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Several introductory set theory books (Hrbacek & Jech, Enderton, Pinter) use very similar reasoning to develop the natural numbers. The argument used in Hrbacek & Jech (minus some supplementary explanatory comments) is pretty much word-for-word as follows:

Definition: The successor of a set $x$ is the set $S(x) = x \cup \{x\}$
We use the more suggestive notation $n + 1$ for $S(n)$ in what follows

Definition: A set $I$ is called inductive if
(a) $\emptyset \in I$
(b) If $n \in I$, then $(n + 1) \in I$

The Axiom of Infinity: An inductive set exists.

Definition: The set of all natural numbers is the set
   $\mathbf{N} = \{x \mid x \in I$ for every inductive set $I\}$.

The issue I see with this is in the last step defining the set $\mathbf{N}$ of natural numbers (which is presumably supposed to yield an $\mathbf{N}$ equal to the set of all finite von Neumann ordinals): The Axiom of Infinity guarantees the existence of an inductive set (let's call it $I_0$). But the Axiom of Infinity does not guarantee that $I_0$ doesn't contain some other elements besides ordinals, nor does it (by itself) guarantee that any other inductive set besides $I_0$ exists. If $I_0$ is the only inductive set and it contains elements that are not ordinals, then "the set of elements belonging to every inductive set" is just $I_0$ itself, which is NOT what you want (given that $I_0$ contains non-ordinals).

In order for the above-quoted development of the natural numbers to work, it seems like you need another step in the argument to get from the existence of an inductive set to the existence of other inductive sets, so that their intersection will be guaranteed to yield just the finite ordinals. Presumably the Powerset Axiom could do this for you? Perhaps the use of the Powerset Axiom is assumed but not explicitly stated? But I would think that's not a step you should skip in an *introductory* text.

So, bottom-line question: Is this development of the natural numbers indeed incomplete, or am I just missing something here?

NikS
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    One can take a particular inductive set $X$, and define $\mathbb{N}_X$ as the intersection of all inductive subsets of $X$. Then one proves the resulting set does not depend on $X$. See here. – Arturo Magidin Aug 14 '22 at 03:53
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    you write that you are worried about "guaranteeing to yield just the finite ordinals". what is your definition of "finite ordinal" here? generally "finite ordinal" would be defined as "a set contained in every inductive set", so there is no issue; it is a tautology/definition – Atticus Stonestrom Aug 14 '22 at 03:58
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    if by "finite ordinal" you mean something of the form $S(S(\dots(S(\varnothing)\dots)$ where there are finitely many applications of the successor operation, then there is no way to guarantee that, since by the compactness theorem any model of ZFC can be extended to one in which every inductive set contains "non-standard" elements – Atticus Stonestrom Aug 14 '22 at 04:00
  • Some models of the natural numbers have "other elements" than our intuition of the natural numbers. The Peano axioms do not fully define the natural numbers, nor can any other axiom system. You can show the Peano axioms apply to this set theory definition. That's the minimum we'd want for a set theory natural numbers. – Thomas Andrews Aug 14 '22 at 04:20
  • Thanks @ArturoMagidin . Yeah, that confirms my hunch that the "missing" step (or at least "unspoken/implicit" step) is an appeal to the Powerset Axiom to guarantee that from our "original" inductive set we can infer the existence of inductive subsets whose intersection will produce the intended set of "natural numbers." Curious that so many textbooks skip this step, as it seems fairly essential to the argument. – NikS Aug 17 '22 at 05:29
  • BTW, by "finite von Neumann ordinal" I mean (informally speaking) a set whose construction consists of starting with $\emptyset$ and then applying the successor operation finitely many times ($\emptyset, S(\emptyset), S(S(\emptyset))$, and so on). The best rigorous formalization of this that I've seen is in Azriel Levy's Basic Set Theory, where he starts with $\emptyset = 0$ as the least/smallest ordinal and then defines "$\alpha$ is a finite ordinal, or a natural number, if $\alpha=0$, or $\alpha$ is a successor and every ordinal $\beta<\alpha$ is $0$ or a successor". – NikS Aug 17 '22 at 05:31
  • Equivalent definitions of "finite ordinal" (as well as the definition of "natural numbers" to be the set comprised of every such "finte ordinal") appear also in the set theory books of Halmos and Jech. AFAIK this is part of the particular formalism introduced by von Neumann in his 1923 paper rigourizing the concept of ordinal numbers. – NikS Aug 17 '22 at 05:31

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