How do Peano's axioms make it clear what the successor is equal to?

Question

Most likely this subject has been covered many times here, still I fail to grasp this.

I can't understand how do we know that the successor of $1$ is $2$ based on Peano's axioms, given that we start from $1$ (for convenience, since the book I am reading starts from $1$)? It is all good to the point where we demonstrate that there is a unique function such that

• $f(x, 1) = x'$
• $f(x, y') = (f(x, y))'$

However, it all falls apart for me, when somehow magically $f(x, y')$ becomes $x+y'$ or particularly $f(x, 1) = x+1$, hence $x+1 = x'$. My problem is that, as we have never been told by axioms what successors really are, just defining the function $f$ through some plus-sign notation doesn't make it clear how to perform (calculate, evaluate) that operation. In my eyes, it is the same as if it was written like $f(x, 1) = x ? 1 = x'$ (yes, the operation is a question mark). So, how does $x ? 1$ imply that the successor of $1$ should be $2$? Where does it come from that we assign $2$ to the successor of $1$, $3$ to the successor of $2$, etc. besides the intuition that it should be this way?

P.S. It reminds me of a square root function $r(x)=\sqrt{x}$ with fancy notation $\sqrt{x}$, definition of which doesn't really tell us how to calculate one but rather defines a property of the result that whatever it is, its square should be equal to the $x$. Similar feeling about $x+1$ is also present here:

• either I have to know where it comes from that the successor of $1$ is $2$, then $x+1$ becomes a mere conventional notation for the successor, on top of which more general operation $x+y$ is defined
• or how to actually perform $x+1$ to get $2$, when $x=1$, which is not obvious to me from the axioms. Although I have an intuition how to define $x+1$ through set-theoretic notation, where natural numbers are defined through empty set ($0 = \varnothing,1 = \{\varnothing, \{\varnothing\}\}, \dots$), I am interested how it derives from Peano's axioms, not from the set theory.

Answer 1

The general concept of an axiom system does not tell you what the objects are that it refers to, it tells you only properties of those objects and of the interactions between them. The axioms do not prescribe what things actually are, they merely describe things.

So, for example, the Peano Axioms do not tell you what $2$ is. Also, they do not tell you what $+$ is. Nor do the tell you how to go about actually performing $x+1$.

In fact that's the whole point of the Peano Axioms. They are not authoritative. Their sole purpose is to give one a tool for building the natural numbers on a simpler foundation.

Our experience of the natural numbers includes many named objects like $1, 2, 3$ and $485096$, and complicated binary operations such as $+$ and $\times$, and complicated binary relations such as $<$, and long lists of useful identities such as $a \times (b+c) = a \times b + a \times c$. The point of the Peano Axioms is that they give us a much less complicated system of objects and properties to accept, from which we, using our powers of logic, can construct the much more complicated system of objects and properties which comprise the theory of natural numbers. Once you accept this simpler foundation (based on your intuition, or on whatever basis you choose to accept), you can then then apply the Peano Axioms to define $2$ and $3$ and so on.

Have you accepted the successor function? Great, now define $2=1'$. Next, define $3=2'$ and $4=3'$ and $5=4'$ and $6=5'$ and $7=8'$ and $9=8'$. That was boring, onwards.

Have you accepted the axioms of the successor function? Great, now define addition, and prove it is commutative and associative, then define multiplication and prove is it commutative and associative and that multiplication distributes over addition, and so on.

---

What if you don't want to accept the Peano Axioms? What if you really, really want someone to tell you what $2$ is and $3$ is and $+$ is and so on?

Well, there are alternatives.

For example, many modern textbooks of analysis will start instead with axioms for the real numbers, and from those will construct the natural numbers satisfying Peano's Axioms; Fitzpatrick's Advanced Calculus textbook is an example, which I used in my Advanced Calculus class.

For another example, in modern set theory one starts with the ZFC axioms, and then using Von Neumann ordinals one constructs the natural numbers satisfying Peano's Axioms.

However, in both cases it's just a matter of pushing the ball backwards: you have to accept some axioms as your starting point: Peano's axioms; the axioms of the real numbers; the ZFC set theory axioms; or something.

Answer 2

`how do we know that the successor of $1$ is $2$ based on Peano's axioms'

$2$ is defined to be the successor of $1$, so $2=1'$. Define addition by the following two rules:

1. $x+1=x'$
2. $x+y'=(x+y)'$

So, $1+1=1'$ by the second rule, and this is $=2$ by the definition of $2$.

Let's do a more complicated example: define $3=2'$, $4=3'$, $5=4'$, ... $8=7'$, and $9=8'$. Then if I wanted to show $3+2=5$

\begin{align*} 3+2&=3+1' & \textrm{ definition of }2\\ &=(3+1)' & \textrm{ + rule }2\\ &=(3')' & \textrm{ + rule }1\\ &=4' & \textrm{ definition of }4\\ &=5 & \textrm{ definition of }5 \end{align*}

`what the successors really are'

This is not the right question, and I will try to explain why. The Peano axioms, and axioms in general, are not meant to tell you what something 'really is'. Rather, the purpose of axioms is to tell you how they are meant to behave. Above we have proofs of $1+1=2$ and $3+2=5$. Notice that the proofs I gave don't use Peano's axioms. They only use the definitions I gave for the symbols $1,...,9$ and the two rules for $+$. Peano's axioms would be needed to prove things like $1\ne 2$ or $6\ne 9$, or to prove that there are infinitely many natural numbers.

If you want to know what natural numbers `really are', you could try a definition like the one you gave with sets. $0=\emptyset$ and $x'=\{x\}$. This gives what logicians would call a model for Peano's axioms. The collection of sets $\emptyset$, $\{\emptyset\}$, $\{\{\emptyset\}\}$, etc. will act as the natural numbers $0$, $1$, $2$ etc. should. This means they are a model of the naturals. But, they are not the only things that behave the right way, that is they are not the only model. Instead of sets, there other types of objects which behave like the natural numbers enough to act as a model for them. Peano's axioms tell you what natural numbers should behave like without having to say what natural numbers 'really are'.

Peano's axioms exist for a purpose, and it is not for people to learn what naturals are or how to add. Students learn what numbers are and how to add them long before this. The axioms reduce the notion of `natural numbers' to a more basic collection of statements which can be expressed with (second order) logic, and that is their purpose. Reasoning about natural numbers can then be carried out using only logic.

I will sketch how we can prove arithmetical statements involving larger numbers in the same way.

The proof of $3+2=5$ above relied on the fact that I had defined which natural numbers are meant by the symbols $1,...,5$. Since I only defined the symbols $1,...,9$ a natural question is how I could do something like $4+7=?$. The answer should be $11$ which I haven't even defined yet.

The way forward is to define two digit numbers as follows: define $\times$ as a function which takes two natural numbers and returns a natural number subject to the rules

1. $x\times 1=x$
2. $x\times y'=(x\times y)+x$

Note that this is the familiar multiplication we know. For $a,b\in \{0,...,9\}$, define $ab$ to be $9'\times a +b$. Now that $11=9'\times 1+1$ is defined, there is a similar argument to show that $4+7=11$.

One can define (using recursion) numbers with more digits by saying that if $x$ is an $n$-digit number, and $d\in \{0,...,9\}$ is a digit, one can define an $n+1$-digit number $xd$ to be $9'\times x+d$. This provides a decimal representation for all numbers!

Now you can prove things like $999+2=1001$.