Set Theory and the Laws of Thought

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The null set is a difficult, perhaps even impossible, concept. We can
perceive existence - indeed we cannot help but perceive existence as
existence and perception are inextricably linked - to each of us,
individually. The fact that we perceive means that we exist, as
Descartes noted so long ago^{3.21} . How can can we even begin to
understand *non*-existence in the deepest sense? Why is it needed
when there is already an empty set which can be proved to be singular
and unique^{3.22}.

I would argue that the empty set is there for a very specific reason -
so that the *algebra* of set theory closes under intersection. If
we simply consider the empty set to be an abstract ``container''
of all sets and hence a ``member'' of all sets in the Universe, then we
*no longer even have to specify* that the intersection of two sets
with no members in common is the empty set, we can simply note that *ordinary* intersection of two sets with no members *other than the
empty member* in common is of course the empty set. In the Universe of
Fruit (which lives in a really big box), the intersection of a *box*
of apples and a *box* of pears is an *empty box*. In a positive
set theory with at least one pair of disjoint sets within, one doesn't
*need* the axiom of the empty set, one just points to it.
Similarly, in an existential hierarchy, the grouping of all
the objects none at a time just happens in and is permuted in
turn in the various higher order power sets.

However, the strong idea of non-being that is expressed in the laws of
thought above seems to be a *different* concept than the emptiness
of a box. Let me see if I can clearly express the way I view the
difference, although we shouldn't be too disturbed if the words to do so
elude me or if I fail to achieve clarity. An *entire philosophical
tradition* holds that the concept cannot be placed into words but is
nonetheless one of the most *important* concepts in the theory of
knowledge.

Set theory is all about boxes, about Venn diagram containers, about
categories. The empty set in an existential set theory is an empty box,
but is not *nothing* because the *box still exists within the
theory*^{3.23}. We can, in fact, perform *all* the algebraic operations of
the set theory on the empty set as an ``object'' in the Universe as long
as we insist that its complement (as one of those actions) is precisely
the Universe itself so that the set theory *closes*. If we relax
this condition we open the door to many paradoxes, a situation which the
null set is introduced to avoid.

This concept of non-null emptiness extends into mathematics and physics.
It is fairly straightforward to imagine an empty Universe^{3.24} - an
infinite set of points represented by some set of abstract coordinates
with no ``objects'' located at any of the points. Hmmmm, isn't that
what mathematics is all about for the first umpty years one studies it?

Note well that (empty or not) we can *imagine* putting things at
those coordinates, using those coordinates to label the things and help
to sort them into sets (including disjoint ``identity sets''), just as
we can *imagine* putting things ``into'' the empty set (via the
Union operation) and creating a non-empty set^{3.25}.

There is, however, a deeper notion of ``emptiness'', that of
``non-being''. The notion of *no box at all*. In physics, this
might be the notion of *no Universe at all*, not even an empty one
consisting of a perfect vacuum at a single mathematical point. Of
course this is an odd statement and it makes us vaguely uncomfortable
even to read it. This is a concept that can be expressed in English
(and *within* a system of logic) only as an oxymoron or a kind of
example of what has come to be called a *Russell
Paradox*^{3.26}. We will consider
it in a somewhat different context than that which is usually presented,
because in the case of our naive existential set theory, we cannot
actually make any sets at all that do not contain themselves by virtue
of Existential Identity, where we do not ``make'' sets at all, only
identify them or choose them from the set of all sets that *exist*
within the set Universe (including the empty set) and of course *all* sets in such a universe *precisely* contain themselves.

So, consider the set of all things that are not in *any set in the
set Universe including the empty set*. Let's see, all ``things'' from
our set Universe are minimally contained in their own identity sets, so
no things can be members of this set. However, the empty set is *also* explicitly excluded, so the result of trying to create this set
cannot be just a set with no members. It is not a set at all, it is
*nothing*, the absence of *even emptiness* as a capacity to be
algebraically manipulated or ``filled''. Yet you can perfectly well
understand what I say when you read the English. This is an ``empty
set'' *without a box* - it isn't, by definition, a set; it is
rather an intrinsic contradiction of the *concept* of set. It is
the absence of any set Universe at all.

Note that this is a *self-referential definition* of a ``set'' and
is *precisely the kind of set* that ties one into Gödelian
knots^{3.27} or produces Barber
paradoxes in logic/set theory^{3.28}. Yet we can
perfectly well understand what this refers to and *use it all the
time in common language*. Obviously there are *no* things that are
not in some set (minimally the identity set), and since the empty set
has no members at all these nonexistent things aren't there either. Our
minds can create a ``class'' of ``things that are not'' while juggling
the word ``things'' and the concept of ``non-being'' (not-things) back
and forth like a hot potato and somehow end up with a meaningful idea
out of a contradiction that isn't just ``the absence of trees'' but is
the ``absence of even a Universe in which things that aren't necessarily
trees that I cannot imagine do not exist.'' Our minds can empty a
hypothetical Universe, shrink it to a point, and then *throw out the
point*, as long as we don't think too carefully about just what the junk
heap we throw it out upon really ``is'', since one might well argue that
``nothing'' is literally inconceivable - certainly not *directly*
conceivable - to a conscious mind.

