Regarding Peano's Axioms

Question

According to the Wikipedia entry on the Peano axioms: "the number 1 can be defined as $S(0)$, 2 as $S(S(0))$ (which is also $S(1)$), and, in general, any natural number n as the result of n-fold application of $S$ to $0$, denoted as $S^n(0)$." (where $S(n)$ is the successor function).

The issue I have is with the statement "n-fold application". With these axioms, we are trying to define what the natural numbers are axiomatically. Therefore, we cannot use the natural numbers to define themselves (or so I think). However, using something like "n-fold application" within the axioms--where n is a natural number--is doing precisely that, is it not (using numbers to define what numbers are)?

Answer 1

You have identified the circular nature of definitions in mathematics. When you are building up the fundamental concepts in mathematics, there is no lower level of foundation on which to build. So you have to use "naive mathematics" to boot-strap the initial structures. (When I say "naive", I mean innate or in-born, or whatever you learned before formal mathematics.)

In this case, the number $n$ is used in two senses: the naive sense and the axiomatic sense. The wikipedia explanation is using naive language to explain formal definitions. As André Nicolas has written above, it is an "informal comment", which is sometimes viewed as metamathematical. Thus the formal $n$ may be explained as $S^n(0)$, where the superscript $n$ is an informal $n$. It is a constant struggle in the foundations of mathematics to try to keep the formal and the informal separate.

Answer 2

Before we have the Peano axioms, we already recognize certain natural numbers, such as 1, 8, 100, and 234124. When we write down the Peano axioms, we use these numbers (among other things) to label the variables that appear in the formulas. Let's call these numbers "metafinite" just to have a name for them. They are the numbers that we use in the meta langauge in which Peano arithmetic is expressed.

For each metafinite number $n$, we can introduce an abbreviation into Peano arithmetic: $S^n(x)$ means $S(S(S(\cdots S(x)\cdots))$ where $S$ appears $n$ times. This helps us identify, within each model of Peano arithmetic, a copy of the metafinite numbers - each metafinite number $n$ corresponds to the element $S^n(0)$ of the model of Peano arithmetic. The fact that every model of Peano arithmetic already includes an element for each metafinite number is a good sign that Peano arithmetic is a useful axiomatic system for studying the natural numbers.

There is a difference between these metafinite numbers and the numbers that may appear in an arbitrary model of Peano arithmetic. For example, there are models of Peano arithmetic in which there are "nonstandard" numbers, which are numbers in the model that do not correspond to any standard natural numbers. These nonstandard numbers correspond, informally, to "infinitely large" natural numbers. If $m$ is a nonstandard element of some model of Peano arithmetic, we do not have an abbreviation $S^m(x)$ in the metalanguage: we only have abbreviations in the metalanguage for metafinite numbers.

There is no way to avoid having a sense of the metafinite naturals when we axiomatize Peano arithmetic. For example, the formal language of Peano arithmetic includes formulas, which are strings of symbols. Each string of symbol has a length. What kind of number is that length? A little reflection shows that the metafinite naturals are exactly the lengths of strings of symbols.

Answer 3

There is no circularity in Peano's Axioms. They define $\mathbb{N}, S, 0$ such that:

1. $0\in \mathbb{N}$
2. $S: \mathbb{N}\to \mathbb{N}$
3. $S$ is injective
4. $\forall x\in\mathbb{N}:S(x)\ne 0$
5. $\forall P\subset \mathbb{N}:[0\in P \land \forall x\in P:S(x)\in P\implies P=\mathbb{N}]$

Note that the only number named here is $0$. Technically, we could have

$\mathbb{N}=\{ 0, S(0), S(S(0)), \cdots\}$

From (2), however, we have equivalently

$\forall x\in\mathbb{N}:\exists y\in \mathbb{N}: y=S(x)$

which we can use to assign names to the numbers other than $0$.

For $x=0$, we have $\exists y\in \mathbb{N}: y=S(0)$, $1\in \mathbb{N}$ and $1=S(0)$

For $x=1$, we have $\exists y\in \mathbb{N}: y=S(1)$, $2\in \mathbb{N}$ and $2=S(1)$

and so on.

The bit about $S^n(x)$ is not part of the axioms. In my opinion, it was completely unnecessary.