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(Warning: When referring to Russell's paradox I mean in the philosophical sense of informal collections, and not the "ZFC+Comprehension is inconsistent" sense. Russell's paradox applies to more situations than formal set theory.)

Mathematicians (of the non-finitist, Platonic sort) like to think of the set of natural numbers as existing as a completed object. Russell's paradox tells us that arbitrary "collections" cannot be collected into a completed object (on pain of contradiction). And yet, it is common in set theory texts to see the class of all sets, $V$, used freely. We are told that $V$ is not a set, but still a(n informal) collection of some kind. Yet, we are never told why we can't collect $V$, or other classes, into bigger classes. (And, indeed, in some versions of set theory, you can!)

Moreover, a significant portion of modern set theory deals with constructing models of set theory, using forcing. Yet, from a philosophical view, if sets were meant to formalize the general notion of a "collection", then a true/Platonic set theory should have no "models" at all. Wouldn't the domain of the model be a collection?

My question is thus: What are the philosophical positions of modern set theorists, on how to reconcile their subject with the philosophical view of Russell's paradox and the notion that sets are supposed to formalize arbitrary collections?

In particular, I suppose that some set theorists are secretly type theorists, and they view set theory as really set theory restricted to a very large type. Does set theorists secretly view type theory as more fundamental to formalizing mathematics? Or are there other resolutions, that are common among the set theory elite, but not necessarily conveyed in standard texts on the subject?

Edited to add: It appears that people have misunderstood some basic parts of my question. So let me try to clarify.

First, I am familiar with the class/set distinction. I'm familiar with the fact that in some formalizations of set theory (such as ZFC) classes don't really exist (per se) but are shorthand for formulas in the formal language, which one could think of as referencing certain meta-collections. I'm also familiar with the fact that in some formalizations of set theory (like NBG) classes are treated as actual objects, similar to sets, but limited in certain ways.

Second, it was my impression that set theory was created as a formalization of the informal concept of "collection". Sets were originally meant to refer to arbitrary collections (where order and repetition are ignored). There was no set/class distinction.

Now, it could be that modern set theorists no longer think of set theory as formalizing arbitrary collections (which seems to be what people in the comments are suggesting). If that's the answer to my question, then it begs the follow-up question: What is it, exactly, that sets are formalizing then?

Many mathematicians like to think of the set of natural numbers as existing as a completed object. But it is difficult to make set theory as concrete, because Russell's paradox, in conjunction with some type of separation principle, tells us that arbitrary "collections" cannot be collected into a completed object. I view this as telling us that the metaphysical idea of "collection" has some natural limitations that we might not have realized, a priori.

Now, in terms of the formal mathematics of collections---known as set theory---there seem to be two standard fixes to address the paradox.

Class and set distinction First is the idea of creating a new level of collection called "proper classes". In some set theories like ZFC, classes are an informal notion referring to the formulas of the language. Some mathematicians still view those classes as referring to meta-collections in the metatheory. They even use set-builder notation to refer to them. In other versions of set theory, like NBG or KM, classes are also formal objects. Sometimes they are of a different type than sets, and sometimes sets are classes with extra properties.

Those theories with classes can often be reinterpreted inside the theories without classes, and vice versa. Thus, it seems that Russell's paradox does not prescribe the existence, Platonically speaking, of two distinct types of collections---the set and the proper class. Yet this language has also become very useful to mathematicians. My question is somewhat philosophical in nature. Do modern set theories view proper classes as a necessary, true concept? Do they favor the view that proper classes are only informal, or are they formal?

I have a follow up question, for those set theorists that believe a "true Platonic set theory" exists. How do you view that completed set theory in light of Russell's paradox? It seems that a "true set theory" couldn't be like a collection itself (hence not like a set, nor like a proper class even). In particular, "true Platonic set theory" would be unlike any model of formal set theory, since the domain of a model is a collection.

Type theory Another solution, which I am much less familiar with, is using type theory to limit collection principles. Are there many modern set theorists who favor this resolution? Or has the proper class idea overriden this solution?

Mathematicians (of the non-finitist, Platonic sort) like to think of the set of natural numbers as existing as a completed object. Russell's paradox tells us that arbitrary "collections" cannot be collected into a completed object (on pain of contradiction). And yet, it is common in set theory texts to see the class of all sets, $V$, used freely.

Moreover, a significant portion of modern set theory deals with constructing models of set theory, using forcing. Yet, from a philosophical view, if sets were meant to formalize the general notion of a "collection", then a true/Platonic set theory should have no "models" at all (whether set sized, or class sized, since classes are collections too).

My question is thus: What are the philosophical positions of modern set theorists, on how to reconcile their subject with Russell's paradox?

In particular, I suppose that some set theorists are secretly type theorists, and they view set theory as really set theory restricted to a very large type. Does set theorists secretly view type theory as more fundamental to formalizing mathematics? Or are there other resolutions, that are common among the set theory elite, but not necessarily conveyed in standard texts on the subject?

Edited to add: It appears that people have misunderstood some basic parts of my question. So let me try to clarify.

First, I am familiar with the class/set distinction. I'm familiar with the fact that in some formalizations of set theory (such as ZFC) classes don't really exist (per se) but are shorthand for formulas in the formal language, which one could think of as referencing certain meta-collections. I'm also familiar with the fact that in some formalizations of set theory (like NBG) classes are treated as actual objects, similar to sets, but limited in certain ways.

