Perhaps we are now discussing different matters, so my response might be of limited use. The emphasis I made was that "$x$ is the smallest inductive set" is a well-formed formula in the language of ZFC. Really, the formula says that "$x$ has the inductive property and amongst such $x$ has the property of being smallest". There is no "is a set" predicate in the language. So all this talk about models and different natural number sets is irrelevant, at least when doing formal proofs with ZFC.

"$x$ is smallest inductive set" is a specific well-formed formula (up to some renaming of individual variables in the language of ZFC), and that's it.

This formula is derivable. You can define auxiliary shorthand using $\omega$ for convenience, with the express understanding that this shorthand is either (1) actually expanded out in a formal proof; (2) proven in a minimal metatheory way to behave in certain ways (proven by expanding out in a formal proof), and then used in these certain ways (because this use has been justified beforehand); or (3) definitionally expanded and proven to behave as intended.

Various philosophical positions on ZFC and mathematics in general usually respect what I have said before. In that we can always resort to syntactical proofs in ZFC if necessary, and that for actually writing and verifying the proof (without additional meaning given) the various positions are equivalent.

Nevertheless, there are ways in which the philosophical positions differ. Namely that they have unlike consequences for our psychology. This psychological effect manifests in the behaviour of the mathematician: what results are considered important to study, which topics even come to mind to study, which results seem intuitive, which proofs are inspired, etc, etc. So the output and ways of thinking will be different.

So there is some association between the results that present themself and the philosophy of the cameraman, sort of speak. With this association in mind and positing the existence of a subjective meter stick on types of result one considers interesting or useful or worthy, one may loosely judge the various positions at a given point in time. Regardless of the fact that they are equivalent in the sense explained previously. There is no need to be loyal to one position only.

You say "it is dangerous to talk about the set of integers" because different models of ZFC will (think to) have a set of integers of various seemingly conflicting properties.

Solutions:

(1) Consider transitive epsilon-models of ZFC (or even Z – P). Then $x = \omega$ is absolute between such models. Finiteness ("$x$ is finite") is also absolute for transitive epsilon-models of ZFC. So this particular problem no longer occurs.

(a) The (classic) universe view is that there is only one true set of integers (up to isomorphism). Various strange models of set theory might think that what they have is (a representative of the class of) the true set of integers, but these models might be said to be mistaken from the point of view of the universe. The "universe" is the jury, the arbiter.

(b) You might be interested in the multiverse (pluralist) view on set theory, recently popuralised by J. D. Hamkins, see, e.g., arxiv.org/pdf/1108.4223.

In conclusion, you arrive at oxymora or seeming paradoxes because (i) you seem to be dissatisfied with the purely "$x$ is smallest inductive set" is a well-formed formula, which is provable in ZFC approach. From the philosophical side, (ii) you are at the same time comparing two different models of ZFC, attempting to judge which is "better" but not completely following through with this judgment.

If you want to judge "betterness", the universe is a good judge. But if you do not want to harshly? judge models against one another, a multiverse view can help.

Finally, no need to be dogmatic or fiercely loyal to one position throughout time. Some malleability can be useful, and one position more useful in one exercise than another.

@user267839

let me try to summarize: so this well formed formula (=syntactically correct in object language grammar) $\exists^! x: \varphi(x)$ strictly saying these two things that (a) x inductive set & (2) x minimal wrt this property and it is derivable in object theory.

Nextly, you mentioned that one can use in this context $\omega$ "as "auxilary shorthand" as long as one treats it with caution elaborated in (1)-(3). But of "what" is $\omega$ acutually shorthand? I'm not sure, is this exactly answered in (1)-(3)? Eg on (1), in which sense a formal proof expand a "shorthand" notation? Isn't a shorthand notation tautologically declared by definition? Therefore the meticulous question for "which object/structure" do we actually introduce this shorthand $\omega$?

Therefore the notorious question for "what/which object/structure" do we actually introduce this shorthand $\omega$?

Adressing the derivibility of the formula, just to avoid confusions could you skim through if my understanding of "a derivable formula in object thery" ist correct? Here how I think about it: Starting from the object theory which itself consists of a collection of formulas phrased in underlying object language (alphabet +grammar). Certain subcollection of the formulas of the object theory are called axioms if other formulas can be "derived" from these. And so far I understand in order to make

the meaning of "deriving" precise here exactly enters the metatheory the game (which formally is again a collection of formulas but phrased in meta language; which may or may not identical wi obj language but in general unrelated) : So far I understand the formulas from meta theory encode completely the deductive rules which for the object theory "are allowed to apply" to deduce formulas within from its axioms. For instance, we can only deduce something from axioms in

object theory via modus ponens (which I regard as archetypcal exple for a formula in meta theory) if actually the modus ponens formula is actually contained in meta theory. So to apply this to our situation the formula $\exists^! x: \varphi(x)$ is derivable from axioms of objects theory according to the deduction rules piled up in meta theory? Is my attept to reproduct my understanding of the mechanism of the interplay between meta and object theories on deriving formulas from axioms correct?

#UPDATE towards my first question above concerning $\omega$ as "shorthand": Or, do you introduce it just as shorthand of the formula $\exists^! x: \varphi(x)$ itself? (...just a guess)