## Frege’s Numbers

Posted 19th November 2021 by Holger

In a previous post, I started talking about natural numbers and how the Peano axioms define the relation between natural numbers. These axioms allow you to work with numbers and are good enough for most everyday uses. From a philosophical point of view, the Peano axioms have one big drawback. They only tell us how natural numbers behave but they don’t say anything about what natural numbers actually are. In the late 19th Century mathematicians started using set theory as the basis to define the axioms of arithmetic and other branches of mathematics. Two mathematicians, first Frege and later Bertrand Russell came up with a definition of natural numbers that gives us some insight into the nature of these elusive objects. In order to understand their definitions, I will first have to make the little excursion into set theory.

You may have encountered the basics of set theory already in primary school. Naïvely speaking sets are collections of things. Often the object in a set share some common property but this is not strictly necessary. You may have drawn Venn diagrams to depict sets, their unions and intersections. Something that is not taught in primary school is that you can define relations between sets that, in turn, define the so-called cardinality of a set.

## Functions and Bijections

One of the central concepts is the mapping between two sets. For the following let’s assume we have two sets, \(\mathcal{A}\) and \(\mathcal{B}\). A function simply defines a rule that assigns an element of set \(\mathcal{B}\) to each element of set \(\mathcal{A}\). We call \(\mathcal{A}\) the domain of the function and \(\mathcal{B}\) the range of the function. If the function is named \(f\), then we write \[

f: \mathcal{A} \to \mathcal{B}

\] to indicate what the domain and the range of the function are.

**Example:** For example, if \(\mathcal{A}\) is the set of uppercase and lowercase vowels, \[

\mathcal{A} = { A, E, I, O, U, a, e, i, o, u },

\] and \(\mathcal{B}\) is the set of all uppercase letters in the alphabet, \[

\mathcal{B} = { A, B, C, D, \ldots, Z}.

\]

Now we can define a function that assigns the uppercase letter in \(\mathcal{B}\) to each vowel in \(\mathcal{A}\). So the mapping looks like shown in the figure.

You will notice two properties about this function. Firstly, not all elements from \(\mathcal{B}\) appear as a mapping of an element from \(\mathcal{A}\). We say that the uppercase consonants in \(\mathcal{B}\) are not in the *image* of \(\mathcal{A}\).

The second thing to note is that some elements in \(\mathcal{B}\) appear twice. For example, both the lowercase e and the uppercase E in \(\mathcal{A}\) map to the same uppercase E in \(\mathcal{B}\).

### Definition of a Bijection

The example shows a function that is not a bijection. In order to be a bijection, a function must ensure that each element in the range is mapped to by exactly one element from the range. In other words for a function \[

f: \mathcal{A} \to \mathcal{B}

\]

- every element in \(\mathcal{B}\) appears as a function value. No element is left out.
- no element in \(\mathcal{B}\) appears as a function value more than once.

A bijection implies a one-to-one relationship between the elements in set \(\mathcal{A}\) and set \(\mathcal{B}\).

## Equinumerosity and Cardinality

Intuitively, it is clear that you can only have a bijection between two sets if they have the same number of elements. After all, each element in \(\mathcal{A}\) is mapped onto exactly one element in \(\mathcal{B}\). This can be used to define a relation between any two sets.

Two sets are called **equinumerous** if there exists a bijection between the two sets. Equinumerous literally means *“having the same number”*. But we have to be careful here because we don’t yet know what the term *“number”* is supposed to mean. That is the reason why we define the term by using bijections and not referring to any *“amount”* or *“number of elements”*. Instead of saying that two sets are equinumerous, we can also say that they have the same **cardinality**.

Now comes the clever bit that Frege proposed. Let’s create a class of sets that all share the same cardinality. We can do that because equinumerosity is an equivalence relation but I won’t go into detail about what that means. We will call this cardinality class \(N\), so \[

N(\mathcal{A})

\] is the class of all the sets that are equinumerous to \(\mathcal{A}\).

Intuitively we now have a class with all the sets that contain exactly one element, another class with all the sets that contain exactly two elements, and so forth. But we don’t know anything about numbers yet, so we also don’t really know what *one* and *two* are supposed to mean.

## Constructing Natural Numbers

Now we have all the tools to construct the natural numbers \(\mathbb{N}\). Of course, we want our numbers to obey the Peano axioms, so we need two things. We need a *zero* element and we need a successor function \(S(n)\) that produces the next number from any given number.

### The Zero Element

The zero-element is easily defined. We can construct the empty set, \[

\emptyset = \{\}.

\] This is the set with no elements in it. Now the zero-element is simply the cardinality class of the empty set, \[

0 = N(\emptyset).

\] This means that zero is a class of sets that all share the same cardinality as the empty set. You can show that this class consists of only one element, the empty set, by I won’t go into that here.

### The Successor Function

Given that we have defined the zero element, \(0\), we can now define a set that contains zero as a single element, \[

\{0\}.

\] Intuitively, this set has one element and we can thus define the natural number \(1\) as the cardinality class of this set, \[

1 = N(\{0\}).

\] In general, given any natural number \(n\) we can define the successor \(S(n)\) by creating the cardinality class of the set that contains \(n\) together with all its predecessors, \[

n+1 = S(n) = N(\{0, 1, \ldots, n\}).

\] You might think that this definition is somewhat circular. We are defining the successor function by using the concept of the predecessors. But this is not as problematic as it might seem at first sight. We know that the predecessor of \(1\) is \(0\) and each time we construct the next natural number, we can keep track of all the predecessors that we have constructed so far.

## Conclusion

The zero and the successor function defined above are enough to define all the natural numbers \(\mathbb{N}\). I will not go into the proof that all the Peano axioms are satisfied by this construction. It is relatively straightforward and not very instructive in my opinion. If you want, you can try doing the proof as an exercise.

I personally find the Frege definition of natural numbers the most satisfying. It tells us that a number is not just some random symbol that doesn’t relate to the real world. A natural number is the class of all sets that share the same property. Each set in the class has the same cardinality and we can identify the cardinality with that number. It means that any set of objects in the real world can be thought of as an instance of a number. The number itself is the collection of sets and the concrete set is contained within it as an element. For example, if you see five apples on a table, you can think of them as a manifestation of the number \(5\).

Another consequence of the definition of cardinality is that it gives us the ability to speak about infinities. A set might have an infinite number of elements. We already encountered \(\mathbb{N}\), the set of all natural numbers. Using the cardinality, we can compare infinite sets and create a hierarchy of infinities. I might talk about this more in a later post.

It would not be fair, however, if I didn’t mention some serious problems with the definition that I Frege came up with. The main problem arises because we are creating classes of sets without explicitly saying which elements we are allowing to be in those sets. This allows sets to contain arbitrary elements, including other sets. A set can even include itself as an element. This leads to the famous paradox by Russel which can be summarised as follows. Construct a set \(\mathcal{R}\) of all the sets that do not include themselves as an element. Then ask the question, does \(\mathcal{R}\) include itself? There are mathematical frameworks that attempt to save the essence of Frege’s definition of the natural numbers without running into these problems. In my personal opinion, they always lose some of the beauty and simplicity. But this is a necessary concession to make if you want to end up with a mathematical framework that doesn’t contain internal contradictions.

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