If 0 is an empty set

The nothing in mathematics

Although we are confronted with the empty crowd in school today, its definition and place in mathematics and philosophy has been controversial for centuries

It was in 1939, on the eve of World War II, that a mathematical term came of age and its final name: it is the empty set, represented by Ø, letter of the Danish and Norwegian alphabets. The new symbol suddenly became the standard notation in set theory.

The suggestion for this came from André Weil (1906-1998), a scientist from Strasbourg and one of the most important members of the Bourbaki group, an association of like-minded French mathematicians who took on the task of formulating all mathematics anew and relentlessly strict. In the very first book on analysis, with the introduction of set theory1, Ø is defined as the empty part of a set in order to "fix the set notation once and for all" .2 Generations of students at universities have since lost positions in a creeping battle against the The dryness and severity of the too many Bourbaki volumes resulted.

In the 21st century it all seems like a distant past. And yet: Although we are already confronted with the empty crowd in school today, its definition and place in mathematics and philosophy has been controversial for centuries. Nothing is more complicated than nothing, at least from a philosophy point of view.

When we talk about numbers, we need a starting number like 1. By gradually increasing it by one unit, we move on to 2, 3, 4, and so on. The starting number can be any, you could also have started with zero, for example. But the zero as a symbol was not available to all cultures. In the Roman notation for numbers there is no sign for it and so we number the years since the birth of Christ from year one. It was only with the spread of positional systems that it became necessary to operate symbolically with the zero.

But back to set theory. When dealing with sets of objects there are two basic ways of building them. There is the predicative way, where we simply linguistically determine which elements a set contains (such as when we say, "Let T be the set of all Telepolis readers"), or the constructive way, in which new sets are created from given sets become. Historically, the predicative path was taken first and brought to maturity in the nineteenth century - but then something terrible happened: the Russell Paradox hit the mathematical stage.

In the paradox the set M of all sets that do not contain themselves is investigated. The definition of M would, linguistically, be a perfectly legitimate thing. But then the question is asked whether this set M is also an element of itself. If it were so, we would have a contradiction, since M as an element of M must not contain itself. But if M is not an element of itself, it fulfills the condition to be an element of M - again a contradiction and thus an inconsistency in the science of abstract structures.

The British mathematician and philosopher Bertrand Russell gave the fatal blow to what is now known as "naive set theory" with his paradox in 1901. The German theorist Ernest Zermelo had discovered the inconsistency a year earlier, and Georg Cantor in Halle much earlier, but both were prudent enough to keep their mouths until a redemption could be found for the system.

The problem here is actually that we can cover far too much territory with language and, for example, talk about crowds that eat themselves, like Ouroboros, the Egyptian snake that bites its tail. It is also like the paradox of the barber, who cuts all the people in the village who do not cut their own hair. The poor barber then doesn't know whether he can cut his own hair or not.

But nobody was more clouded by Russell's discovery than the father of German logic, Gottlob Frege, who was preparing his book "The Basic Laws of Arithmetic" for the second edition of 1903, when Russell told him about the contradiction in his book. Frege wrote to Russell:

Your discovery of the contradiction surprised me and, I would almost like to say, dismayed me, because it shakes the reason on which I thought arithmetic ... was to be built. (...) I have to think about the matter further. It is all the more serious as with the elimination of my Law V, not only the basis of my arithmetic, but the only possible basis of arithmetic seems to have sunk.

Frege to Russell, June 22, 1902.

George Boole and the first empty lot

So although mathematicians have been operating implicitly with sets of objects for centuries, there was no real theoretical formalization for them until well into the nineteenth century. Frege himself was in a certain sense just a pioneer, as he created the logical language with which one could work mathematically clean from then on (the so-called predicate logic).

Frege's opponent was the British mathematician George Boole, whom we now understand as a kind of pioneer of the computer because of his Boolean algebra. Apparently he was the first to assign an explicit symbol to the empty set. This happened in his book "Mathematical Analysis of Logic" from 1847.

The step to an explicit symbol for the empty set is by no means trivial. We can always cite nothing purely linguistically, but when we start operating with sets, we would also like to have sets as results. If we combine two sets, like {1,2} with {3,4}, we get the set of numbers from 1 to 4. But if we now "intersect" both sets, ie look at the common elements, there are none and the result is "empty". We could then continue linguistically, but not symbolically. Much more convenient than saying "A and B have no common elements" is therefore to write "A∩B = ∅" (i.e. the intersection of A and B is the empty set) and we can then continue to work algebraically.

It is then the case that Boole used the symbol 0 for the empty set. This double use of the symbol for the number zero was convenient for him, since Boole developed his logical operations with the help of the truth values ​​0 and 1. Many other mathematicians then adopted Boolean notation, i.e. almost the current one, but without the dash through the vowel O.

All this basic research was joined by an Italian mathematician, Giuseppe Peano, who wanted to develop arithmetic entirely without words. Instead of long sentences he only wanted to use chains of symbols in order to keep the whole formalism free from intuitive, uncheckable linguistic prejudices. Peano's notation for the empty set was extremely original and it is actually a pity that we no longer use it today: Following Georg Cantor, who used the capital letter O (instead of the zero like Boole) as a symbol for the empty set, Peano decided in 1888 for a black circle as a representation of the universal set (everything) and an empty circle as a representative of the empty set or nothing (see fig.)

