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I'll try to make this as short as I can. I'm just beginning to read Thomas Jech's Set Theory (3rd Millenium Edition) where he starts with the axioms of ZFC. Pardon the naivete of my question that follows, but some of these axioms sound vague to me. Take the Axiom of Separation: If $P$ is a property with parameter $p$, then for any $X$, there exists a set $Y$ $= \{u \in X : P(u,p)\}$.

The same with Axiom of Power Sets: For any $X$, there exists a set $Y$ of the subsets of $X$.

The key thing I have an issue with is that this "any $X$" is vague and does not help the reader make sense of the axioms. What is $X$ to be? It's not a trivial question when you consider the assertions these axioms make: namely, that the objects $Y$ in my two examples above are actually sets (and thus do not belong to the cesspool of proper classes that tend to breed those Russell-type paradoxes).

For example, in the power set axiom, since there are no restrictions on $X$, can I take $X$ to be a proper class? If so, this axiom tells me that the power set of $X$ (which has strictly larger cardinality than $X$ itself) is a set and not a proper class. How is that to be?

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    $\begingroup$ If you're familiar with first order logic the situation is analogous to the axioms of groups, we just say $\forall x\forall y\forall (z (x+y)+z=x+(y+z))$, without the need to specify that $x,y,z$ range over "elements of groups". In $\mathsf{ZFC}$ we write axioms $\forall x(\cdots)$ without the need to specify that $x$ ranges over "sets" $\endgroup$ – Alessandro Codenotti Aug 8 '18 at 20:26
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    $\begingroup$ I understand that that's what you're thinking, but that's simply not the case. Just like the quantifiers in a group theoretic statement do not range over the entire universe of sets and distinguish which are group elements and which are not; the quantifiers in a group theoretic statement range only over the elements of the group. You've developed a very misleading picture of what theories actually do. $\endgroup$ – Malice Vidrine Aug 8 '18 at 21:06
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    $\begingroup$ @marcus66502 I think you need to seriously consider the possibility that you do have a misleading picture of what axiomatic set theory does. (Back and composing my followup, as promised). $\endgroup$ – spaceisdarkgreen Aug 8 '18 at 21:20
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    $\begingroup$ A) ZFC does not guarantee us any such thing, and B) it does not pick out the "safe" classes from a prior universe of proper classes, it tells us what holds internally to the universe of sets. ZFC is a first order theory like any other, and if you understand the semantics for first order theories, then you should know the quantifiers do not do what you're trying to say they do. $\endgroup$ – Malice Vidrine Aug 8 '18 at 21:25
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    $\begingroup$ @KeyFlex: Really??? You edited just two variables into $ signs? You're that badge hungry that you won't even spend the extra three minutes to edit this post properly??? $\endgroup$ – Asaf Karagila Aug 9 '18 at 5:25
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In ZFC the domain of discourse is all sets. So when we say 'for all $X$' we mean for all sets $X$. So the power set axiom can be written in more detail as "for every set $X$ there is a set $P(X),$ whose elements are all sets that are subsets of $X.$"

(What 'all' sets means is a different can of worms and varies from model to model. But in any event, proper classes are not sets by definition, so are not included. In fact they can't be directly expressed as objects in ZFC's language. Rather, they are talked about indirectly, via formulas. So a class is just a formula, whose extension may or may not correspond to a set.)

Edit (In response to your comments)

It is wrong to think of the ZFC axioms as 'defining what a set is' although I can see why you might be under that impression since most of the axioms essentially tell you how to define new sets from previously defined ones. Rather, the ZFC axioms define what a set theoretical universe is, in that a set theoretical universe is a collection of objects and a membership relation $\in$ on those objects for which the axioms hold.

