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 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 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.
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».
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
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, meta-objects||
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
|1.8. Binders in set theory||Time in set
Expansion of the set theoretical universe
Can a set contain itself ?
The relative sense of open quantifiers⇨Interpretation of classes
Classes in an expanding universe
| 1.10. Formalization
of set theory
1.11. Set generation principle
Justifying the set generation principleConcepts of truth in mathematics
Alternative logical frameworks