Successor in Peano Axioms I am an engineer who is learning real analysis myself. I read Peano axioms which are about the set of natural numbers $\mathbb{N}$. However, I find the statements, especially the word "successor" is not well defined, and cannot be converted into well-formulated formulae in first-order logic. Could anyone explain how to accept the word "successor"?
 A: In the context of Peano, we use the language of first-order logic with equality, enhanced by two symbols: $0$ and $S$, which are intended to stand for the number zero and for the successor operation. It is the addition of these symbols to the language that makes the respective axioms (e.g., $\forall n\colon \neg(Sn=0)$ well-formed. There is nothing to be defined about $S$ (or $0$). The meaning of these symbols is comprised by the axioms.
A: I am a physicist, so my perspective is a not quite that of either an eningeer or a mathematician. As usual, my empathy lies with the engineer, but my aim is toward the mathematician.
The 5 Peano axioms at the beginning of Kenneth Ross's excellent text $\it{Elementary~ Analysis: The Theory~of~Calculus}$ could/should be called (for teaching purposes) the Successor Axioms. This is because the notion of a successor is foundational to them. I would define a successor operator $\texttt{++}$ with notation that evokes computer science, and organize axioms differently. To be clear, the successor of $\texttt{n}$ is denoted $\texttt{++n}$ and has the value $\texttt{n+1}$. (So somewhere along the line, we have learned what "add 1" means.) Then, as Ross states, the natural numbers $\mathbb{N}=\{1, 2, \dots \}$ have the following "properties" (which we treat as axioms)

*

*If $\tt{n}$ is an element of $\mathbb{N}$, then its successor $\texttt{++n}$ is an element of $\mathbb{N}$. (Ross, Peano N2)



*

*

*a) 1 is an element(Ross, Peano N1),  but

*b) 1 is not the successor of any element of $\mathbb{N}$. (Ross, Peano N3)





*If $\texttt{++m = ++n}$, then $m=n$. (There is no convergence of separate branches of successors in $\mathbb{N}$; it is one long string.) (Ross, Peano N4.)

*Any subset of $\mathbb{N}$ that contains 1 and contains $\texttt{++n}$ if it contains $\texttt{n}$ constitutes the entire set $\mathbb{N}$. (Ross, Peano N5)

So I have four axioms, with the second one being a joining of two of the Ross Peano axioms. Ross Peano N1 and N3 as separate axioms may be logically indicated, but their recombination is warranted, I believe, since each axiom should be a separate statement about a successor property of $\mathbb{N}.$ Separating them disguises the centrality of successors to the axiomatic basis of $\mathbb{N}.$
Perhaps this appeals more to an engineer (and perhaps also to a computer scientist?) without distressing mathematicians too, too much. (?)