Although we certainly want such a theory to be consistent, and paradox avoidance is what motivated the restricted form of comprehension that ZFC has, it's also important to realize that ZFC makes some 'arbitrary' choices that have nothing to do with paradox avoidance. For instance, the axiom of extensionality implies that everything in our universe is a set and, for instance, there are no 'pure elements' (usually called urelements) that are not set-like in nature. (It does so by implying that the empty set is the only thing that doesn't contain anything else.) Also, the axiom of foundation guarantees that there are no strange sets that contain themselves, e.g. so-called Quine atoms obeying an equation like $x=\{x\},$ which are not inherently paradoxical despite not being as natural as urelements and superficially smelling like Russell's paradox. It is useful since it implies that the universe of sets is arranged in a Von Neumann hierarchy The other axioms (including perhaps choice) are less arbitrary, in the sense that they formalize our intuition from naive set theory of what kinds of sets we should be able to define based on others.

It is also important to note that ZFC is not a full characterization of a unique universe of sets. For instance, you have probably heard that the continuum hypothesis is independent of ZFC and that the axiom of choice is independent of ZF. This means that there are models of ZF/C that differ on these questions, and thus we haven't uniquely specified a universe of sets.

(In fact we could have never hoped to do anything close to this in a first order theory, with results like Lowenheim-Skolem. And yet if we try to move to a second order theory we run into another problem that we need set theory in our metatheory so set theoretical questions our theory 'answers' may well depend on what the answer is in the metatheory.)

As other have noted in the comments, this is not a lot different from how we usually proceed in axiomatizing math. The group axioms don't 'define what a group element is', they tell you the properties all groups have (and note there are many groups just as there are many models of ZFC). Same goes for the axiomatization of any type of mathematical object. Sometimes, as in the second order Peano axioms, there is only one model (up to isomorphism) obeying them, but that is a rare case.

This issue of circularity that you bring up is called impredicativity and a lot has been written about it. Essentially, it is the problem of circularity when the definition of an object involves quantifying over a universe containing that object. So for instance, when you define the empty set as 'the set that doesn't contain anything,' this definition includes an implicit clause that the empty set doesn't contain the empty set, and thus seems to require we know what the empty set is prior to our defining the empty set. Of course this is a silly example since it's reasonably clear that the notion of the empty set is self-consistent (see also the example of 'the tallest person in the room' from the Wikipedia page), so not all impredicative definitions are problematic.

Russell originally believed impredicativity was the ultimate source of his paradox and thus something that should be eliminated from formal foundations mathematics. However he was unable to develop mathematics satisfactorily on a predicative basis. There have been more successful attempts since then, but in general, much of classical mathematics is either impossible or intolerably difficult to develop only using predicative definitions. (Compare to the situation with constructive mathematics.)

Still, like constructivity, predicativity has its proponents, and consensus seems to be that it is at least something worth paying attention to as an organizing principle. For instance, in a second order theory, once can restrict comprehension principles to only apply to predicative formulas (i.e. ones that do not quantify over properties). ZFC is "hopelessly" impredicative, in the sense that if you think predicativity is a requirement in some sense, you probably reject ZFC foundations. You are correct to highlight the power set axiom as a weak point here, and there is also essential impredicativity in the separation and replacement axioms. (Kripke-Platek set theory (which is often viewed as a predicative fragment of ZFC) eliminates power set and choice and restricts both separation and replacement.

Notice, however, according to the the view I expressed at the outset (which is the view Jech and the majority of modern set theorists hold), impredicativity is not a problem. To be a little more specific, way to think about it is we have a (perhaps hypothetical) collection $V$ of objects related by the membership relation $\in$ that obey the ZFC axioms. This universe is fixed (though we don't necessarily specify it fully other than that it obeys the ZFC axioms). Then it makes sense that the axioms say 'for all sets, yada yada'. They are true statements about this universe of sets. They do tell us some that sets with certain properties have to exist and for certain properties (like $\forall x (x=x)$ by Cantor's paradox) there can't be a set with that extension, but again, this is a description of the universe.