In our semantic conceptualization of all things that *are*, that are
minimally in their own identity set, we can ``fill'' the empty set by
taking the union of the empty set with a nonempty set. We can consider
the intersection of the empty set with any nonempty set and of course
get the empty set. However, we *cannot* take the intersection of
all things that belong to no set at all including the empty set with any
set. If the result were the empty set, then the set we intersected was
*not* in fact the set of all things not in any set including the
empty set. The null set is therefore the absence of any box - it lies
*outside the algebra of the set* where the empty set is within the
algebra. Similarly the union of any real set (including the empty set)
with the null set is undefined, is itself null.

We can imagine joining a box of apples and a box of pears and ending up
with a box of mixed fruit, or filling an empty box with the box of mixed
fruit (forming unions of sets of fruits). We can imagine looking for
apples in a box of mixed fruit (forming the intersection of the ``subset
of all apples in the Universe'' with the ``subset of mixed fruit'') to
put into an empty fruit box - the result can be some apples, a
non-empty intersection added to the empty box to make it a *box of
apples* - or no apples at all, empty intersection added to the empty
box, leaving one *with an empty box*.

Can we even *imagine* combining a box of apples and a null set?

We can! At least sort of, metaphorically, kinda. Physics to the
rescue. If we dump a box of apples into a *black hole*, then
Poof^{3.29}! It is
gone! No more apples, *no more box*. So we can *conceptually*
think of the null set as a ``black hole'' of set theory^{3.30}.

This concept of set theoretic (and other) contradictions are actually
explored and developed more in *Eastern* philosophy and logic than
in the West, and Zen logic^{3.31} is perhaps more
suited to the sorts of oxymoronic construction that one associates with
nonbeing as opposed to emptiness. For example, ``the sound of one hand
clapping'' in a rather famous Zen koan is not the sound of clapping in
the limit that the noise being produced by a clap goes to zero, it is
not even *no clapping at all* - two hands sitting at rest. It is
``impossible'', or ``undefined'', or ``self-contradictory''. Not
clapping. Not the absense of clapping. It is null.

This concept pervades Buddhism and Eastern philosophy and culture. It
is referred to in e.g. Musashi's *Book of Five Rings*, for example,
as the *Void*. One essential component of Zen and meditation (often
meditation on paradoxical Zen Questions) leading to Enlightenment is the
realization of the null, the no-thing, Mu^{3.32} . It is a concept that is inconceivable and hence
openly contradictory in language. It *cannot be spoken of* because
words are *symbols* and live in an information-theoretic set
Universe where things *exist*. Zen masters therefore refuse to
speak of it but rather force you to perform exercises that provide you
with at least the opportunity to wrap your mind all the way around the
blind spot to the point where you can see it by considering what isn't
there, to *resolve* the paradox of existence and our imperfectly
imagined versions of death, of impermanance and permanence and change,
of *non-*existence. This resolution, whenever and however it is
managed, brings about a state of remarkable mental clarity^{3.33}.

The null set is conceptually similar to the role of the number ``zero''
as it is used in quantum field theory. In quantum field theory, one can
take the *empty* set, the *vacuum*, and generate all possible
physical configurations of the Universe being modelled by acting on it
with creation operators, and one can similarly change from one thing to
another by applying mixtures of creation and anihillation operators to
suitably filled or empty states. The anihillation operator applied to
the *vacuum*, however, yields *zero*.

Zero in this case is the null set - it stands, quite literally, for no
physical state in the Universe. The important point is that it is not
possible to act on zero with a creation operator to create something;
creation operators only act on the vacuum which is empty but *not*
zero. Physicists are consequently fairly comfortable with the existence
of operations that result in ``nothing'' and don't even require that
those operations be contradictions, only operationally non-invertible.

It is also far from unknown in mathematics. When considering the set of
all real numbers as *quantities* and the operations of ordinary
arithmetic, the ``empty set'' is algebraically the number zero (absence
of any quantity, positive or negative). However, when one performs a
division operation algebraically, one has to be careful to *exclude*
division by zero from the set of permitted operations! The result of
division by zero isn't zero, it is ``not a number'' or ``undefined'' and
is *not in the Universe* of real numbers.

