On pg no. 3 of this article, the author says
let’s consider this version of $\Bbb{N}$ that satisfies all the above axioms, but is not the usual natural numbers we know: $\Bbb{N}=\{0,1,2,3,...,\} ∪\{a,b\}$. That is, this version of $\Bbb{N}$ contains all the natural numbers and also includes two other symbols, $a$ and $b$.
My question is if $a, b \notin \Bbb{N}$, then where do $a$ and $b$ belong? Also, if the above argument in the article is wrong, can you please provide another argument to show why the induction axiom is necessary?
Thanks in advance.