I am learning about classical set theory and how $\mathbb{N}$ can be constructed from sets satisfying ZFC in such a way that the Peano arithmetic axioms are satisfied.

Although the construction is clever, it seems to me to be quite convoluted (particularly the way $\mathbb{R}$ is then constructed from Dedekind cuts!) and then a bunch of effort is needed to show that the basic properties we want $\mathbb{N}$ are still satisfied.

Even then - we do not claim that $\mathbb{N}$ is actually the construction from ZFC but rather than ZFC allows us to construct something which satisfies the properties we want in $\mathbb{N}$ (e.g. "quasi-$\mathbb{N}$"). The $\mathbb{N}$ we know and love is still sort of undefined, and the most succinct definition, if one is needed, is arguably Peano arithmetic (at least, that any model satisfying Peano axioms has the same key properties as $\mathbb{N}$).

My question: what was gained, mathematically, by constructing a quasi-$\mathbb{N}$ from ZFC?

Suppose instead that mathematicians simply accepted that $\mathbb{N}$ was fundamental and axiomatic, and it has the properties we like, and that the Peano axioms provide sufficient definition (inarguably simpler than ZFC!). In both cases we are left with a set of axioms which are unproven.

So why bother constructing quasi-$\mathbb{N}$ from ZFC at all?

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    $\begingroup$ This answer to a related question is very relevant. $\endgroup$ Jul 29, 2022 at 2:50
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    $\begingroup$ The construction of $\mathbb N$ as an initial infinite segment of ordinal numbers is but one topic involved in showing ZFC suffices as a foundation for doing mathematics in a recognizable fashion. $\endgroup$
    – hardmath
    Jul 29, 2022 at 2:59
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    $\begingroup$ "any model satisfying Peano axioms has the same key properties as $\mathbb N$" - not true, for example there is model of PA that has an element encoding proof of inconsistency of PA. I would consider lack of such number pretty important property of $\mathbb N$. $\endgroup$
    – mihaild
    Jul 29, 2022 at 8:17


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