Just as one can easily ``prove'' that 1 = 2 if one does algebra on this
set of numbers as *if* one can divide by zero
legitimately^{3.34}, so in logic one gets into trouble if one assumes that the
set of all things that are in no set *including the empty set* is a
set within the algebra, if one tries to form the set of all sets that do
not include themselves, if one asserts a Universal Set of Men exists
containing a set of men wherein a male barber shaves all men that do not
shave themselves^{3.35}.

It is not - it is the *null set*, not the empty set, as *there
can be* no male barbers in a non-empty set of men (containing at least
one barber) that shave all men in that set that do not shave themselves
at a *deeper level* than a mere empty list. It is not an empty set
that *could* be filled by some algebraic operation performed on Real
Male Barbers Presumed to Need Shaving in trial Universes of Unshaven
Males as you can very easily see by considering any particular barber,
perhaps one named ``Socrates'', in any particular Universe of Men to see
if any of the sets of that Universe fit this predicate criterion with
Socrates as the barber. Take the empty set (no men at all). Well then
there are no barbers, including Socrates, so this cannot be the set we
are trying to specify as it clearly must contain at least one barber and
we've agreed to call its relevant barber Socrates. (and if it contains
more than one, the rest of them are out of work at the moment).

Suppose a trial set contains Socrates alone. In the classical rendition
we ask, does he shave himself? If we answer ``no'', then he is a member
of this class of men who do not shave themselves and therefore must
shave himself. Oops. Well, fine, he must shave himself. However, if
he *does* shave himself, according to the rules he can only shave
men who *don't* shave themselves and so he *doesn't* shave
himself. Oops again. Paradox. When we try to apply the rule to a
potential Socrates to *generate* the set, we get into trouble, as we
cannot decide whether or not Socrates should shave himself.

Note that there is no problem at all in the existential *set theory*
being proposed. In that set theory either Socrates must shave himself
as All Men Must Be Shaven and he's the only man around. Or perhaps he
has a beard, and all men do *not* in fact need shaving. Either way
the set with just Socrates does not contain a barber that shaves all men
because Socrates either shaves himself or he doesn't, so we shrug and
continue searching for a set that satisfies our description pulled from
an *actual* Universe of males including barbers. We immediately
discover that adding more men doesn't matter. As long as those men,
barbers or not, either shave themselves or Socrates shaves them they are
consistent with our set description (although in many possible sets we
find that hey, other barbers exist and shave other men who do not shave
themselves), but in *no* case can Socrates (as our proposed *single* barber that shaves *all* men that do not shave themselves)
be such a barber because he either shaves himself (violating the rule)
or he doesn't (violating the rule). Instead of concluding that there is
a paradox, we observe that the criterion simply doesn't describe any
subset of any possible Universal Set of Men with *no* barbers,
including the empty set with no men at all, or any subset that contains
at least Socrates for any possible permutation of shaving patterns
including ones that leave at least some men unshaven altogether.

That is we don't end up concluding that the set described by our predicate
criterion is the *empty set* (a set with no men) or any other
possible subset of the Universe of Men. We conclude that the predicate
leads to a *null* result. There is *no Universal Set of Men*
(including one with no members at all) for which the predicate describes
a set or subset or empty set as the answer.

We therefore dump the proposed statement, Socrates and all, into the
null, or undefined, ``set'' (which is *not* a set). It is an
algebraic placeholder for all algebraic set theoretic results that do
not consistently lie within the algebra even as an empty set and which
(among other things, such as overt contradictions and English words such
as ``nothing'' or ``nonbeing'' or mathematically ``undefined'' results)
lead to paradoxes, incorrect propositions, undefined results. Set
theory (and language and logic and mathematics) have *always* had
this ``black hole'' around, it just needs to be formalized.

To make this understandable at a very simple level, there is a very real
difference between the sentences: ``Honey, could you take this *empty* list and stop by the store on the way home and pick up nothing
today?'' and ``Honey, could you fail to take this nonexistent list and
not stop by the store on the way home and not pick up nothing today?''.
The first describes something that could really happen. We can easily
imagine tearing off the wrong piece of paper (the blank one) and taking
it to the store, only to be frustrated and end up buying nothing.
Mathematically, one can perform all of the operations permitted with the
algebra (stopping by the store to pick up items on a list to create a
new list called ``items I got at the store'') where an *empty list*
in leads to an *empty cart* out.

In the second case, there *is no list* - not even a blank one or
piece of paper that *might* hold a list - and this sentence really
*makes no sense*. You cannot pick up a list that doesn't exist.
Without a list (even a mental list or a *possible* mental list that
you could perhaps *fill in* at the store itself) you would never
*go* to the store motivated to buy items from a list (even if the
list turned out to be empty). Basically, if there is no list at all you
*cannot perform algebraic operations on what is not there*. List
oriented computer languages do not just spontaneously start up and run
themselves not just on empty list pointers but on *no pointer at
all*. I don't even know what such a thing would *mean*.

