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I was watching a set of lectures on introductory real analysis. LINK. Brazilian portuguese only unfortunately.

The lecturer starts defining (his own words):

Peano's axioms

Let $\mathbb{N}={1,2,3,\dots}$:

  1. There exists an injective function $s:\mathbb{N}\rightarrow\mathbb{N}$;
  2. There exists an element $1$ such that $1 \in \mathbb{N}$ but $1 \notin s(\mathbb{N})$, i.e. $1 \notin Im(s)$;
  3. (induction) Let $A$ be a set such that $A \subset \mathbb{N}$. If $1 \in A$ and ($n \in A \implies s(n) \in A$) then $A = \mathbb{N}$.

I'm still learning the very (very) basics of this, so I'm not sure whether this a proper way to define this stuff neither I can tell if these are right or wrong.

The question that bugs me is:

  1. He assumed that there is such a thing as a "function" and also assumed "advanced" concepts like "injection". Can he do that given we are in the terrain of very primitive mathematical foundation, i.e. things that all other things build upon in math? In my beginner's math mind he should have used more primitive mathematical concepts (I'm not sure which ones exactly).
user0102
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daniel.franzini
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    Yeah, Peano usually requires only first order logic, not set theory. This is a reformulation of Peano in a kind of naive set theory. Still, even in first order logic, we have "functions" they just aren't set theory functions, and only in the models of the theory do we talk about them as set functions. – Thomas Andrews Jul 22 '21 at 16:02
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    So in first order theory, there's be a constant $1$ and a function $S$ defined on our whole domain such that $S(x)=S(y)$ implies $x=y$ and $\forall x:S(x)\neq 1.$\ – Thomas Andrews Jul 22 '21 at 16:05
  • @daniel.franzini In fact, it is an introduction to real analysis. – user0102 Jul 22 '21 at 16:20
  • Yes, @user0102, it is! My bad. Fixed! – daniel.franzini Jul 22 '21 at 19:57
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    How to present Peano's Axioms with notions of "first" and "next":

    1. $1$ is a natural number ('$0$'is often used instead).

    2. Every natural number has a unique "next" natural number. (Functionality of the successor relation)

    3. Different natural numbers have have different next natural numbers. (Injectivity of the successor relation)

    4. No natural number has $1$ as its next natural number, i.e. 1 is the "first" natural number.

    5. Every natural number but $1$ can be reached by a process of repeatedly going to the next natural number starting at $1$. (Equivalent to induction)

     – Dan Christensen Jul 23 '21 at 03:07
  • Using "first", "next" and "previous": 1. $1$ is a natural number ('$0$'is often used instead). 2. Every natural number has a unique "next" natural number. (Functionality of the successor relation) 3. If a natural number has a "previous" natural number, then that previous number is unique. (Injectivity of the successor relation) 4. $1$ is the "first" natural number, i.e. it has no previous natural number. 5. Every natural number but $1$ can be reached by a process of repeatedly going to the next natural number starting at $1$. (Equivalent to induction) – Dan Christensen Jul 23 '21 at 16:25
  • For what it is worth, here is my formal proof of the above mentioned equivalence to induction using a form of natural deduction: http://www.dcproof.com/InductionIffAccessible.htm (228 lines) – Dan Christensen Jul 23 '21 at 16:54
  • @daniel.franzini: You are right that your lecturer is using less primitive concepts than are needed. Do note that this approach essentially requires a background set theory, otherwise you have nothing that you can apply the so-called induction axiom to. Peano came way before FOL was properly understood, so his axiomatization was unclear. This is why in modern logic PA is not the same, and spaceisdarkgreen has adequately pointed that out. However, for your purposes (undergraduate mathematics) you already must work within some set theory, otherwise you cannot express many things. – user21820 Jun 07 '22 at 21:31
  • @daniel.franzini: So just keep track of all the primitive notions that you need to use, and only in the future if you are interested in foundations of mathematics do you need to think about how little you want to assume, for which you can read this post. By the way, Dan's linked website is useless; its system cannot be used for real mathematical work, and some articles are based on conceptual errors. – user21820 Jun 07 '22 at 21:37

3 Answers3

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As others have pointed out, things like "injective function" can be rephrased in more elementary way. The induction axiom is where all the action is. In order to make sense of the induction axiom, we need to have a notion of arbitrary subsets of $\mathbb N,$ which requires some set theoretical commitments. In other words, we need second-order and not merely first-order logic to state and interpret this.

One way to "get around" this is to replace the induction axiom with a first-order schema. So we have for each first-order formula $\varphi(x,\vec y)$, a first-order induction axiom Ind$_\varphi$: $$ \forall \vec y\;((\varphi(0,\vec y)\land \forall x\;(\varphi(x,\vec y)\to \varphi(Sx,\vec y)))\to \forall x\;\varphi(x,\vec y)).$$

Now our theory is first-order and we can make sense of it without any set theory. However, it's also much weaker and does not determine the natural numbers up to isomorphism (in other words, it has nonstandard models, which is pretty much inevitable with purely first-order axiomatizations).

In my experience "Peano Axioms" refers to the five axioms in the second-order formulation. Whereas "Peano Arithmetic" (aka PA) refers to a first-order axiomatization with an infinite schema of induction axioms that also includes the arithmetical operations $+$ and $\cdot$ in the language.

spaceisdarkgreen
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Peano arithmetic is usually defined in a way to avoid this. It is not hard to change. Instead of defining successor as a function, you say "Every integer has a successor." And instead of saying it is an injective function, you say "If $n, m$ have the same successor, then $n = m$."

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I like very much the way Terence Tao describes the Peano's axioms.

Complementing the answer given by @MichaelBarz and the comments of @ThomasAndrews, I advise you to consider the following formulation:

  1. $0$ is a natural number.
  2. Every successor of a natural number is a natural number.
  3. $0$ is not the successor of any natural number.
  4. Different natural numbers have different successors.
  5. Induction axiom.

In his book "Analysis I", he discusses the motivation for each of them. It is very enlightening.

Hopefully this helps!

user0102
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