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Peano Axioms

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---
title: "Peano Axioms"
date: 2019-03-18
---
* Introduction
I have recently bought the book /Category Theory/ from Steve
Awodey [[ref-1][(1)]] (which is awesome, but probably the topic for another
post), and a particular passage excited my curiosity:
#+begin_quote
Let us begin by distinguishing between the following things:
i. 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.”
#+end_quote
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 [[https://en.wikipedia.org/wiki/Peano_axioms][Peano axioms]] and the [[https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory][Zermelo-Fraenkel]]
axioms. Also, I should steer clear of the axiom of choice if I
could, because one can do [[https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox][strange things]] with it, and it is
equivalent to many [[https://en.wikipedia.org/wiki/Zorn%27s_lemma][different statements]]. Finally (and this I knew
mainly from /Logicomix/, I must admit), it is [[https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems][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
[[ref-2][(2)]], 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 [[https://en.wikipedia.org/wiki/First-order_logic][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 [[https://en.wikipedia.org/wiki/Predicate_(mathematical_logic)][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
#+begin_proposition
$\forall a, \forall b,\quad a+b = b+a$.
#+end_proposition
#+begin_proof
First, we prove that every natural number commutes with $0$.
- $0+0 = 0+0$.
- For every natural number $a$ such that $0+a = a+0$, we have:
\begin{align}
0 + s(a) &= s(0+a)\\
&= s(a+0)\\
&= s(a)\\
&= s(a) + 0.
\end{align}
By Axiom 5, every natural number commutes with $0$.
We can now prove the main proposition:
- $\forall a,\quad a+0=0+a$.
- For all $a$ and $b$ such that $a+b=b+a$,
\begin{align}
a + s(b) &= s(a+b)\\
&= s(b+a)\\
&= s(b) + a.
\end{align}
We used the opposite of the second rule for $+$, namely $\forall a,
\forall b,\quad s(a) + b = s(a+b)$. This can easily be proved by
another induction.
#+end_proof
** Associativity
#+begin_proposition
$\forall a, \forall b, \forall c,\quad a+(b+c) = (a+b)+c$.
#+end_proposition
#+begin_proof
Todo, left as an exercise to the reader 😉
#+end_proof
** Identity element
#+begin_proposition
$\forall a,\quad a+0 = 0+a = a$.
#+end_proposition
#+begin_proof
This follows directly from the definition of $+$ and commutativity.
#+end_proof
From all these properties, it follows that the set of natural
numbers with $+$ is a commutative [[https://en.wikipedia.org/wiki/Monoid][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 [[ref-3][(3)]] 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!
#+begin_quote
Mathematics cannot be defined without acknowledging its most obvious
feature: namely, that it is interesting --- M. Polanyi [[ref-3][(3)]]
#+end_quote
* References
1. <<ref-1>>Awodey, Steve. Category Theory. 2nd ed. Oxford Logic
Guides 52. Oxford ; New York: Oxford University Press, 2010.
2. <<ref-2>>Gowers, Timothy, June Barrow-Green, and Imre Leader. The
Princeton Companion to Mathematics. Princeton University
Press, 2010.
3. <<ref-3>>Wigner, Eugene P. ‘The Unreasonable Effectiveness of
Mathematics in the Natural Sciences’. In Mathematics and Science,
by Ronald E Mickens, 291–306. World
Scientific, 1990. https://doi.org/10.1142/9789814503488_0018.