If the axioms simply describe a universe of sets rather than 'building one from the ground up', then impredicative definitions pose no risk. It's just like the example with the tallest person in the room: if you really have a room with some people in it and aren't defining the people in the room into existence, there can't be any problems brought on by the circularity here. This isn't philosophically bulletproof since of course who says these set theoretical universes even exist, but still, given the difficulties with predicative mathematics and the general belief amongst set theorists that ZFC is consistent and that set theoretical universes either really exist in some sense, or are at least worth reasoning about as hypothetical objects, it is the prevailing attitude.

The idea that taking a Platonist view of the domain makes predicativity less of a concern leads readily to the idea of relative predicativity, where certain objects are taken to 'exist at the outset' and everything else is defined predicatively. An example that came up in the comments was the constructible universe $L$. If you are given the ordinals, $L$ is a completely predicative construction where every set (except the ordinals you start with) can be regarded as 'defined into existence' from the bottom up.

Edit 2 This is already overlong and a bit unfocused (though I think the commenters-- including a few professional logicians in case authority makes a difference-- have done a really good job of hammering home the main take-aways) but I thought it would be useful to actually spell out how the definition of the power set should be thought of here.

First of all, there is the subset property $y\subseteq x$ which is an abbreviation for $\forall z (z\in y \to z\in x),$ which hopefully already seemed sharp and unproblematic here. We don't even really quantify over 'all sets', just the elements of $y.$ Then, the power set axiom says $$\forall x \exists!z \forall u (u\in z \leftrightarrow u \subseteq x)$$ which informs us we are assuming that a set theoretical universe has, for any set $x,$ a unique power set $z$ whose elements are exactly those sets $u$ that obey the relation $u\subseteq x$ (just as we assume a group has an identity element with certain properties).

(Note we don't actually have to put all this in the axiom... for instance extensionality already guarantees the uniqueness part, and if we wish, we can weaken the $\leftrightarrow$ to a $\leftarrow$ and use subset comprehension to pick up the slack.)

Now, since we know there is a unique set with this property, we can define an operation $\mathcal P:V\to V$ such that given a set $x,$ $\mathcal P(x)$ is the unique set with this property. So we have argued that within any set-theortical universe, that this operation is well-defined and thus we have a power set for every set.

If you are a bit unsatisfied and think it's tautoglogical and we rigged that to happen, well, that's the point of the power set axiom. It's a thing that guarantees the existence of a power set for every set.

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    $\begingroup$ @marcus66502 'you don't think so' with regard to what? I don't think anything I've said here is controversial, but I'm not sure what you are disagreeing with. $\endgroup$ – spaceisdarkgreen Aug 8 '18 at 20:21
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    $\begingroup$ @marcus66502 I agree this is confusing and I see the apparently circularity, but I assure you that is exactly what he means. I'm a little busy at this exact moment but I will elaborate in a few minutes if someone else doesn't get to it first. $\endgroup$ – spaceisdarkgreen Aug 8 '18 at 20:29
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    $\begingroup$ @marcus66502 "The whole point of Axiomatic Set Theory is to provide a framework for proving that a certain object is a "set"" No. The whole point of axiomatic set theory is to provide a framework for describing how sets behave. Amongst other things, this lets us prove that certain formulas don't define sets (that is, that certain classes are proper classes), but the axioms need to be understood as describing a (hypothetical) existing universe rather than trying to create one from scratch. That's why quantifiers range over all sets. $\endgroup$ – Noah Schweber Aug 8 '18 at 22:51
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    $\begingroup$ ... exactly the same way that the axioms of a group don't provide a framework for telling whether any particular object is a "group element". The group axioms tell whether a collection of elements together with an operation form a group; the ZFC axioms tell whether a collection of objects together with a membership relation form a model of ZFC. When we talk about a model of ZFC, we have to do so in some metatheory (which could also be ZFC, or it could be some other set theory, or even a theory like second order arithmetic). $\endgroup$ – Carl Mummert Aug 8 '18 at 22:53
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    $\begingroup$ @Noah I wouldn't be able to say what "The whole point of axiomatic set theory is". $\endgroup$ – Andrés E. Caicedo Aug 8 '18 at 23:06

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