2.B. Interpretation of classes
The relative sense of open quantifiers
As the set theoretical universe expands,
the values of decidable statements
(under the given axioms) must of course stay
constant. However, any undecidable statement, as well as any other formula containing
open quantifiers and given with fixed values of its free
variables (let us call these "open formulas"), may change value from one universe to another.
But if any value of an open formula can only be "the current one" relative to the choice of
this universe, the question whether it is constant or variable, that is how
things go «elsewhere», is relative to a choice of multiverse. But the realistic view of set theory means to
neither assume a fixed universe nor a fixed multiverse. Thus, the natural way for it is to
abstain from giving any value to open formulas (beyond decidable statements with known
proof or refutation). Indeed, tries to do better would be rather hopeless due to how messy
the situation is. Let us review its diverse aspects.
In a multiverse of «all universes» (standard or not) fitting a given
consistent axiomatic set theory, the situation was explained in Part 1 :
- The variability of
value of a statement there, is equivalent to its undecidability (completeness theorem);
- However, this undecidability, when true, cannot anyway be proven by the
same axiomatic set theory (incompleteness theorem).
The realistic view refering to standard universes, leads to another messy
situation, first by lack of formal definition for the ideal concept of standardness.
Trying to materialize the concept of "real indefiniteness" of statements as their variability
in a standard multiverse M, the behavior of this M is determined
by its union U, as M is the range of all small sub-universes of U;
this allows to express relevant questions as statements in U.
For example, the variability in M of an existential
statement (∃x, A(x) where A is a bounded formula)
means there exists in M a universe U where it is false (∀x∈U,
¬A(x)), and another universe U' where it is true (∃x∈U',
A(x)). But then it is also true in any standard universe containing both U and
U', among which some other members of M, and U itself. Such a
statement may be called «ultimately true».
But since some statements may vary between acceptable standard universes
U, some of these
may still vary between the U of acceptable standard multiverses.
Indeed, U is just another universe which can be axiomatically described,
but only incompletely, for both formal reasons (the incompleteness theorem) and
realistic ones (there is no ultimate standard multiverse).
Just more statements can be made decidable for U than for U
by describing U with a stronger theory T than the theory T
given to describe U. Such a stronger T can already be obtained by
formalizing this description of U as the union of a quasi-standard multiverse
whose members satisfy T (while the standardness of U escapes formalization).
Behind the undecidability of a statement (existential or other), multiple situations
may occur. It might (I guess) be always true in one quasi-standard multiverse, and
always false in another, while, depending on the statement, either of them
(but not both) may be standard. It may vary or not vary between standard universes,
but this variability, itself expressible as a statement in the union of a standard
multiverse, may be itself undecidable, and eventually vary between standard multiverses.
The undecidability of some statements such as AC, usually reflects the variability,
which may also happen among (ST')-standard universes, of the values of the open
quantifiers used in these statements when
re-written
without ℘. In particular, a quantifier ∀x∈℘(E) appears
as an open quantifier on the class of subsets of
E, while other uses of ℘ either also involve this once translated, or cannot even
be translated when the powerset axiom does not hold.
The indefiniteness of classes
Unlike objects which can be compared in formulas by the = symbol,
the meta-relation of equality between classes is as
indefinite as the open ∀ since both concepts are definable from each other :
- The equality of classes A = B would be
defined by ∀x, A(x) ⇔ B(x);
- The statement (∀x, A(x))
means A = 1 (the universe, class of all objects).
This indefiniteness can be understood by remembering that, depite being
usable as ranges, classes are only meta-objects, basically given as predicates,
i.e. synctactically (as a formula with parameters).
Like with open quantifiers, this leaves us with both concepts
of provable equality (or proven equality), and provable
inequality, according to the status (provable or refutable) of this
statement of "equality" of classes
(∀x, A(x) ⇔ B(x)).
Each universe U interprets each class C as a meta-set of
objects P = {x∈U | C(x)}, and sees it
as a set when P ∈ U. This condition is expressible by set theory in
U as a statement S(C), written equivalently as
∃P, ∀x, C(x) ⇔ x ∈ P
∃E, ∀x, C(x) ⇒ x ∈ E
The equivalence is because
P can be defined from such an E by P = {x∈E | C(x)}.
Having two open quantifiers, it is "more indefinite" than the equality between classes.
Classes and sets in expanding universes
Aside this formulation S(C) in a fixed universe, let us analyze this
distinction of sets among classes (and thus what makes other classes indefinite),
from the perspective of a quasi-standard multiverse.
There, this concept has 3 possible versions, ordered only approximately by implications which
"often work" but have exceptions.
- The "weak" version is to claim S(C) true in all members of this multiverse.
