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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)?

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    $\begingroup$ The quoted passage should be thought of as an informal comment. $\endgroup$ – André Nicolas Apr 30 '15 at 1:23
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    $\begingroup$ S(0) is called "the number 1"; S(S(0)) is called "the number 2"; S(S(S(0))) is called "the number 3". What do you want to call S(S(S(S(0))))? We can call it whatever want ("Red lip gloss", ...). So I take it the intent of the Wikipedia comment is to say: suppose we have S applied some number of times to 0, what do we want to call that? Well, the common English definition of that number of times is as good a label as any. Those labels are completely arbitrary. What matters for the Peano axioms is that S(0) is distinct from S(S(0)), and both of those are distinct from S(S(S(0))). And so on. $\endgroup$ – Simon S Apr 30 '15 at 1:33
  • $\begingroup$ Less ponderous might have been, "The number $1$ can be defined as $S(0)$, $2$ as $S(1)$, $3$ as $S(2)$ and so on." Leaving out the "n-fold application" bit. $\endgroup$ – Dan Christensen Apr 30 '15 at 15:32
  • $\begingroup$ BTW, $S^n(x)$ is just $x+n$. $\endgroup$ – Dan Christensen Apr 30 '15 at 15:46
  • $\begingroup$ @ Dan. We can define the addition operation before defining the natural numbers? $\endgroup$ – Johny Diala May 1 '15 at 0:00
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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.

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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.

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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.

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  • $\begingroup$ Okay, thank you as well for clarifying Peano's Axioms. It's possible that Wikipedia's explanation was simply somewhat obscure. $\endgroup$ – Johny Diala May 1 '15 at 0:05
  • $\begingroup$ Unfortunately, the Wiki article does not describe the most modern version of Peano's Axioms (which I list). The axioms listed there look more like the original 1889 version, most axioms of which are now either the domain of formal logic (the equality axioms) or the domain of set theory (the construction of addition -- with inequalities -- multiplication and exponentiation functions). Of those listed at Wiki , only axioms 1 and 6 through 9 remain. (Not sure about the first-order axioms.) $\endgroup$ – Dan Christensen May 1 '15 at 2:33

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