Interpretation of classes

Classes in an expanding universe

Unlike objects which can be compared by an equality symbol (that can be used in formulas), the meta-relation of equality between classes is as indefinite as the open ∀ since both concepts are definable from each other: Like with open quantifiers, this indefiniteness leaves us with both concepts of provable equality (or proven equality), and provable inequality, according to whether the statement of this equality (∀x, A(x) ⇔ B(x)) is provable or refutable.

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, 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) ⇔ xP

or equivalently

E,∀x, C(x) ⇒ xE

since from such an E we can restore P as P={xE |C(x)}.
Otherwise (if PU), this P is arising to existence with U and will exist as a set in future universes (those which see U as a set).

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 = {xU |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) ⇒ xE) 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).
But the ideally intended standard multiverse, that is the range of U, would have to be itself a class instead of a set. Thus, both perspectives (a constant vs. a variable universe) alternatively encompass each other, endlessly along the expansion. Meanwhile, a class defined by some special formula may alternatively gain and lose the status of set ; but if in some growth range, «P perpetually alternates between variability and constancy», then it would ultimately not be constant there, thus C would not be a set. So the alternation of its status would end... if we stopped checking it at wrong places. But how ?

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, 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.

Justifying the set generation principle

Let Q* be the quantifier defined by a given bounded formula using an extra unary predicate symbol.
Let E be any fixed universe (seen as a set in a larger universe) containing the values of all parameters, so that the formula can be interpreted there. It must contain all values ever taken by the argument y of A(y) when interpreting Q*y, A(y). (This range may depend on implicit parameters of Q*y but does not depend on A. It must be a set because this formula only has definite, fixed means (variables bound to given sets, fixed parameters) to provide these values).

Assume that ¬(Q*y,0), and let C(x) defined as (Q*y, y = x). The hypothesis of the set generation principle means that we have a proof of (Q* ⇔ ∃C) which, coming by second-order universal introduction, remains valid in any universe where it is interpreted. For any x, the value C(x) of Q* 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) ⇔ ((Q*y, y = x) ⇎ (Q*y,0)) ⇒ (∃yE, y=x ⇎ 0) ⇔ xE

thus C is a set. ∎
For classes satisfying the condition of the set generation principle (being indirectly as usable as sets in the role of domains of quantifiers), are they also indirectly as usable as sets 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 definers and evaluators for functions having these classes as domains ? The answer would be yes, but we shall not develop the justifications here.

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, families
2.4. Boolean operators on families of sets
2.5. Products, graphs and composition
2.6. Uniqueness quantifiers, functional graphs
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
2.13. Galois connection
Time in set theory
Interpretation of classes
Concepts of truth in mathematics
3. Algebra - 4. Arithmetic - 5. Second-order foundations

Other languages:
FR : Interprétation des classes