2.A. Time in set theory

The expansion of the set theoretical universe

Given two universes UU', the universe U will be called standard in U', or a sub-universe of U', if its interpretation of set theoretical structures (values they give for all values of arguments inside U) coincides with their meta-interpretation (that of U'). Precisely, let us require the preservation of the following data: Thus, as structures in U' are fixed, U only needs to be specified as a meta-subset or class in U'. Let us call it a small sub-universe if it is more precisely a set (UU').
If we have 3 universes UU'U" where U' is standard in U", then we have the equivalence:
(U is standard in U') ⇔ (U is standard in U").
Thus the idea to consider the standardness of a universe as an absolute property, independent of the external universe in which it is described... provided that this external universe is itself standard. This does not formally define standardness as an absolute concept (which is impossible), but suggests that such a concept may make ideal sense.

Let us call standard multiverse any collection (range) of standard universes, where any 2 of them are small sub-universes of a third one. Let us say that the set theoretical universe expands when it ranges over a standard multiverse.
From any standard multiverse M, we can rebuild an external universe containing all its universes, defined as their union U=⋃M, where they are all standard. Indeed, any expression with free variables in U takes its meaning from at least one UM containing the values of all these variables, and thus where the expression can be interpreted. This U is still another specific standard universe, but it cannot belong itself to M, as its presence there would contradict the concept of multiverse which does not admit any biggest element. So, no single standard multiverse can ever contain all possible standard universes.

We can understand the intended sense of set theory as of a different kind than that of generic theories, in this way : Unlike standard universes, not all non-standard universes can be small sub-universes of some larger universe (itself non-standard), as extensions may be unable to preserve the power set operators. We may also have a multiverse of non-standard universes looking like a standard multiverse (its members are small sub-universes of each other), but whose union U (with the same structures, letting these universes appear standard), no more behaves as a good universe, as it no more satisfies the axiom schema of specification over formulas with open quantifiers (which is a necessary quality for a universe to be a small sub-universe of another one). Namely, there may be a set EU and a formula A such that
{xE|∀U y, A(x,y)}∉U.

Two universes will be called compatible if they can both be seen as sub-universes of a common larger universe. All standard universes are compatible with each other. So when 2 universes are incompatible, at least one of them is non-standard ; they may be both parts of a common larger universe, only by representing there at least one of them as non-standard.

Can a set contain itself ?

Let us call a set reflexive if it contains itself.
By the proof of Russell's paradox [1.8], the class (Set(x) ∧ xx) of non-reflexive sets, cannot coincide with a subset F of any set E, since this F would then be a non-reflexive set outside E, giving a contradiction. But can reflexive sets exist ? this is undecidable; here is why.

From a universe containing reflexive sets, we can just remove them all: these sets cannot be rebuilt from the data of their elements (since each one has at least an element removed from the universe, namely itself), but for the rest to still constitute a universe (model of set theory), we need to manage the case of all other sets that contained one (and similarly with functions): Another way is to progressively rebuild the universe while avoiding them: each set appears at some time, formed as a collection of previously accepted or formed objects. Any set formed this way, must have had a first coming time: it could not be available yet when it first came as a collection of already available objects, thus it cannot be reflexive.
As the reflexivity of a set is independent of context, a union of universes each devoid of reflexive sets, cannot contain one either.

On the other hand, universes with reflexive sets can be created as follows:

Riddle. What is the difference between
Answer: the role of the set containing x but not y, played by y in the former universe, is played by x in the latter.

The absence of reflexive sets, is a special case of the axiom of foundation (or regularity), to which the above arguments of undecidability, will naturally extend. Its formulation will be based on the concept of well founded relation. But this axiom is just as useless as the sets it excludes.

The relative sense of open quantifiers

When the universe expands, the values of statements (first-order formulas, admitting open quantifiers) may change.
Of course, if a statement is formally provable from given axioms then it remains true in all universes satisfying these axioms; similarly if it is refutable (i.e. its negation is provable, so that it is false in all universes). But set theory does not give sense (a Boolean value) to undecidable ground statements, and similar non-ground statements (with open quantifiers and given values of free variables), as any given value would be relative to how things go «here and now» : if a universal statement (∀x, A(x) for a bounded formula A) is true «here», it might still become false (an x where A is false might be found) «elsewhere».

But if the value of an indefinite statement is relative to how things go «here», then the actual variability of this value between places (to motivate its indefiniteness status) remains relative to how things turn out to go «elsewhere». Namely, it is relative to a given range of possible coexisting «places» (universes) where the statement may be interpreted, that is a multiverse. But to coexist, these universes need the framework of a common larger universe U containing them all. In fact, the mere data of U suffices to essentially define a multiverse as that of «all universes contained in U». Or rather, 2 multiverses, depending on whether we admit all universes it contains, or only the standard ones.

A standard multiverse, as defined above. There, the variability of an existential statement (∃x, A(x)) for a bounded formula A, means the existence of universes U, U'U such that ∀xU, ¬A(x) but ∃xU', A(x). That is, A(x) only holds at some x outside U. We can get a U' such that UU' by taking any universe containing both U and the old U'. In particular, (∃x, A(x)) is also true in U (we may call this statement «ultimately true»). Intuitively, the x where A is true are out of reach of the theory : they cannot be formally expressed by terms, and their existence cannot be deduced from the given existence axioms (satisfied by U).
But since (∃x, A(x)) was not definitely true for the initially considered universe U actually unknown and expanding, its chances may be poor to become definitely true for U which is just another axiomatically described universe, that is unknown as well. So, when a statement aimed for U is indefinite, it may be varying when U expands, but it may also be that the very question whether it indeed varies (that can be translated as a question on U), remains itself an indefinite question as well. Just more truths can be determined for U than for U by giving more axioms to describe U than we gave for U.
The incompleteness theorem will imply that a formalization of this description of U (as the union of a standard multiverse, whose universes satisfy given axioms) is already such a stronger axiomatization, but also that neither this nor any other axiomatic theory trying to describe U (as some kind of ultimate standard universe), can ever decide (prove or refute) all ground statements in U; in particular, the question of the variability of a ground statement in the expanding U cannot be always decided either.

A multiverse of «all universes» no matter if they are standard or not. The completeness theorem will show that for any generic theory, the interpretation of the indefiniteness of a ground formula as variability of its boolean value in this multiverse, coincides with their formal undecidability. Strange things may happen there for an undecidable (∃x,A(x)), as the universes where it is true and those where it is false may be incompatible : Intuitively, different possible universes with different properties do not necessarily "follow each other" in time, but can belong to separate and incompatible growth paths, some of which may be considered more realistic than others.

So, while the formal undecidability of a ground statement makes it automatically variable in any "multiverse of all universes", this still does not say how it goes for standard multiverses. In conclusion, the indefiniteness of statements should only be treated by avoidance, as a mere expression of ignorance towards the range of acceptable universes, partially selected by axioms, where they may be interpreted.

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 : Temps en théorie des ensembles