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Can you make something out of nothing?

We explore the concept of emptiness in set theory, and explain how zero went from “nothing” to “something”

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Image: public domain.

none30@example.com'

20 September 2018

The equation:

$$\emptyset + \{0\} = 1$$

looks a little bit like:

$$0 + 0 = 1.$$

According to Aristotle, nature could not create emptiness at all. Public Domain

Doesn’t it? The main point is: How to deal with zero. You can start with nothing and say: something is the opposite of zero.  On the other hand, one can say: zero is already something, I can count it: one element. But first things first. I would like to start with an old fear of people: the fear of emptiness. The Romans had no zero. For centuries physicists were afraid of the so-called horror vacui based on the idea of Aristotle that nature could not create any emptiness at all (Aristotle).  Even in art, there is the horror vacui as criticism of the art of the Victorian era (Mario Praz, 6/09/1986-23/03/1982).

Let us take one step back and start with sets. Why sets? Before we can handle this let us ask something else: How can a science be justified?  Religions work with dogmas. You need one example? Here it is: God is good and almighty. There is no further proof of this, for it has been established as a statement. Mathematics does not work with dogmas. You can start with something else: axioms.

What is an axiom? The word axiom comes from the Greek and means “perceived principle” and is in a theory, a science or an axiomatic system something that is not founded or deductively derived within this system. One amazing hint: Insufficiently strong systems, such as arithmetic, there must be statements that can neither be formally proven nor refuted (this is called the G\”odel incompleteness theorem).

Georg Cantor, a German mathematician that founded set theory. Public Domain

For a long time, mathematicians tried to design a clean axiomatic system. After many trials and errors, they decided to try it with quantities. So-called sets and finally we are in set theory. Set theory is a fundamental branch of mathematics that deals with the investigation of sets, i.e. summaries of objects. Set theory was founded by Georg Cantor and deals with representing (mathematical) objects as sets. By a set, we mean every summary M of certain well-different objects in our intuition or thinking (which are called the elements of M) into a whole. You need some examples?

$$ \mbox{Patrick Set} = \{A,B,C\}$$

This is the set of the letters A, B and C. The set is named after me. You can also write:

$$\mbox{Super Animal Set} = \{Dog,Cat,Duck\}$$

Another set with animals in it. And now an exciting question: What is an empty set?

The first symbol for an empty set comes from Andre Weil and is the letter of the Danish or Norwegian alphabet:

$$\emptyset.$$

Weil, a scientist from Strasbourg and one of the most important members of the Bourbaki group, a union of like-minded French mathematicians who undertook the task of formulating all mathematics in a new and relentlessly strict manner.

When dealing with sets of objects, there are two basic ways to make them. There is the predicative way, where we simply lay down verbally which elements contain a set or the constructive way in which new sets are made from given sets. Historically, the predicative path was first taken and brought to maturity in the nineteenth century but then something happened: the Russell paradox entered the mathematical stage.

The Russell paradox was discovered by Bertrand Russell in 1901. Public Domain

According to naive set theory, any definable collection is a set. Let A be the set of all sets that are not members of themselves. If A is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. It is like a barber who is allowed to cut all the hair of those who do not cut their own hair. Can the barber now cut his own hair or not?

So although mathematicians have operated implicitly with sets of objects for centuries, there was no real theoretical formalization until the nineteenth century. Frege (08/11/1848-26/07/1925) himself created the so-called predicate logic. So let us ask again: What is nothing in the set theory?

The return of zero

If you create a set of all even numbers you have the first set. All odd numbers are the second set but the cut of both sets is empty. George Boole said that the set is empty and wrote simply 0 (zero) for this. This is one possibility to create a nothing out of sets.

Ernst Zernelo. Public Domain

The Berliner mathematician Ernst Friedrich Ferdinand Zermelo (27/07/1871- 21/05/1953) studied mathematics, physics, and philosophy at the Universities of Berlin, Halle (Saale) and Freiburg. He tried to prove that sets have a so-called “Wohlordnung” (well-ordering).  A simple example of well-ordering is the normal arrangement of natural numbers. A totally ordered set is called well-ordered if every subset that is not empty has a smallest element. The set of integers with the usual order is not well ordered, because it does not have even the smallest element.

Why is this good for quantities? The so-called “Wohlordnungssatz” (the theorem of the well-ordering):
There is a well-ordering on every quantity. Apparently, every subset of a well-ordered set is well ordered. It is one idea to make sets like this beautiful and usable. It brings us closer to the constructive way.

