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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.

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$a$ and $b$ are not among the natural numbers that we're used to, but they are part of these non-standard natural numbers, specifically to demonstrate that without the axiom of induction, we allow some really strange notions of what "natural number" means.

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