I'm a first year student in mathematics, and we recently defined Natural numbers as following:
"For all numbers, a number is an element of the set Natural numbers, if and only if it is an element of every inductive set". An inductive set is defined as "A subset of the Real numbers that contains 1 and for each element, that element + 1 is also an element of the set'.
One's natural instinct is to ask why not define $\mathbb{N}$ = {1, 2, 3 ...} which is easy to dismiss because we cannot define an infinite set in Roster notation when there is no set builder notation. We have not yet defined what natural numbers are and therefore these numbers are not related by any means (The only property that they share is being Natural which yet has no definition in our stack. even if one wishes to define this set inductively, through something call a successor function if I'm not mistaken, the domain of the said function must be the natural numbers which...defeats the purpose? Someone might argue for a definition through sequences, which...can one have sequences before defining Natural numbers?).
I suppose the reason why we are being so rigorous in defining natural numbers in such way is that we wish to prove things by induction. When we prove through induction we basically define a set like M to be the subset of $\mathbb{N}$ of all numbers who satisfy P(n) which P(n) is usually our the algebraic conclusion in our theorem. We prove 1 $\in$ M and $\forall n (n+1 \in M)$. From this we conclude M = $\mathbb{N}$.
This triviality is in need of proving, so we naturally (no pun intended) resort to making a more rigorous definition of $\mathbb{N}$ in order to be able to use induction. We define inductive sets, then define a Natural number as said above, see that according to definition $\forall \hspace{1mm} inductive A, \hspace{1mm}(\mathbb{N} \subseteq A)$.
So it is sufficient to know a subset of Natural numbers is inductive to prove they are equal (which we apply in M for the above). It is easy now to prove that 1, 2, 3 and so on are natural numbers individually, and I don't know if I'm being nitpicky about this, but I have the following concern: when we prove things through induction we essentially have in mind that we want to prove that a theorem holds for {1, 2, ...} and that happens to be called $\mathbb{N}$. We can't have {1, 2, 3, ...} because of the reasons above so we take $\mathbb{N}$ as nothing but a name and define it as we did above. we can prove that 1 $\in \mathbb{N}$ and so on, but that is a stretch of it being equal to the infinite set of {1, 2, 3, ...}. Instantly, it seems ridiculous to call it a stretch for it not being equal to something that we cannot define or is defined through something we earlier defined. does it suffice to say {n| n $\in \mathbb{N}$} = {1, 2, 3, ...}, or we should find a separate way to define the algebraic form of Natural numbers and then prove it to be equal to the $\mathbb{N}$ we defined above through inductive sets. It also seems that we can prove 1 $\in \mathbb{N}$, 2 $\in \mathbb{N}$ and ... . But it seems that proving {1, 2, ...} to be equal to $\mathbb{N}$ is (in addition to needing to define {1, 2, ..} in a way it makes sense) in itself in need of induction for proving.
I would be most thankful if someone would help clarify my perspective.