2.B. Interpretation of classes

The relative sense of open quantifiers

As the set theoretical universe expands, the values of decidable statements (under the given axioms) must of course stay constant. However, any undecidable statement, as well as any other formula containing open quantifiers and given with fixed values of its free variables (let us call these "open formulas"), may change value from one universe to another.
But if any value of an open formula can only be "the current one" relative to the choice of this universe, the question whether it is constant or variable, that is how things go «elsewhere», is relative to a choice of multiverse. But the realistic view of set theory means to neither assume a fixed universe nor a fixed multiverse. Thus, the natural way for it is to abstain from giving any value to open formulas (beyond decidable statements with known proof or refutation). Indeed, tries to do better would be rather hopeless due to how messy the situation is. Let us review its diverse aspects.

In a multiverse of «all universes» (standard or not) fitting a given consistent axiomatic set theory, the situation was explained in Part 1 :

The realistic view refering to standard universes, leads to another messy situation, first by lack of formal definition for the ideal concept of standardness.
Trying to materialize the concept of "real indefiniteness" of statements as their variability in a standard multiverse M, the behavior of this M is determined by its union U, as M is the range of all small sub-universes of U; this allows to express relevant questions as statements in U.

For example, the variability in M of an existential statement (∃x, A(x) where A is a bounded formula) means there exists in M a universe U where it is false (∀xU, ¬A(x)), and another universe U' where it is true (∃xU', A(x)). But then it is also true in any standard universe containing both U and U', among which some other members of M, and U itself. Such a statement may be called «ultimately true».

But since some statements may vary between acceptable standard universes U, some of these may still vary between the U of acceptable standard multiverses. Indeed, U is just another universe which can be axiomatically described, but only incompletely, for both formal reasons (the incompleteness theorem) and realistic ones (there is no ultimate standard multiverse). Just more statements can be made decidable for U than for U by describing U with a stronger theory T than the theory T given to describe U. Such a stronger T can already be obtained by formalizing this description of U as the union of a quasi-standard multiverse whose members satisfy T (while the standardness of U escapes formalization).
Behind the undecidability of a statement (existential or other), multiple situations may occur. It might (I guess) be always true in one quasi-standard multiverse, and always false in another, while, depending on the statement, either of them (but not both) may be standard. It may vary or not vary between standard universes, but this variability, itself expressible as a statement in the union of a standard multiverse, may be itself undecidable, and eventually vary between standard multiverses.

The undecidability of some statements such as AC, usually reflects the variability, which may also happen among (ST')-standard universes, of the values of the open quantifiers used in these statements when re-written without ℘. In particular, a quantifier ∀x∈℘(E) appears as an open quantifier on the class of subsets of E, while other uses of ℘ either also involve this once translated, or cannot even be translated when the powerset axiom does not hold.

The indefiniteness of classes

Unlike objects which can be compared in formulas by the = symbol, the meta-relation of equality between classes is as indefinite as the open ∀ since both concepts are definable from each other : This indefiniteness can be understood by remembering that, depite being usable as ranges, classes are only meta-objects, basically given as predicates, i.e. synctactically (as a formula with parameters). Like with open quantifiers, this leaves us with both concepts of provable equality (or proven equality), and provable inequality, according to the status (provable or refutable) of this statement of "equality" of classes (∀x, A(x) ⇔ B(x)).

Each universe U interprets each class C as a meta-set of objects P = {xU | C(x)}, and sees it as a set when PU. This condition is expressible by set theory in U as a statement S(C), written equivalently as

P, ∀x, C(x) ⇔ xP
E, ∀x, C(x) ⇒ xE

The equivalence is because P can be defined from such an E by P = {xE | C(x)}.
Having two open quantifiers, it is "more indefinite" than the equality between classes.

Classes and sets in expanding universes

Aside this formulation S(C) in a fixed universe, let us analyze this distinction of sets among classes (and thus what makes other classes indefinite), from the perspective of a quasi-standard multiverse. There, this concept has 3 possible versions, ordered only approximately by implications which "often work" but have exceptions.

But the ultimate realistic meaning of set theory is to refer to the range of "all standard universes", which differs from the concept of standard multiverse in that this range is not a set but a class. So, the concept of union cannot apply to this range, and yet its impossible result would form the ideal interpretation of set theory. This ideal can be well approximated by interpretations of set theory by either fixed universes or quasi-standard multiverses (whose existence is ensured by the completeness theorem). Both perspectives, in terms of a fixed vs. a variable universe, alternatively transcend each other endlessly along its expansion.

