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From Jacobson's Basic Algebra I on P. 16, the Peano axioms are stated as:

  1. $0 \neq a^+$ for any $a$ (that is, $0$ is not in the image of $\mathbb{N}$ under $a \to a^+$).
  2. $a\to a^{+}$ is injective.
  3. (Axiom of induction). Any subset of $\mathbb{N}$ which contains $0$ and contains the successor of every element in the given subset coincides with $\mathbb{N}$.

First, $a$ does not seem to be defined to be anything (at least from what I can see on the current and previous page), though it seems like it is supposed to be representing some arbitrary element in $\mathbb{N}$? It is not clear. So my first question is what is the notation $a$ and $a^+$ referencing?

The other question I have is regarding that of the successor map. Why do I need two axioms for a map? Why can I not just point to the mapping $f: \mathbb{N} \to \mathbb{N}$ given by $f(n) = n+1$. When I think more about it, couldn't I define the map to be $f(n) = n+k$, where $k$ is a positive integer? What is the relation of the axiomatization to the successor map? Does the successor map not exist without an axiomatization?

Additional Context

There are only two other places where $a$ is mentioned before this, which is on P.15 of the text: "For example, in a structure $S$ in which an associative binary composition and a unit are defined, any element $a \in S$ defines a map $n \to a^n$ where $a^0=1, a^1 = a,$ and $a^k = a^{k-1}a$."

"From this point of view, we begin with a non-vacuous set $\mathbb{N}$, a particular element of $\mathbb{N}$ designated as $0$ and a map $a \to a^+$ of $\mathbb{N}$ to itself, called the successor map."

Richard K Yu
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  • Please link or copy all the relevant information about Peano axioms from Jacobson. Surely he has something like "there is a set $A$ and a map $a \mapsto a^+$ such that..." – coffeemath Jul 22 '23 at 02:19
  • Here is a pdf which is the same as the text I have: http://www.math.toronto.edu/~ila/Jacobson-Basic_algebra_I%20(1).pdf – Richard K Yu Jul 22 '23 at 02:22
  • Yes that's a link. But it is to the entire text. For questions here, I think users would appreciate more effort to summarize relevant definitions than simply to link to a large pdf file. – coffeemath Jul 22 '23 at 02:31
  • I felt it was less cumbersome to link (the pages are 15 and 16, towards the beginning) than to either type out the entire section or guess at what I missed. I typed out what I believed was relevant, I am asking for help because I do not know what is relevant beyond this and because to the best of my knowledge there was not a previous definition of $a$. – Richard K Yu Jul 22 '23 at 02:41
  • ricard: OK I'll try to check it when I get time. Hopefully someone here can respond who has a copy of Jacobson. Good luck. – coffeemath Jul 22 '23 at 02:49
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    Richard: I found the basic setup in the second paragraph of section 0.4 [The natural numbers]. That explanation is actually in the same paragraph from which you quote the rules. Also Gerry Myerson has outlined things in his answer below. – coffeemath Jul 22 '23 at 03:18

1 Answers1

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The Peano axioms are a way to define the natural numbers; to introduce the concepts of addition when trying to define the natural numbers would be circular.

So: "for any $a$" means "for any $a$ in $\bf N$". It is not assumed that $\bf N$ has any structure or properties other than those we can prove from the Peano axioms. After we have proved a few theorems, we can identify $a^+$ with $a+1$, but when we're just getting started, we don't have any "$+$" and we don't have any "$1$".

Gerry Myerson
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