So far I know it is not possible to express the natural numbers $\Bbb N$ in terms of a first order sentence in language from ZFC. More precisely, one can in ZFC prove only it's existence as "witnessing set" of abstractly derivable proof from ZFC axioms of proposition $\exists^! x\, \varphi(x)$ where the predicate $\varphi(x)$ states "$x$ is minimal inductive set".
Recall, inductive sets $A$ a characterized by properties
(1) $\{ \emptyset\} \in A$, and (2) $(a \in A) \to (S(a):= a+1:=a \cup \{a\} \in A)$
$S(-)$ called from plausible reasons the successor function, as this allows to define integers inductively: $0:= \{\}, 1:=S(0),2= S(1),...$.
Now the witness of truth of $\exists^! x\, \varphi(x)$ is the set
$$ \bigcap_{A: \ A \ \text{inductive set}} A $$
But note that this set cannot expressed as set determined by membership relation in terms of a first- order formula in language of set theory.
Firstly part of my concern is - from pure didactical reasons - what is wrong with following naive attempt: Can't we pin down the integers $\omega$ as
$$ \{ x \ \vert \exists n: x= S^{\circ n}(\{ \emptyset\})\} $$
Question #1: Where is precisely the problem with this attempt?
Motivation #1: In this answer is briefly stated that it not works, but I would like to understand better the concrete reasons for it. What breaks down there?
Motivation #2: In comments below this answer Alex Kruckman remarks that it's not possible to form a first order sentence in ZFC language refering to "external" integers $\Bbb N $.
And I was wondering if this imposibility is precisely "at heart of the reason" that there exist models with non-standard integers. And if yes, can this be remedied if we allow second-order sentences?
Firstly observe what is meant by "external" intergers, what most undergrade students regard as "the" integers $\omega$.
Recall that the statement $\exists^! x\, \varphi(x)$ that there exist a minimal inductive set is provable in ZFC, but so far I see the problem is that the set witnessing this - which we would call "external" integers $\omega$ - cannot be explicitly expressed/written in language of set theory; what we only have "in our hands" is a proof that it exists. Hope, I phrased this point correctly.
This has as consequence that every model $M$ of ZFC has "its own" unique "internal" witness $\omega^M \in M$ with $M \models \varphi(\omega^M)$, as ZFC cannot "dictate" which one it should be, so there is no "universal" $\omega$, and the "internal" integers $\omega^M$ have a priori nothing to do with "external" integers $\omega$ as sets.
This leads me to several questions about relation between imposibility to write a first order ZFC formula refering to external $\omega$ and existence of such non standard models of integers.
Question(s) #2: Is it equivalent to say that no first-order sentence can refer to "external" integers to that there exist non-standard integers (eg uncountable)? Maybe the core of confusion is to phrase what it precisely means to say that a formula "refers to external integers?" Or to some other particular set. For instance, there should be so far I see no problem to produce a fol formula refering to a particular integer - say $n$ - right? What is here the "principal obstruction" to find such formula refering to external integers? That such hypothetical formula would neccessarily have to comprise of infinitely many symbols?
What, if we would allow to second-order sentences? Is it possible to construct such 2nd order sentence refering to "external" intergers ( ...what 1st order can't)
#Edit: As David Gao pointed out it's a fact that in second order logic there exist only a unique model of integers. But the question is if this is equivalent to that then in second order logic it is possible to formulate a sentence in language of set theory "refering to integers as set"? ( what is non possible in fol)
