2.A. Time in set theory
The expansion of the set theoretical universe
Given two universes U⊂U', 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:
- The content of every set (every set coincides with the meta-set of the
same elements): not only ∈ is preserved, but every set in U is
interpreted by U' as included in U (this is not a direct
consequence: if not requiring the sub-universe to be a set nor a class,
a counter-example would be given by internal set theory).
Therefore the set
builder is also preserved (then translated in 1.11 into an
infinite list of operator symbols in first-order logic).
- The
function evaluator and the domain functor (1.4). Therefore
the function
definer is also preserved (translated in 2.1 into an infinite
list of operator symbols).
- Tuples (2.3.)
- The powerset (2.7).
- Thus also, any other structure defined from the above by expressions and
bounded formulas, such
as finiteness (definable
from the power set): there actually exist non-standard universes with a
different interpretation of finiteness, thus having as «set of integers» a non-standard
model of arithmetic.
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 (U∈U').
If we have 3 universes U ⊂ U' ⊂ 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 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.
We can understand the intended sense of set theory as of a different
kind than that of generic theories, in this way :
- Interpretations of a generic theory require to fix a model, in
which its variables and expressions may take values;
- Set theory aims to locally give to its expressions with fixed values
of the free variables, their standard interpretation, independently of
a standard universe containing them, which may keep expanding.
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 E∈U
and a formula A such that {x∈E|∀_{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) ∧ x
∉ x) 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):
- Either reduce them to urelements
- Or eliminate them too (and so on for other sets that contained
the latter).
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
- a universe with an urelement x and a set y
such that x ∈ y ∧ y ∉ y,
- and a universe with a set x and an urelement y
such that x ∈ x ∧ y ∉ x ?
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 ∀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.
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 :
- A universe U where it is true might not contain any of those
where it is false as a sub-universe : {x∈U|
¬A(x)} may not have any meta-subset behaving as a
good universe;
- A universe where it is false might not be a small sub-universe
of any another one where it is true.
- It might be only true in non-standard universes.
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.
Other languages:
FR : Temps en
théorie des ensembles