Why is the language of arithmetic usually $(+, \cdot, 0, s)$, not $(+, \cdot, 0, 1)$?

Question

The formalized theory of arithmetic has usually $(+, \cdot, 0, s)$ as its language. However, from what we usually do in ring theory, it seems natural to use $(+, \cdot, 0, 1)$ as the language of arithmetic.

Why do we use the successor function, not the constant 1, when we formalize arithmetic? Are there any practical, philosophical or historical reasons behind this?

Answer 1

There is no significant difference between $(+,\cdot,0,1)$ and $(+,\cdot,0, s)$. They will be interchangeable, modulo rewriting some terms, for essentially every purpose. If there is any preference between the two, it is likely personal preference or tradition, rather than some technical mathematical reason.

There is a much more important difference between $(0,s)$ and $(+,\cdot, 0,s)$. The former is sufficient when we formalize the natural numbers in an ambient theory, like set theory, that allows us to define $+$ and $\cdot$ from $s$. This was the context of Peano's original work.

But, if we formalize arithmetic on its own, the signature $(+,\cdot, 0, 1)$ (or equivalent) is needed, because the first order theory of the natural numbers in the signature $(0,s)$ is not able to define either $+$ or $\cdot$, and neither of those operators is definable from the other. The theory with just 0, successor, and addition is known as Presburger arithmetic, and is a well known example of a weak system of arithmetic. The first-order theory Peano arithmetic is axiomatized in a language that includes $+$ and $\cdot$ as basic operations.

Answer 2

Peano (in the context of the question) was interested in the natural numbers, and not in general rings (which did not even exist at the time, but you can say that Peano was certainly aware of other rings, even if he did not call them rings).

A fundamental study of the natural numbers must identify certain aspects of the natural numbers upon which an axiomatization will be given. For the naturals it is the concept of the successor of an element that is fundamental. After all, what are the natural numbers $\{0,1,2,3,\}$ if not: Start with $0$, then apply the successor operation again and again and again. This is far from a trivial thing to turn into something formal and useful, but it already shows that the successor is more important (or more fundamental, or more primitive) than, say, addition or multiplication. Indeed, how to you even define addition or multiplication without the successor operation?

So, even today with our elaborated machinery of abstract rings and other algebraic structures, if you are interested in the naturals foundationally, then you don't want the language of rings, but rather of Peano arithmetic. The existence of various non-standard models of $PA$ shows how non-trivial the notion of "the successor of" is for the naturals.