Let us call standard multiverse any collection (range) of
standard universes, where any 2 of them are small subuniverses
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 U∈M 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.
Two universes will be called compatible if they can both be seen as subuniverses 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 nonstandard ; they may be both parts of a common larger universe, only by representing there at least one of them as nonstandard.
When the universe expands, the values of statements (firstorder
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 nonground 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».
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 ∀x∈U,
¬A(x) but ∃x∈U', A(x).
That is, A(x) only holds at some x outside
U. We can get a U' such that U∈U'
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.
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 
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 
The metaphor of the usual
time
The finite time between expressions 

1.6. Logical connectives 1.7. Classes in set theory 
The infinite time between
theories
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

