Peano Axioms

Introduction

I have recently bought the book Category Theory from Steve Awodey (Awodey 2010) is awesome, but probably the topic for another post), and a particular passage excited my curiosity:

Let us begin by distinguishing between the following things:

1. categorical foundations for mathematics,

ii. mathematical foundations for category theory.

As for the first point, one sometimes hears it said that category theory can be used to provide “foundations for mathematics,” as an alternative to set theory. That is in fact the case, but it is not what we are doing here. In set theory, one often begins with existential axioms such as “there is an infinite set” and derives further sets by axioms like “every set has a powerset,” thus building up a universe of mathematical objects (namely sets), which in principle suffice for “all of mathematics.”

This statement is interesting because one often considers category theory as pretty “fundamental”, in the sense that it has no issue with considering what I call “dangerous” notions, such as the category \(\mathbf{Set}\) of all sets, and even the category \(\mathbf{Cat}\) of all categories. Surely a theory this general, that can afford to study such objects, should provide suitable foundations for mathematics? Awodey addresses these issues very explicitly in the section following the quote above, and finds a good way of avoiding circular definitions.

Now, I remember some basics from my undergrad studies about foundations of mathematics. I was told that if you could define arithmetic, you basically had everything else “for free” (as Kronecker famously said, “natural numbers were created by God, everything else is the work of men”). I was also told that two sets of axioms existed, the Peano axioms and the Zermelo-Fraenkel axioms. Also, I should steer clear of the axiom of choice if I could, because one can do strange things with it, and it is equivalent to many different statements. Finally (and this I knew mainly from Logicomix, I must admit), it is impossible for a set of axioms to be both complete and consistent.

Given all this, I realised that my knowledge of foundational mathematics was pretty deficient. I do not believe that it is a very important topic that everyone should know about, even though Gödel’s incompleteness theorem is very interesting from a logical and philosophical standpoint. However, I wanted to go deeper on this subject.

In this post, I will try to share my path through Peano’s axioms (Gowers, Barrow-Green, and Leader 2010), because they are very simple, and it is easy to uncover basic algebraic structure from them.

The Axioms

The purpose of the axioms is to define a collection of objects that we will call the natural numbers. Here, we place ourselves in the context of first-order logic. Logic is not the main topic here, so I will just assume that I have access to some quantifiers, to some predicates, to some variables, and, most importantly, to a relation \(=\) which is reflexive, symmetric, transitive, and closed over the natural numbers.

Without further digressions, let us define two symbols \(0\) and \(s\) (called successor) such that:

1. \(0\) is a natural number.
2. For every natural number \(n\), \(s(n)\) is a natural number. (“The successor of a natural number is a natural number.”)
3. For all natural number \(m\) and \(n\), if \(s(m) = s(n)\), then \(m=n\). (“If two numbers have the same successor, they are equal.”)
4. For every natural number \(n\), \(s(n) = 0\) is false. (“\(0\) is nobody’s successor.”)
5. If \(A\) is a set such that:
   • \(0\) is in \(A\)
   • for every natural number \(n\), if \(n\) is in \(A\) then \(s(n)\) is in \(A\)
   then \(A\) contains every natural number.

Let’s break this down. Axioms 1–4 define a collection of objects, written \(0\), \(s(0)\), \(s(s(0))\), and so on, and ensure their basic properties. All of these are natural numbers by the first four axioms, but how can we be sure that all natural numbers are of the form \(s( \cdots s(0))\)? This is where the induction axiom (Axiom 5) intervenes. It ensures that every natural number is “well-formed” according to the previous axioms.

But Axiom 5 is slightly disturbing, because it mentions a “set” and a relation “is in”. This seems pretty straightforward at first sight, but these notions were never defined anywhere before that! Isn’t our goal to define all these notions in order to derive a foundation of mathematics? (I still don’t know the answer to that question.) I prefer the following alternative version of the induction axiom:

• If \(\varphi\) is a unary predicate such that:
  • \(\varphi(0)\) is true
  • for every natural number \(n\), if \(\varphi(n)\) is true, then \(\varphi(s(n))\) is also true
  then \(\varphi(n)\) is true for every natural number \(n\).

The alternative formulation is much better in my opinion, as it obviously implies the first one (juste choose \(\varphi(n)\) as “\(n\) is a natural number”), and it only references predicates. It will also be much more useful afterwards, as we will see.

Addition

What is needed afterwards? The most basic notion after the natural numbers themselves is the addition operator. We define an operator \(+\) by the following (recursive) rules:

1. \(\forall a,\quad a+0 = a\).
2. \(\forall a, \forall b,\quad a + s(b) = s(a+b)\).

Let us use these rules to prove the basic properties of \(+\).

Commutativity

Associativity

Identity element

From all these properties, it follows that the set of natural numbers with \(+\) is a commutative monoid.

Going further

We have imbued our newly created set of natural numbers with a significant algebraic structure. From there, similar arguments will create more structure, notably by introducing another operation \(\times\), and an order \(\leq\).

It is now a matter of conventional mathematics to construct the integers \(\mathbb{Z}\) and the rationals \(\mathbb{Q}\) (using equivalence classes), and eventually the real numbers \(\mathbb{R}\).

It is remarkable how very few (and very simple, as far as you would consider the induction axiom “simple”) axioms are enough to build an entire theory of mathematics. This sort of things makes me agree with Eugene Wigner (Wigner 1990) when he says that “mathematics is the science of skillful operations with concepts and rules invented just for this purpose”. We drew some arbitrary rules out of thin air, and derived countless properties and theorems from them, basically for our own enjoyment. (As Wigner would say, it is incredible that any of these fanciful inventions coming out of nowhere turned out to be even remotely useful.) Mathematics is done mainly for the mathematician’s own pleasure!

Mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting — M. Polanyi (Wigner 1990)

References