We thus see that this is *not* a silly issue; that even a naive
existential set theory requires *both* an empty set, defined to be
``a set'' and required so that the intersection operation in particular
closes within the ``Universe'' of objects being listed/grouped/placed in
sets, *and* a null set which is *not* a set^{3.36} - it is algebraically the undefined result of
operations that *might* be defined within the set theory that result
in no set within the theory including the empty set, and semantically it
is nullity of the *concept* of ``thing'' or ``existence'' (set
object) so great that not even the absence of a thing is permitted
within the language. ``Inconceivable'' is perhaps the right term for
it, as opposed to ``imaginary''^{3.37}.

Of course, the word ``inconceivable'' itself is a walking, talking
oxymoron waiting to happen. When I say that it is inconceivable that
space aliens^{3.38} control the President of the
United States, what I really mean is that I've already conceived of the
notion but consider it to be pretty unlikely^{3.39}. Only when one uses it in a sentence containing a
null construct does it really make literal sense. It is *inconceivable* that there exists a male barber who shaves all men who do
not shave themselves.

We cannot (by definition) even *imagine* that which is inconceivable
and will get a nasty headache from even trying - it is the ``set of all
sets that are not sets'', which leads the imagination into unresolvable
knots if one tries to conceive of it, at least as a set. It is *nonbeing*. It is No-Thing. Let's call it Mu, and write it symbolically
as .

This symbol is selected quite deliberately to make an entirely relevant
trilingual Zen Pun. By strange chance the word for No-Thing in Japanese
is Mu. Note that this isn't an exact translation - it can equally well
be thought of as meaning ``That does not compute!'' or ``Say what?'' or
``That is bullshit''. We will use it quite happily in all of these
senses when we make *it* the idea of non-existence in our
existential set theoretic Laws of Thought.

As you should know by now from having followed the previous Wikipedia
link for Mu^{3.40}, one of
the most famous Zen koans is: ``Does the dog have Buddha-nature?''
This is a damned-if-you-do, damned-if-you-don't sort of question - if
you say yes it indicates that you are just parroting scripture (which
also says yes), and if you say no then you are disagreeing with
scripture which is if anything even worse as we'll see when we study the
axioms of religion.

When asked this `are you still beating your wife' sort of question by a wandering monk in a Zen Shootout (see above) Joshu replied ``Mu''. The generally accepted interpretation of this reply is that Joshu was indicating that this wasn't a question, it was a transparent ploy to make him look bad. The alternative way to demonstrate this might have been to beat his opponent about the head with a banana, but Joshu doubtless didn't have a banana handy at the time. He was acclaimed instant winner of the shootout and his opponent's very name is long since lost in the mists of the past while his is still remembered and revered.

Zen students *now* aren't permitted to answer any of ``yes'',
``no'', or ``Mu'' any more. My own favorite reply to this question is
to fire back at the questioner ``Does Buddha have Dog nature?''. This
neatly traps the trapper. If they reply ``That's not an answer!'' or
really say pretty much anything at all, you can slam a book down or
otherwise make a loud noise and whack them with a banana. They are *almost certain* to be Enlightened, and you are very likely to have the
questioner follow you around fawning at your feet and calling you
`master' (something, hmmm, that you should think about before trying
this in public). You should feel free to try this at home instead, by
the way - it isn't necessary that you strike *someone else* with
the banana for it to work, and you are less likely to be annoyed by your
Self fawning at your own metaphorical feet.

At any rate, this and many other Koans make it very clear that the
discovery of the null ``set'' (where it is not a set but rather the lack
of any set, even empty) quite probably occurred no later than the very
beginning of Zen, if not thousands of years *still* earlier as
captured by the writers of the Vedas and Upanishads so that it was
merely *refined* in Zen. Many of the odd customs and stories and
Koans of Zen - for example the recurring statement that Zen
Enlightenment cannot be stated and that to reduce it to words is to lose
it - are reflections of the fact that (Mu) is the ultimate *null* semantic construct and hence *cannot be stated in words* or
other symbols^{3.41}.

in Zen therefore cannot be defined, only demonstrated, and that
only by semantic contradiction of direct experience - from the *metaphor* of ``holes'' No-Thing leaves in (experiential) Things of all
sorts. Symbolic representations or visualizations to help you come to
terms with all consist of strange exercises such as writing a
perfectly lovely complete and consistent set theory (or a pithy little
koan) down on a piece of paper and then *burning the paper*. Or by
the bottom unexpectedly falling out of a bucket of water being carried
on a moonlit night so the *reflection of the moon* vanishes along
with the water, leaving one carrying - No-Thing, a *hole* where the
*illusion* of the moon once danced on the *illusion* of the
water.

Hmmm, pretty heady stuff, but can it be worked into an actual set theory? I think so. Let's try.

A Bit of Formalism

Set Theory and the Laws of Thought

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