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The Null Set

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 ago3.21 . 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 unique3.22.

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 $\Pi^n(S)$ hierarchy, the grouping of all the objects none at a time just happens in $\Pi(S)$ 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 theory3.23. 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 Universe3.24 - 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 set3.25.

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 Paradox3.26. 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 knots3.27 or produces Barber paradoxes in logic/set theory3.28. 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 Poof3.29! It is gone! No more apples, no more box. So we can conceptually think of the null set as a ``black hole'' of set theory3.30.

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 logic3.31 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, Mu3.32 . 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 clarity3.33.

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 legitimately3.34, 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 themselves3.35.

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 set3.36 - 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 aliens3.38 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 unlikely3.39. 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 $\mu$.

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 Mu3.40, 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 $\mu$ (Mu) is the ultimate null semantic construct and hence cannot be stated in words or other symbols3.41.

$\mu$ 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 $\mu$ 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.

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Next: A Bit of Formalism Up: Formal Set Theory Previous: Set Theory and the   Contents
Robert G. Brown 2007-12-17