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What Are Natural Numbers?

We use natural numbers 1, 2, 3,… (sometimes including zero) in everyday life so obviously, so effortlessly, and so automatically that it hardly ever occurs to us to ask what they actually are. What is a natural number? How do you define twenty-seven? Let us take a brief look at three approaches, ranging from Plato to the present day, that try to set down a formal definition of the fundamental term number.

Plato’s Derivation of The Existence of Number

In the Parmenides dialogue, Plato derives the existence of numbers on the basis of two ideas: first, there is “the One”, and second, this “One” has a “being” that is its attribute but is distinct from it. These assertions themselves implicate the existence of “two”, since “the One” and its “being” are two different objects. But then “three” exists as well, since “the One”, its “being”, and “two” are three distinct objects. In this way, the existence of any finite number can be established.

If we label “the One” as 1 and the “being” as <$B$>, the principle can be illustrated as follows:

<$$ 2 = \{B, 1\},\quad 3 = \{B, 1, 2\} = \{B, 1, \{B, 1\}\},\quad 4 = \{B, 1, 2, 3\} = \cdots $$>

Of course, basing the existence of a mathematical object on fuzzy metaphysical notions like the abstract “One” and its existence, or “being”, is not acceptable in modern mathematics. Let us move on.

Bertrand Russell’s Idea of a Number

In Introduction To Mathematical Philosophy and other Russell’s works one can read about defining the natural number <$n$> as the set of all sets having exactly <$n$> members. (The idea is ascribed to Gottlob Frege as well.) It might seem strange to define numbers as sets, but it proves to be quite convenient for the later definition of set cardinality and other things. While the modern definition (see below) is slightly different, it still defines numbers as sets. Let us note that those sets having exactly <$n$> members need not consist of “real” objects only, such as elephants, people, or cars. On the contrary, they can encompass purely “virtual” things, e.g. other mathematical structures. Every natural number might therefore easily happen to be an infinite set.

Russell’s definition is not circular if formulated precisely. It corresponds to a profound idea about the nature of number. For what else is a number than that common characteristic shared by all groups having that particular number of members? What do three giraffes have in common with three stones? And with three rivers? And three women? Three sets? Three Hilbert spaces? We can somehow generalize the idea of “three-ness”, and Russell says that number three is precisely this “three-ness”. It is a genuine, philosophically profound idea if you think about it for a moment.

There is a legitimate philosophical concern about this definition: how can we be sure that there actually exist as many objects as we need to construct a number? For instance, if the whole universe consisted of 10 distinct objects, we could construct all numbers from 0 up to 10, but the number eleven would not exist. (More precisely, it would equal to zero since, by the definition, <$11 = \{A: A\, \text{has 11 members}\} = \emptyset = 0$>.) There is an elegant answer, though. All we really need is an empty set, since from it we can generate its power set <$\mathcal{P}(\emptyset) = \{\emptyset\}$> which is a singleton distinct from <$\emptyset$> itself. Now we have a set of two objects, <$\{\emptyset, \{\emptyset\}\}$>, whose power set in turn contains 4 distinct objects, and so on (<$|\mathcal{P}(X)| = 2^{|X|}$>). We can thus generate any finite number by this procedure.

Standard Construction Using Set Theory

This approach may seem strange at first, but it is in fact very elegant and convenient. We construct the natural numbers from almost nothing. We identify zero with the empty set and then inductively define <$n + 1$> as <$n \cup \{n\}, n \in \mathbb{N}$>.

<$$ 0 = \emptyset $$> <$$ 1 = \{\emptyset\} $$> <$$ 2 = \{\emptyset, \{\emptyset\}\} $$> <$$ 3 = \{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\} $$> <$$ \vdots $$> <$$ n + 1 = n \cup \{n\} $$>

It is apparent that <$1 = 0 \cup \{0\} = \{0\}$>, <$2 = 1 \cup \{1\} = 0 \cup \{0\} \cup \{0 \cup \{0\}\} = \{0, \{0\}\}$> and so on. It is easily proved that <$n = \{0, 1,\cdots, n-1\}\,\, \forall n \in \mathbb{N}$>.

Note that in order to construct the whole set of natural numbers, all we need are two primitives: the empty set <$\emptyset$> and the notion of a successor, i.e. the idea of what <$n + 1$> is in relation to <$n$>. In more general terms, we would drop the notion of “adding one” and define the successor function <$S: \mathbb{N} \to \mathbb{N}$> instead: <$S(n) := n \cup \{n\},\, n \in \mathbb{N}$>.

The actual existence of the set of all natural numbers so defined is guaranteed by the axiom of infinity which postulates that there exists at least one infinite set.

February 25, MMXI — Mathematics, Philosophy.