With the help of Abraham Fraenkel (17/02/1891-15/10/1965), Zermelo published an axiomatic theory known as Zermelo-Fraenkel set theory. Zermelo (like Boole) used zero as the empty set sign, and with this convention, one can, for example, in Zermelo’s system construct a new set $\{0\}$. This symbol means an amount that contains only the empty set and it equals 1. In other words: $\{ \}$ means: let us count the number of elements in the set. A zero is one element. Therefore $\{0\} = 1$. It is another way of thinking about zero. Zero is something and at least one. It is not an emptiness at all. You can count it.

The idea behind sets is that you can create everything from your axiomatic system. Sets of sets or other theorems based on your created sets. You can then create a new set of two sets, such as $\{0, \{0\}\}$. Zermelo lends this trick to another mathematician named Dedekind.

Through such operations as inserting an element into a set, or through combinations of pairs, union, and cut of sets, as well as additional rules, one can always make new sets. Now we are here: that is the constructive approach to set theory.

The interesting fact is that we start from nothing the so-called “Urelement” (you can translate it with the primary element). You begin with a single object, with an empty set. In other variants of the set theory, an arbitrary set is taken and from this, the empty set and from this set everything will be created. New elements can then be identified with newly produced quantities. You start from any point and say what is true for “ n ” will be also true for “n + 1” and so on. In Zermelo’s logic, this is the axiom of infinity and you are able to create all natural numbers.

It looks strange: In the Zermelo-Fraenkel theory, there is no need to introduce individual elements. For example, one speaks not about the amount of all letters. Instead, one talks exclusively about quantities and quantities of quantities that are built up over the permitted operations. You start with the Urelement which is an empty set, so it contains no elements. If you count the number of elements in the set “a “or the empty element in a set you will get one (element). In the Zermelo world, we just need the empty set and axioms but we start from the pure empty. You can even think of the equation as:

$$\emptyset + \{0\} = 0 + a = 0 + 1 = 1.$$

Fantastic! You can write shortly:

$$\emptyset + \{0\} = 1.$$

In the set theory of Zermelo, you can find 10 axioms creating everything from empty or nothing. There is just one thing you have to decide. If you want to create all natural numbers you can include zero or not:

$$\mathbb{N} = {1,2,3,\dots}$$

or:

$$\mathbb{N} = {0,1,2,3,\dots}$$

Both conventions are used inconsistently. The older tradition does not count zero as a natural number. The zero became common in Europe only from the 13th century. Today one writes $$\mathbb{N}_{0}$$ for all natural numbers including zero and $$\mathbb{N}_{+}$$ for all natural numbers without zero (you may find $\mathbb{N}_{>0}$ or $\mathbb{N}^{+}$ as well).

John von Neumann (16/10/1831-12/02/1916) had the idea of a stepwise construction of the entire set universe with the help of ordinal numbers and the iteration of raizing a number to a given power. Written in empty sets the natural numbers look like this:

$$\mathbb{N} = \emptyset,\{ \emptyset \},\{ \emptyset,\{ \emptyset \} \}, \dots. $$

So Zermelo was not the only one with that idea. Once again: How to deal with zero? You can start with nothing and say: something is the opposite of zero. On the other hand, one can say: zero is already something, I can count it. It is not just about the different spelling: $\emptyset \, \mbox{or} \, \{0\}$.

The return to the void

Outer space has been seen for some as complete emptiness. Is it? Public Domain

Criticizing the financial world often means: how can you get something (for example, interest) from nothing? And nothing means securities that are based on other securities that are merely bets or only worthless loans. On the other hand, in philosophy, science, and mathematics we find a return to nothing or emptiness. Physicians found out that the vacuum is full of fluctuations from which even particles can arise. So particles from the nothing. In modern art, you can find a Renaissance of empty space. There is Yves Klein’s empty room of 1961 for example. There is a piece of music in which musicians think about the silence. The name is 4’33”, since it takes 4 minutes and 33 seconds in three musical movements by John Cage.

In philosophy, there is a long discussion about the nothing coming from pre-socratic ideas that something is the negation of nothing to modern thoughts like Bloch’s Not-Yet-Being (Noch-Nicht-Seienden).

So the nothing never really disappeared.

Or is it the return of empty and you can interpret this for our time? It might be the idea that something can come from nothing.  And finally, we can say (with a wink of the eye) that $0 + 0 = 1$ with a lot of curly braces and a slash going through of course.

none30@example.com'

Patrick Hedfeld is a doctor of philosophy and diploma physicist. He has been working as a senior project manager for many years in the finance industry. He is interested in interdisciplinary topics such as notation as a tool of thought, economic ethics, gamification and digitisation. As a private lecturer at FOM university, he supervises a new IT masters on the border of theory and practice. In the business ethics lecture, weapons dealers are found, the chainsaw unpacked and people run over with the trolley. In his spare time he is very nice.
All articles by Patrick

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