Each universe U interprets each class C as a metaset of objects P={x∈U C(x)}, and sees it as a set when P ∈ U. This condition, that C has the same elements in U as an object (set) P also in U, is expressed by set theory in U as
∃P,∀x, C(x) ⇔ x ∈ P
or equivalently∃E,∀x, C(x) ⇒ x ∈ E
since from such an E we can restore P as P={x∈E C(x)}.From the perspective of an expanding universe, a class C (given as a formula with parameters), «is a set» (equal to P) if the part P = {x∈U C(x)} that this formula defines in each U (formally depending on U), turns out to remain constant (the same set) during the expansion of U. More precisely, it is known as a set (proven equal to P) if we could prove this independence, i.e. refute the possibility for any object x outside the current universe (but existing in a larger universe), to ever satisfy C(x). On the other hand, a class C is not considered a set, if it remains eventually able to contain «unknown» or «not yet existing» objects (in another universe), that would belong to some future value of P, making P vary during the growth of U.
In a given expansion of U, the interpretation of the formal condition for C to be a set (∃E,∀x, C(x) ⇒ x ∈ E) in the union U of these U, means that in this growth, «there is a time after which P will stay constant». Compared to our last criterion to distinguish sets among classes in an expanding universe (the constancy of P), this one ignores any past variations, to focus on the latest ones (those between the largest universes, with size approaching that of the U where it is interpreted).A set: Is there any dodo left on Mauritius ? As this island is well known and regularly visited since their supposed disappearance, no surviving dodos could still have gone unnoticed, wherever they may hide. Having not found any, we can conclude there are none. This question, expressed by a bounded quantifier, has an effective sense and an observable answer.
A set resembling a class: Bertrand Russell raised this argument about theology: «If I were to suggest that between the Earth and Mars there is a china teapot revolving about the sun..., nobody would be able to disprove my assertion [as] the teapot is too small to be revealed even by our most powerful telescopes. But if I were to go on to say that, since my assertion cannot be disproved, it is intolerable presumption on the part of human reason to doubt it, I should rightly be thought to be talking nonsense.» The question is clear, but on a too large space, making the answer practically inaccessible. (An 8m telescope has a resolution power of 0.1 arcsec, that is 200m on the moon surface)
A class: the extended statement, «there is a teapot orbiting some star in the universe» loses all meaning: not only the size of the universe is unknown, but Relativity theory sees the remote events from which we did not receive light yet, as not having really happened for us yet either.
A metaobject: how could God «exist», if He is a metaobject, while «existence» can only qualify objects? Did apologists properly conceive their own thesis on God's «existence» ? But what are the objects of their faith and their worship ? Each monotheism rightly accuses each other of only worshiping objects (sin of idolatry): books, stories, beliefs, teachings, ideas, attitudes, feelings, places, events, miracles, healings, errors, sufferings, diseases, accidents, natural disasters (declared God's Will), hardly more subtle than old statues, not seriously checking (by fear of God) any hints of their supposed divinity.
A universal event: the redemptive sacrifice of the Son of God. Whether it would have been theologically equivalent for it to have taken place not on Earth but in another galaxy or in God's plans for the Earth in year 3,456, remains unclear.
Another set reduced to a class... The class F of girls remains incompletely represented by sets: the set of those present at that place and day, those using this dating site and whose parameters meet such and such criteria, etc. Consider the predicates B of beauty in my taste, and C of suitability of a relationship with me. When I try to explain that «I can hardly find any pretty girl in my taste (and they are often unavailable anyway)», i.e.
(∀_{F} x, C(x)
⇒ B(x)) ∧ {x∈ F  B(x)}≈Ø,
What,(∀x ∈ F, C(x) ⇔ B(x)) ????
then «If you find a pretty girl but stupid or with bad character, what will you do ?». Formally : (∃x ∈ F, B(x) ⇏ C(x) !!!). And to conclude with a claim of pure goodness: «I am sure you will find !», ( « ∃ ∃ x ∈ F, C(x)). Not forgetting the necessary condition to achieve this: «You must change your way of thinking».C(x) ⇔ ((Q*y, y = x) ⇎ (Q*y,0)) ⇒ (∃y∈E, y=x ⇎ 0) ⇔ x ∈ E
thus C is a set. ∎Set theory and
Foundations of mathematics 

1. First
foundations of mathematics 
Philosophical
aspects 
1.1. Introduction
to the foundations of mathematics 1.2. Variables, sets, functions and operations 
Intuitive representation and
abstraction
Platonism vs Formalism 
1.3. Form of theories: notions, objects, metaobjects 
Realistic vs. axiomatic
theories

1.4. Structures of mathematical systems 1.5. Expressions and definable structures 
Time
in model theory
The time of interpretation
The metaphor of the usual time The infinite time between models 
1.6. Logical connectives 1.7. Classes in set theory 
Truth
undefinability
The Berry paradox
Zeno's Paradox 
1.8. Binders in set theory  Time
in Set theory ⇦
Expansion of the set
theoretical universe
Can a set contain itself ? 
1.9. Quantifiers 
The relative sense of open
quantifiers
Interpretation of classesClasses in an expanding
universe
Concrete examples 
1.10. Formalization
of set theory 1.11. Set generation principle 
Justifying the set
generation principle
⇨ Concepts
of truth in mathematics Alternative logical
frameworks
