Why do we take the axiom of induction for natural numbers (Peano arithmetic)?

Question

More precisely, when we define the set of natural numbers $\mathbb{N}$ using the Peano axioms, we assume the following:

1. $0\in\mathbb{N}$
2. $\forall n\in\mathbb{N} (S(n)\in\mathbb{N})$
3. $\forall n\in\mathbb{N}(0\neq S(n))$
4. $\forall m,n (m\neq n\to S(m)\neq S(n))$
5. If $P(n)$ denotes the fact that $n$ has property $P$, then $\Big(P(0)\wedge \forall n\in\mathbb{N}\big(P(n)\to P(S(n))\big)\Big)\implies \forall n\in \mathbb{N} (P(n))$

I understand that using these axioms we can derive everything about the natural numbers, but I also think it's helpful to know why the axioms were chosen the way they are. So my question is why we choose to accept the axiom of induction ((5.) above), which in a way makes this more of a metamathematical question.

For example in Tao's Analysis I, it says that the axiom of induction keeps unwanted elements (such as half-integers) from entering the set.

Wikipedia says:

"Axioms [1], [2], [3] and [4] imply that the set of natural numbers is infinite, because it contains at least the infinite subset $\{ 0, S(0), S(S(0)), \ldots \}$, each element of which differs from the rest. To show that every natural number is included in this set requires an additional axiom, which is sometimes called the axiom of induction. This axiom provides a method for reasoning about the set of all natural numbers."

But I find this tautological: $\mathbb{N}$ is defined as the set of natural numbers so "$n$ is a natural number" means "$n\in\mathbb{N}$", right? So isn't every natural number included in $\mathbb{N}$ by definition?

Suppose we want to show $\mathbb{N}=\{0,1,2,3,\ldots\}$ using all five of the Peano axioms.

If we let $P(n)$ denote $n\in\{0,1,2,3,\ldots\}$, then $P(0)$ is true. Suppose $n$ is in $\{0,1,2,3,\ldots\}$. Then (informally) the dots indicate that $S(n)$ is in $\{0,1,2,3,\ldots\}$. So $\mathbb{N}\subseteq\{0,1,2,3,\ldots\}$, i.e., our defined set contains no "extra" elements (as in Tao's Analysis I).

Yet I still do not see how to show $\{0,1,2,3,\ldots\}\subseteq\mathbb{N}$ (in order to complete the "proof" that $\mathbb{N}=\{0,1,2,3,\ldots\}$) without just assuming it. (I think this is what the Wikipedia article was doing(?))

Thanks in advance for any help and I apologize if this kind of question is unsuitable for this site.

Answer 1

Informally, $\{0,1,2,3,\ldots\}\subseteq\mathbb{N}$ comes from the first and second axioms.

Of course you would need to define what $\{0,1,2,3,\ldots\}$ is. Perhaps writing it $\{0,S(0),S(S(0)),S(S(S(0))),\ldots\}$ makes it clearer. Then you can take a particular member of this set and use the first and second axioms to show it is in $\mathbb{N}$.

Answer 2

You are trying to deduce something about the importance of 5th axiom only from the axioms itself, which is impossible. The axioms by itself do not carry any significance. What is important is that the set defined by the axioms is (a) unique and (b) isomorphic to some real-world object (real-world natural numbers, as in "two apples" and "three oranges").

Giving away the fifth axiom means that:

a) Peano axioms no longer define a set of natural numbers, as there could be two non-isomorphic (and even not of the same cardinality) sets, both compatible with Peano axioms; rather they define a class of sets, each containing a subset, isomorphic to $\{0, 1, 2, 3, \ldots\}$. For example, let us consider the set $\mathbb N$ in its usual sense, and the set $\mathbb C \setminus {\mathbb N}^+$. Both sets fulfill the Peano axioms (except for the fifth one), but they are quite different in itself.

b) Of course, the second set from the previous paragraph has nothing in common with a real=world object called "the natural numbers".

Answer 3

Way of proving some theories by indcution is very importand technique without it mathematics would not be the same so it was added to formalize our intuition. From what I know suprise was that this axiom was not consequence of rest of the axioms but is needed to be stated explicite.

Answer 4

Note: I'm still learning about the Peano Axioms, so if any part of my answer is inaccurate, please let me know.

First, consider why we even bother thinking about the natural numbers. Of course, it's because they obey unique properties; namely, the natural numbers are "natural" in the sense that they represent how we count real-life objects. Clearly, the set of natural numbers is no ordinary set, so we might say that $\mathbb{N}$ is a "set with structure," where the "structure" encapsulates all the unique properties that make the natural numbers special.

Think of the Peano Axioms as a means of formulating this "structure." Any set obeying this "structure" should either be the natural numbers, or a set isomorphic to the natural numbers. Therefore, it doesn't make much sense to try and prove that $\{0, 1, 2, 3, \dots\} \subseteq \mathbb{N}$ (as you do in your answer); instead, you should be trying to prove that the set $\mathbb{N} = \{0, 1, 2, 3, \dots \}$ paired with the canonical successor function obeys the Peano Axioms, and thus represents the natural numbers.

If you view the axioms in this way, your question can be reframed as, "Why are the first four Peano Axioms insufficient in defining the 'structure' of $\mathbb{N}$?" A simple counterexample suffices to answer this, which Tao's half-integers does beautifully. Tao defines the half-integers to be

$$\mathbb{N} := \{0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, \dots\}.$$

and its successor function $S$ to be

$$ S : \mathbb{N} \to \mathbb{N} \\ S(n) = n + 1 $$

where $+$ is the standard real addition operator. It isn't difficult to check that this obeys the first four Peano Axioms. But this set is not the natural numbers, nor is it isomorphic to them! Therefore, we add a fifth axiom, which (in plain English) states that, "All natural numbers are some eventual successor of $0$."