This explains how any intended set theory can be formalized by mere first-order axioms: if any intended property of the universe was only expressible by a second-order statement, or if anyhow its expression involved external objects (regarding this universe as a set), then it could be re-expressed by moving the framework, as the stronger first-order axiom of existence of a sub-universe of this kind, and why not also endlessly many of them, forming a standard multiverse (stating that every object is contained in such a sub-universe).

Justifying the set generation principle

Let B be a quantifier defined by a bounded formula whose parameters take values in a sub-universe U of any other universe we might work in. Let E be the range of all values taken by the argument y of A when interpreting (By, A(y)). It is independent of A and included in U insofar as A is absent from any expression of y in the defining formula. Typical exceptions would be when B involves a formula of the form A(t({xF|A(x)})) for some term t, in which case defining E as the set of possible values of y would require to assume ℘ and the (ST")-standardness of U. Without these assumptions, such exceptional formulas need to be excluded from the definitions of B for the justifications to work. Anyway they are excluded by understanding, as explained in 2.A-2.B, the set-builder as the notation for the operator symbols from an infinite list, one for each formula using the language of set theory to which A does not belong. (The set-builder is needed to put a formula inside a term, unless the conditional operator is accepted as primitive, which is acceptable, but then, function definers using A by the conditional operator are excluded for the same reason)

Let C(x) defined as (By, y = x). By the hypothesis of the set generation principle for the B which was written Q*, we have a proof of (B ⇔ ∃C). This implies ¬(By, 0), and, coming by second-order universal introduction, remains valid in any universe where it is interpreted.

For any x, the value C(x) of B on the predicate (y ↦ (y = x)), can only differ (be true) from its (false) value on (y ↦ 0), if both predicates differ inside E, i.e. if x belongs to E :

C(x) ⇔ ((By, y = x) ⇎ (By, 0)) ⇒ (∃yE, y = x ⇎ 0) ⇔ xE

This inclusion of C in E shows it is essentially a set: For any class satisfying the condition of the set generation principle (being indirectly as usable as a set in the role of domain of quantifiers), is it also indirectly as usable as a set in the role of domains of functions (before using this principle) ? Namely, is there for each such class a fixed formalization (bounded formulas with limited complexity) playing the roles of the definer and the evaluator for functions having this class as domains ? The answer would be yes, but we shall not develop the justifications here.

Concrete examples

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 meta-object: how could God «exist», if He is a meta-object, 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)) ∧ {xF | B(x)}≈Ø,

the common reaction is: «Do you think that beauty is the only thing that matters ?», i.e.

What,(∀xF, C(x) ⇔ B(x)) ????

then «If you find a pretty girl but stupid or with bad character, what will you do ?». Formally : (∃xF, B(x) ⇏ C(x) !!!). And to conclude with a claim of pure goodness: «I am sure you will find !», that is (∃ plenty of xF, C(x)). Not forgetting the necessary condition to achieve this: «You must change your way of thinking».
... by the absence of God...
: F would have immediately turned into a set by the existence of anybody on Earth able to receive a message from God, as He would obviously have used this chance to make him email me the address my future wife (or the other way round).
... and of any substitute: a free, open and efficient online dating system, as would be included in my project trust-forum.net, could give the same result. But this requires finding programmers willing to implement it. But the class of programmers is not a set either, especially as the purpose of the project would conflict with the religious moral priority of saving God's job from competitors so as to preserve His salary of praise.

Another concrete example of a small proper class is given by Laureano Luna.
Set theory and Foundations of Mathematics
1. First foundations of mathematics
2. Set theory
2.1. Formalization of set theory
2.2. Set generation principle
2.3. Tuples
2.4. Uniqueness quantifiers
2.5. Families, Boolean operators on sets
2.6. Products, graphs and composition
2.7. The powerset
2.8. Injectivity and inversion
2.9. Binary relations ; order
2.10. Canonical bijections
2.11. Equivalence relations, partitions
2.12. Axiom of choice
2.A. Time in set theory
2.B. Interpretation of classes
2.C. Concepts of truth in mathematics
3. Algebra 3.1. Galois connection
4. Arithmetic - 5. Second-order foundations

Other languages:
FR : Interprétation des classes