# Confusion regarding definition of Natural Numbers (from book Numbers, english version of Zahlen)

In the book "Numbers" by Ebbinghaus et. al, the Natural numbers are defined as:

The natural numbers form a set $$\mathbb N$$, containing a distinguished element $$0$$, called zero, together with a successor function $$S:\mathbb N \to \mathbb N$$, of $$\mathbb N$$ into itself, which satisfies the following axioms:

(S1) $$S$$ is injective,

(S2) $$0 \notin S(\mathbb N)$$,

(S3) If a subset $$M \subset \mathbb N$$ contains zero and is mapped into itself by $$S$$, then $$M = \mathbb N$$.

Does this definition puts any strict requirement that only $$0$$ can be starting element? In other words, does (S2) excludes any possibility of another starting element, say $$w_0 \notin S(\mathbb N)$$? If it was strict requirement, I think the author would have explicitly mentioned something like :(S2) only $$0 \notin \mathbb N$$

Also, as (S1) says $$S$$ is just injective (and not surjective), so there can be some other elements outside of range of $$S(\mathbb N)$$ similar to $$0$$.

The reason I ask, as if there exists such element $$w_0 \notin S(\mathbb N)$$, we can have a chain of elements in sequence $$w_0 \to w_1 \to w_2$$ and so on with starting element $$w_0$$ independent of the one with the $$0$$ (zero). Then, there can be two different sets, $$M_1$$ and $$M_2$$ for S(3), where $$M_1$$ include $$w_0$$ and $$M_2$$ doesn't, but both still include $$0$$.

then, $$M_1 = \mathbb N$$

and also, $$M_2 = \mathbb N$$

giving two different sets for $$\mathbb N$$ with different sizes, which seems wrong to me.

Can you please let me know what I understood wrong here?

• What do you mean by $w_0$, $M_1$, et cetera? Do you mean subscripts?
– user1134696
Commented Aug 15, 2023 at 22:22
• Also your $(S3)$ condition doesn't make sense. Please edit it.
– user1134696
Commented Aug 15, 2023 at 22:28
• @ShyamalSayak Thanks and apologies from my side. Fixed both now. Commented Aug 15, 2023 at 22:37
• ($S3$) is still not fixed properly.
– user1134696
Commented Aug 16, 2023 at 15:22

Suppose that $$\mathbb N$$ contains an element $$w_0$$ like you describe, that is, $$w_0 \neq 0$$ yet $$w_0$$ is not the successor of any element. Then consider the set $$\mathbb N \setminus \{w_0\}$$. Clearly this set contains 0. And it is also mapped into itself by $$S$$, since by assumption $$w_0$$ is not the successor of any element. It follows by (S3) that $$\mathbb N \setminus \{w_0\} = \mathbb N$$. But this is a contradiction. So no such element $$w_0$$ exists.
More informally, consider the set $$\{0, S(0), S(S(0)), S(S(S(0))), \ldots\}.$$ This set satisfies the requirements of (S3), and therefore it is precisely this set that is $$\mathbb N$$, and there cannot be any other elements.
• Thanks. Yes, this is the exact contradiction I was observing. So the sole purpose of (S3) is to just exclude any such element $w_0$ from $\mathbb N$? Also, I'm curious still, wouldn't then adding only to (S2) i.e. $S(\mathbb N)$ does not contain only $0$ (or rewording it as: $S(\mathbb N)$ contains all elements in $\mathbb N$, except 0) , would have avoided the need for S(3) and made definition simpler? Or does (S3) serves some other purpose too? Commented Aug 15, 2023 at 23:04
• @dheerajSuthar, (S3) serves lots of purposes! For instance, it also rules out that there's a loop: the "regular" natural numbers plus elements $a, b, c$ with $S(a) = b, S(b) = c, S(c) = a$. And it rules out that, besides the "regular" natural numbers, there are infinite chains: $\ldots, x_{-2}, x_{-1}, x_0, x_1, x_2, \ldots$ where $S(x_i) = x_{i+1}$. Commented Aug 15, 2023 at 23:22
• Thanks @mees-de-vries. The loop example was quite helpfu. To confirm, if I understood the infinite chain argument correctly, (S3) would enforce we have a starting point, here 0? In summary, the purpose of (S3) is to give us an intersection of all such possible valid sub sets of $\mathbb N$, thereby ensuring we get a minimum set fulfilling properties expected out of Natural numbers (start with 0, in sequence without loop, infinite in one direction etc.), and nothing else. Commented Aug 15, 2023 at 23:52