are still left with what one might call ``strong'' nonbeing - not
being in any *possible* set Universe, nonbeing in the absolute sense
- and ``weak'' nonbeing where a set *exists*, but just not in the
*right* set Universe, where
exists but where
attempts to reason about it as a natural number involve either some sort
of restriction/projection from the complex plane to the natural numbers
or extension/embedding of the natural numbers in the complex plane. To
do either one requires axioms, many axioms, and many theorems derived
from those axioms besides and hence is far beyond our analysis of the
laws of thought.

There is one more sense in which nonbeing is used, however - and for
better or worse it is one of the most *common* forms of usage and is
*completely different* from the ones associated with identity and
existence in the set Universe. This ambiguity is one major source of
paradox and antimony. In many cases, not being means *not being in
the same identity set*. That is, the law of identity can be interpreted
as saying ``A thing (identifiable in the set Universe for
some existential set S) is either *in* set from somewhere in
that hierarchy or it is in , the complement of (from the
entire hierarchy).'' This is of course a *useful* thing to have
around when trying to decide if an object ``belongs to'' set or if
it doesn't, when trying to define an *axiom of
equality*^{3.19} . It is also the source
of much dark evil when nonbeing in the strong or weak *tautological*
sense are confused with nonbeing in this sense.

The possibility of antimony is apparent when one considers how
differently the *empty set* is treated by the two meanings. The
empty set is is *always* a set *within* any set theory,
existential and permutative or not. Its existence is an axiom in
positive set theory, but one can *also* just ``observe'' it as the
outcome of evaluating a false formula. In our permutative approach, it
is just the set of all set objects selected zero at a time, one of the
possible permutations of objects that exists *even for five year
olds* seeking the various ways of grouping a small pile of pennies on a
table. It is an explicit but often *invisible* member of the
subsets of - I like to think of the empty set at any level
of the hierarchy as being the ``set brackets'' of the hierarchical set
itself, so that
in a manner of
speaking, since of course we technically cannot speak of
all by
itself outside of a set container. If sets are metaphorically objects
in a box, the empty set is the box, which can be empty but always is
there. Its presence is required to that operations like intersection
*close* within the set theory where a full set theory allows set
objects to be manipulated with the operations of union and intersection
as part of its basic definition.

If not being is used in the sense of *not being equal*, or *not
being in the set of true statements* as an essential part of predicate
evaluation, it cannot *also* be used in the sense of *not in the
set Universe*. Russell worked far too hard to define his paradox (which
we will discuss in some detail later). He might just as well have tried
to create ``a set of all things that are not in any set including the
empty set''. Say what? Clearly this kind of ``paradox'' isn't
paradoxical at all in an existential set Universe, it's just a
meaningless statement.

The final law of thought, the law of the excluded middle, tells us that
every thing must either be or not be. Once again, now that we know what
a thing is, we can see that this is a tautology of the law of identity
for the strong or the weak formulation of nonbeing - all things
(objects in the hierarchy) *are*, so sure, they are *or* they are not - not. Once again we see how a wide range of problems
can come from extending over the existential set Universe boundary and
allowing ``things'' to exist (sort of) in sets ``outside'' the
existential set Universe , in which case we can talk about things
that are not, like pink hippogriffs dancing the tango or irrational
numbers in a natural number Universe. We observe that for the strong
version of nonbeing this law sounds rather odd - nothing doesn't exist
where anything or everything do, quite literally.

Finally, we observe that as before this statement has a *different
meaning altogether* when used *inside* a set Universe as another
form of disjunction. In this context we can interpret the law as saying
that every object in is either a member of any *particular* set in or it isn't. There isn't anyplace else
for it to be, after all, because we exclude imaginary Universes or
embeddings of the set and because no Universe at all cannot exist in
the presense of a Universe that does^{3.20}.

If we mix up these different interpretations, and use contradiction or excluded middle on the one hand to refer to actual impossibilities and on the other hand to partitionings of actualities into disjoint sets (be those sets sub or super to the set Universe in question) then we are bound to get ourselves into trouble.

Still, this was a generally successful effort. We note that the law of
identity is a pure tautology when expressed in terms of an existential
set theory, and that the laws of contradiction and excluded middle are
*irrelevant* restatements of the *same* tautology - just
another way of stating the principle of identification that defines the
set Universe as the hierarchy in the first place. We also
see that there are at least two or three *other* ways these laws can
be interpreted with greater or lesser meaning and utility. The
difference is that these interpretations require *axioms* where I
would argue that the law of identity itself and its two strong-form
statements of negation are tautologies of a naive theory that is nearly
axiom free after the assumption ``suppose one has a set of objects
that actually exist''.

It seems worthwhile to see how our new strong definition of nonbeing
enters into existential set theory as the absence of any set *including the empty set* as the empty set is very much a part of
. To make up a formal theory that manages to *sound* like
a set theory extended to ``include nothing'' let us now introduce a new
concept (at least in western thought) - the *null set*.

The Null Set

The Power Set

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