# Is multiplication postulated axiomatically in Peano arithmetic?

I figured this question is better suited to this stackexchange. I give some mathematical details, but this is primarily an HSM question.

According to this post, the existence of multiplication in Peano arithmetic is an axiom rather than a definition. This leads me to the following questions below.

How does this square with the fact that Peano axioms are usually presented as a list of five (or nine) axioms that mainly talk about the successor operation (and the equality relation if we're not presupposing it) with no mention of addition or multiplication?

Was multiplication postulated as an axiom in Peano's original work, or was it given by a definition?

Is the person in the post wrong?

• Why did you choose to believe an answer from some random person (with rather low reputation score) on MSE rather than reading the actual list of Peano Axioms? Nov 21, 2021 at 16:54
• Note: I think I might be the one misrepresenting things. The user was talking about Peano arithmetic rather than Peano axioms. Looking at the post again, it seems that "Peano arithmetic" and "Peano axioms" are used in specific ways, especially in logic. Nov 21, 2021 at 18:23
• In The Principles of Arithmetic (1889) Peano introduces multiplication as a "definition," $a \times 1 = a$ and $a \times (b + 1) = a \times b + a$. Obviously, you could instead call the two lines of the definition "axioms". Peano didn't give rules of what constituted a proper definition.
– abo
Nov 21, 2021 at 21:34
• No. There is a mixup here between object and meta-language. One can define addition and multiplication recursively in the object language (or rather its conservative extension) and then prove some properties. But not even ask whether they thereby "exist". PA need not postulate it any more than it needs to postulate its own consistency, those are meta questions. If we go for that already addition has to be "postulated" because it is also recursively defined from the successor function, see Devlin. Nov 21, 2021 at 22:47

There needs to be a distinction made between the terms Peano axioms and Peano arithmetic. The OP is conflating the two terms, hence the confusion.

In this wikipedia article (as of 11/21/21), it says,

A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema.

The math stackexchange post linked in the OP refers specifically to the theory of first-order Peano arithmetic where multiplication and addition operations are indeed postulated axiomatically. See this link and this link.

As for the original Peano axioms, we should refer to Arithmetices principia, nova methodo exposita by Giuseppe Peano, 1889. The two images below are from the archive here (around pages 23 and 29 of 49).

Later we see this.

A translation of Peano's work is given by From Frege to Gödel: A source book in mathematical logic by Jean van Heijenoort, 1967. We see that there are nine axioms (for natural numbers), and the multiplication is indeed given by a definition in the original work.

It needs to be stressed that there is no contradiction between the post that the OP linked and the historical fact. The post is talking about first-order Peano arithmetic; the Peano axioms refer to Peano's original list of axioms. It's just that Peano axioms are axioms for the theory of second-order Peano arithmetic as opposed to first-order Peano arithmetic. So both are correct.

To answer the questions directly, see below.

How does this square with the fact that Peano axioms are usually presented as a list of five (or nine) axioms that mainly talk about the successor operation (and the equality relation if we're not presupposing it) with no mention of addition or multiplication?

The list of Peano axioms is a list of five/nine axioms that only talk about the successor operation and no mention of addition/multiplication. The axioms for the system of first-order Peano arithmetic requires the induction axiom to be changed to a first-order axiom schema and additional axioms involving addition/multiplication added.

Was multiplication postulated as an axiom in Peano's original work, or was it given by a definition?

It was given by a definition.

Is the person in the post wrong?

No, because he's talking specifically about first-order Peano arithmetic.