That means C always coincides with some set P, but this may be a different
P from one universe to another.
- The "strong" version requires P to stay constant. This can be seen as the
meaning behind the formal choice of naming P by a symbol of set theory, as
was discussed with the powerset (2.7). But just the constancy of P =
{x∈U | C(x)} when U varies, needs not imply the above
weak version, as there can exist some universe U (among the smallest of this
multiverse) such that P ∉ U. This means P is arising to existence
with U, and will anyway exist as a set in all the "future" (larger) universes which see
U as a set. So, a class C is not considered a set in this "strong"
sense, if it remains able to contain «unknown» or «not yet existing» objects (in some future
universe), that would only belong to some larger future value of P, which can thus vary
with the growth of U.
- The middle version (implied by the strong one) is to interpret S(C) in the
union U of the multiverse. Deciphered there, it says that, while U expands,
«there exists a time from which P stays constant» (and thus after which it will exist
as a set). So, this just differs
from the strong version by ignoring any "past" variations (which could occur during some
"beginning" of the expansion), to focus on the ultimate behavior (among the latest, i.e. largest
universes, somehow "closest" to the U where S(C) is interpreted).
But the ultimate realistic meaning of set theory is to refer to the range of "all standard
universes", which differs from the concept of standard multiverse in that this range is not a set
but a class. So, the concept of union cannot apply to this range, and yet its impossible result
would form the ideal interpretation of set theory. This ideal can be well approximated by
interpretations of set theory by either fixed universes or quasi-standard multiverses
(whose existence is ensured by the completeness theorem). Both perspectives, in terms of a
fixed vs. a variable universe, alternatively transcend each other endlessly along its expansion.
This explains how any intended set theory can be formalized by mere first-order axioms:
if any intended property of the universe was only expressible by a second-order statement,
or if anyhow its expression involved
external objects (regarding this universe as a set), then it could be re-expressed
by moving the framework, as the stronger first-order axiom of existence of
a sub-universe of this kind, and why not also endlessly many of them, forming a
standard multiverse (stating that every object is
contained in such a sub-universe).
Let B be a quantifier defined by a bounded formula whose parameters take values
in a sub-universe U of any other universe we might work in.
Let E be the range of all values taken by the argument y of A when
interpreting (By, A(y)). It is independent of A and
included in U insofar as A is absent from any expression of y in
the defining formula. Typical exceptions would be when B involves a
formula of the form A(t({x∈F|A(x)}))
for some term t, in which case defining E as the set of possible
values of y would require to assume ℘ and the (ST")-standardness of U.
Without these assumptions, such exceptional formulas need to be excluded from the definitions
of B for the justifications to work. Anyway they are excluded by understanding,
as explained in 2.A-2.B, the set-builder
as the notation for the operator symbols from
an infinite list, one for each formula using the language of set theory to which A
does not belong. (The set-builder is needed to put a formula inside a term, unless the
conditional operator is accepted as primitive, which is acceptable, but then, function
definers using A by the conditional operator are excluded for the same reason)
Let C(x)
defined as (By, y = x).
By the hypothesis of the set generation principle for the B which was written Q*,
we have a proof of (B ⇔ ∃C). This implies ¬(By, 0),
and, coming by second-order universal introduction, remains valid in any
universe where it is interpreted.
For any x, the value C(x)
of B 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) ⇔
((By, y = x) ⇎ (By, 0))
⇒ (∃y∈E, y = x ⇎ 0) ⇔ x
∈ E
This inclusion of C in E shows it is essentially a set:
- C is a set in the above "strong" sense (C ⊂ U) ;
- Moreover E is a set in the
sense of first-order logic (1.D): the formula of B
only has definite, fixed means (variables bound to given sets,
fixed parameters) to provide its elements. ∎
For any class satisfying the condition of the set generation principle
(being indirectly as usable as a set in the role of domain of
quantifiers), is it also indirectly as usable as a set 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 the definer and
the evaluator for functions having this class as domains ? The
answer would be yes, but we shall not develop the justifications here.
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)) ∧ {x∈ F | B(x)}≈Ø,
the common reaction is: «Do you think that beauty is the only thing
that matters ?», i.e.
What,(∀x ∈ F, C(x)
⇔ B(x)) ????
then «If you find a pretty girl but stupid or with bad character,
what will you do ?». Formally : (∃x ∈ F, B(x)
⇏ C(x) !!!). And to conclude with a claim of pure
goodness: «I am sure you will find !», that is (∃ plenty of x ∈ F,
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 trust-forum.net, 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.
Another concrete example of a small proper class is given by Laureano Luna.
Other languages:
FR : Interprétation des classes