It was probably difficult to enforce these symbols with the stubborn typesetters and so only one year later Peano switched to an inverted lambda (capitalized) as a symbol for the empty set. As you can see, Peano has thought a lot about the right symbolic language for mathematics - but also the language for people. Incidentally, he invented "Interlingua", a simplified Latin without declensions, which, as a universal language, should enable unlimited communication.

In Great Britain, Whitehead and Russell, who continued Peano's program, adopted Peano's symbol for the empty set in their grandiose work Principia Mathematica, which perhaps no mathematician has ever really read from beginning to end. That is how impenetrable and cumbersome the whole treatise is. The reader moves at a snail's pace through a desert of notations until after hundreds of pages 1 + 1 = 2 is proven.

Zermelo and the axiomatic set void

Ernst Friedrich Ferdinand Zermelo (1871-1953) from Berlin did his doctorate on calculus of variations and was then assistant to none other than Max Planck. Under the influence of David Hilbert, he switched from physics to logic. In Göttingen, Zermelo tried to prove that sets could be assigned a so-called "well-order", but needed a set theory free of contradictions and strictly axiomatized. He published a first version of his axioms in 1908, but it was only with the help of other mathematicians (especially Abraham Fraenkel) that he finally succeeded in building a consistent axiomatic structure.

The main difference in the access from Zermelo was to produce quantities from other quantities only through authorized operations. Like Boole, Zermelo used the zero as a symbol for the empty set, and with this convention one can, for example, construct a new set {0} in Zermelo's system, i.e. a set that only contains the empty set. So we get a lot with one element.

You can then create a new set with two elements from two sets, such as {0, {0}}. Operations such as the insertion as an element in a set, or combinations of pairs, union and intersection of sets, and additional rules can be used to create new sets over and over again. That is the constructive approach to set theory. You can use it to set up a model for the natural numbers (i.e. 1, 2, 3, etc.) and prove, for example, Peanos' formalism for arithmetic. This is then the logician's heaven: all mathematics is reduced to set theory and logic, as God, thank Frege, strived for.

It is strange: in the Zermelo-Fraenkel theory there is no need to introduce individuals. For example, one does not talk about the set of all letters. Instead, one talks exclusively about sets and sets of sets that are built up via the permitted operations. You start with the empty set that does not contain any elements. Since the letters a, b, c, etc. also contain no elements, they are actually equivalent to the empty set (!). In other words, besides the operations to build sets from sets, we only need the empty set 0 as an anchor.

The universe of discourse begins with a single object, namely with this empty set (in other variants of the theory an arbitrary set is taken and from this the empty set is generated). New individuals can then be identified with newly produced sets. I could identify an "a" with {0}, a "b" with {0, {0}}, etc. It's like in a computer, I can represent any characters with a string of zeros and ones and so I only need the binary system deep inside the computer.

What is really interesting is that, in order to apply the set operations, we can start with a single very first object. And in the Zermelo-Fraenkel theory, this object can be nothing, the empty set. The whole theory building is erected from nothing as a single solid brick. The other rules are just the "mortar" to attach all the pieces under and together. Russell's monster construct can no longer appear in the system, because the ouroborosic set suggested by him can no longer be constructed using the permitted rules and the whole thing remains consistent, at least at that point.

Overcoming the fear of nothing

If you follow the glamorous history of the empty crowd, it is really strange to see how mathematicians and philosophers were all afraid of this nothing at the beginning of development and then in the end the nothing has earned itself as an anchor for the formation of ever new mathematical objects .

Everyone knows what Parmenides said from Elea, who rejected changes in beings, because beings exist and nothing is not. Change would mean that the being turns into something that it was not before. But that would be a contradiction. His student Zeno tried with similar arguments and the so-called Zeno aporias to prove the impossibility of movement, which the astute Diogenes of Sinope refuted by simply walking up and down in silence.

Many physical phenomena were also seen earlier horror vacui explains, i.e. the view that nature does not allow empty space to be produced. Therefore, water rose in a pipe from which air was pumped out to prevent the vacuum. In this century, however, physicists have long since come to terms with nothing and assigned a new meaning to it. It turns out that the vacuum isn't that empty after all. It is teeming with virtual particles and fields, so that today we will not understand the origin and development of the universe without gaining a better understanding of the vacuum.

Nowadays one cannot practice cosmology without paying attention to the infinitely small. It is also precisely the description of the not-so-empty vacuum, where gravitational physics and quantum mechanics meet and perhaps have the best chance of transitioning into a unified theory.

Some philosophers today also think differently about nothingness. Sartre's turnaround in philosophy, in his concept of existentialism, is similar to that of Zermelo in mathematics. Man, as a knowing subject, can simply rise into the world without hesitation, like a fish in water, or he can transform the world. So man can transform what is not into reality. Expressed with the eloquence of my philosophy teacher Bolivar Echeverria: What Sartre tells us is that man, like "a bubble out of nothing", confronts the world and can shape it anew. So nothing becomes the basis of freedom.

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