Second, it was my impression that set theory was created as a formalization of the informal concept of "collection". Sets were originally meant to refer to arbitrary collections (where order and repetition are ignored). There was no set/class distinction.

Now, it could be that modern set theorists no longer think of set theory as formalizing arbitrary collections (which seems to be what people in the comments are suggesting). If that's the answer to my question, then it begs the follow-up question: What is it, exactly, that sets are formalizing then?

(Warning: When referring to Russell's paradox I mean in the philosophical sense of informal collections, and not the "ZFC+Comprehension is inconsistent" sense. Russell's paradox applies to more situations than formal set theory.)

Mathematicians (of the non-finitist, Platonic sort) like to think of the set of natural numbers as existing as a completed object. Russell's paradox tells us that arbitrary "collections" cannot be collected into a completed object (on pain of contradiction). And yet, it is common in set theory texts to see the class of all sets, $V$, used freely. We are told that $V$ is not a set, but still a(n informal) collection of some kind. Yet, we are never told why we can't collect $V$, or other classes, into bigger classes. (And, indeed, in some versions of set theory, you can!)

Moreover, a significant portion of modern set theory deals with constructing models of set theory, using forcing. Yet, from a philosophical view, if sets were meant to formalize the general notion of a "collection", then a true/Platonic set theory should have no "models" at all. Wouldn't the domain of the model be a collection?

My question is thus: What are the philosophical positions of modern set theorists, on how to reconcile their subject with the philosophical view of Russell's paradox and the notion that sets are supposed to formalize arbitrary collections?

In particular, I suppose that some set theorists are secretly type theorists, and they view set theory as really set theory restricted to a very large type. Does set theorists secretly view type theory as more fundamental to formalizing mathematics? Or are there other resolutions, that are common among the set theory elite, but not necessarily conveyed in standard texts on the subject?

Edited to add: It appears that people have misunderstood some basic parts of my question. So let me try to clarify.

First, I am familiar with the class/set distinction. I'm familiar with the fact that in some formalizations of set theory (such as ZFC) classes don't really exist (per se) but are shorthand for formulas in the formal language, which one could think of as referencing certain meta-collections. I'm also familiar with the fact that in some formalizations of set theory (like NBG) classes are treated as actual objects, similar to sets, but limited in certain ways.

Second, it was my impression that set theory was created as a formalization of the informal concept of "collection". Sets were originally meant to refer to arbitrary collections (where order and repetition are ignored). There was no set/class distinction.

Now, it could be that modern set theorists no longer think of set theory as formalizing arbitrary collections (which seems to be what people in the comments are suggesting). If that's the answer to my question, then it begs the follow-up question: What is it, exactly, that sets are formalizing then?

Mathematicians (of the non-finitist, Platonic sort) like to think of the set of natural numbers as existing as a completed object. Russell's paradox tells us that arbitrary "collections" cannot be collected into a completed object (on pain of contradiction). And yet, it is common in set theory texts to see the class of all sets, $V$, used freely.

Moreover, a significant portion of modern set theory deals with constructing models of set theory, using forcing. Yet, from a philosophical view, if sets were meant to formalize the general notion of a "collection", then a true/Platonic set theory should have no "models" at all (whether set sized, or class sized, since classes are collections too).

My question is thus: What are the philosophical positions of modern set theorists, on how to reconcile their subject with Russell's paradox?

In particular, I suppose that some set theorists are secretly type theorists, and they view set theory as really set theory restricted to a very large type. Does set theorists secretly view type theory as more fundamental to formalizing mathematics? Or are there other resolutions, that are common among the set theory elite, but not necessarily conveyed in standard texts on the subject?

Mathematicians (of the non-finitist, Platonic sort) like to think of the set of natural numbers as existing as a completed object. Russell's paradox tells us that arbitrary "collections" cannot be collected into a completed object (on pain of contradiction). And yet, it is common in set theory texts to see the class of all sets, $V$, used freely.

Moreover, a significant portion of modern set theory deals with constructing models of set theory, using forcing. Yet, from a philosophical view, if sets were meant to formalize the general notion of a "collection", then a true/Platonic set theory should have no "models" at all (whether set sized, or class sized, since classes are collections too).

My question is thus: What are the philosophical positions of modern set theorists, on how to reconcile their subject with Russell's paradox?

In particular, I suppose that some set theorists are secretly type theorists, and they view set theory as really set theory restricted to a very large type. Does set theorists secretly view type theory as more fundamental to formalizing mathematics? Or are there other resolutions, that are common among the set theory elite, but not necessarily conveyed in standard texts on the subject?

Edited to add: It appears that people have misunderstood some basic parts of my question. So let me try to clarify.

First, I am familiar with the class/set distinction. I'm familiar with the fact that in some formalizations of set theory (such as ZFC) classes don't really exist (per se) but are shorthand for formulas in the formal language, which one could think of as referencing certain meta-collections. I'm also familiar with the fact that in some formalizations of set theory (like NBG) classes are treated as actual objects, similar to sets, but limited in certain ways.

Second, it was my impression that set theory was created as a formalization of the informal concept of "collection". Sets were originally meant to refer to arbitrary collections (where order and repetition are ignored). There was no set/class distinction.

Now, it could be that modern set theorists no longer think of set theory as formalizing arbitrary collections (which seems to be what people in the comments are suggesting). If that's the answer to my question, then it begs the follow-up question: What is it, exactly, that sets are formalizing then?