My title probably doesn't explain my worry / concern too well, but it's the best title I could think of. I am researching construction of real numbers for a college project, and as a consequence I am researching the Peano axioms. A couple of sources I have read come to the conclusion that
$$
\{0,1,2,3,…\} \subseteq \mathbb{N}
$$
(where ${1=S(0)}$, ${2=S(1)}$,…) and by the axiom of induction we have
$$
\mathbb{N} \subseteq \{0,1,2,3,…\}
$$
hence
$$
\mathbb{N}=\{0,1,2,3,…\}
$$
(in other words, the Natural Numbers are the set of all possible successors of $0$). The part that makes me feel uneasy is the definition of ${\{0,1,2,3,…\}}$. Doesn't defining this set in the first place use induction in some sense? Like – we start with $0$. Then ${S(0)}$ cannot be $0$, and so we define ${1:=S(0)}$. Then ${S(1)}$ can be neither $0$ or $1$, and so we define ${2:=S(1)}$… I'm not sure how doing this for finitely many elements justifies the existence of the set ${\{0,1,2,3,…\}}$. Does what I'm saying even make sense? Am I overthinking things? Thank you!
EDIT: I should specify, I am following the definition of the axiom of induction as is on Wikipedia (https://en.wikipedia.org/wiki/Peano_axioms)
EDIT 2: source example: http://www2.hawaii.edu/~robertop/Courses/TMP/7_Peano_Axioms.pdf , page $3$
Best Answer
The notation $\{0,1,2,3,\cdots\}$ is being used here to symbolize "A set containing $0$, containing $S(n)$ whenever it contains $n$, and containing nothing else." It is true that the Peano axioms do not justify the existence of this set, but that is no surprise, since the Peano axioms do not deal with sets at all! With this line of thought, your two claims become "For every $n\in \mathbb N$, we have $S(n)\in \mathbb N$" and "If a set contains $0$ and $S(n)$ for every $n$ which it contains, then it contains $\mathbb N$", both of which are directly asserted by the Peano axioms. So your sources, at least as you paraphrase them, are saying nothing at all.