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ABBB. SET-THEORETICAL ANTINOMIES 1
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The most famous set-theoretical antinomy is the Russel's Antinomy
usually formulated as follows:
Set of all sets not being elements of themselves is element of itself
IFF is not element of itself.
Before discussing the Antinomy itself, let us have a look at the concept
"set of all sets", symbolized as s(t), where t refers to the time at
which s is invoked. Let t=p and t=n refer respectively to our epoch and
to Newtons lifetime. s(p) clearly contains lp, the set of laser
printers. What about s(n)? In other terms, does the set theory support
rigorous objective propositions concerning the future? The answer to
this question depends upon the involved foundation of the set theory.
If we assume it to be UH, the answer is clearly negative. Cognition
of the future is incompatible with UH. lp is conceivable for any of our
contemporary more or less intelligent teenager, while, in spite of all
Newton's genius, the very question about lp being conceivable for him is
a nonsense. lp simply does not exist in UH(n), while it exists in UH(p).
Within a set theory founded in UH, s(p) contains lp and s(n) does not
contain it.
Both, the humanist and the theologian would probably agree with us at
this point. However, the theologian would certainly go further and
argue, that existence should not be restricted to objects of human
knowledge, perception and conception. Something not existing for humans
may still exist as such, may exist for God. He would expect from us a
sincere attitude towards this claim, as, after all, our own arguments
changed the apparently paradoxical Theological Antinomy to a legitimate
postulate.
We would answer that, indeed, nothing seems to stand in the way of
asserting that s(n) contains lp, as long as this assertion is founded
not in UH but in UT. A set theory supporting such a s(n) would have
to be a transcendental, theology-like discipline.
At this point the humanist will doubtless raise an objection. For him
science in general and set theory in particular are by definition
founded in UH. We are inclined to accept this definition which excludes
theology from the domain of science. After what we have said about
the Theological Antinomy we can hardly be accused of having an
antitheological bias. However, as long as some Einstein of UT has not
created a theo-logical counterpart of the logical calculus we have to
consider UT as a non-rigorous, non-scientific foundation.
We shall consider henceforth science as by definition founded in UH.
ABBB. SET-THEORETICAL ANTINOMIES 2
Let us introduce a few symbols:
t an instant, the present relative to this instant,
t- the past relative to t,
t+ the future relative to t,
x a set (where x stands for any lower case character),
X a class (where X stands for any upper case character),
AND operator "and",
OR operator "or",
<- operator "is contained" (element in set),
<< operator "is included" (subset in set, subclass in class),
~ negation operator,
: instantiation operator,
[x] proposition type where x="MN": meaningful,
x="RT": rational,
x="RG": rigorous.
(RG necessarily implies RT).
(RT necessarily implies MN).
We shall use them within statements of the form:
[x] : p
which reads: the propostition p is an instance of the type x, or
the proposition p is of type x.
[RG] may be dropped, when clearly implied by the context.
An immediate consequence of founding the set theory in UH is that
any set s invoked at t may contain only elements existing at t.
Rigorous propositions may refer therefore not to s, but to its time
instances s(t). This statement may be called INSTANCE RULE
and written symbolically:
ABBB1. [RG] : e(t-) <- s(t)
(It may be rigorously asserted that a set contains elements
already existing at the assertion instant.)
ABBB2. [~MN] : e(~t-) <- s(t) (where ~t- means "not past".)
(It is meaningless to assert that a set contains elements not
yet existing at the assertion instant.)
Some elements e(t-) may be collected into s(t-) which is clearly a
subset of s(t), so that the Instance Rule implies the Subset Instance
Rule:
ABBB3. [RG] : s(t-) << s(t)
ABBB4. [~MN] : s(~t-) << s(t)
ABBB. SET-THEORETICAL ANTINOMIES 3
Intuitively the Instance Rule appeals to the common sense. The set g
of all Frenchmen (f) is determined only for a specific instant t and
each g(t) clearly differs from one instant to another. However, for
each t, "f(t-) <- g(t)" is rigorous. Indeed, all f(t-) exist at t and
while perhaps not all precisely known, are certainly accessible to
investigation. g(t) has a cognizable domain.
We have specified the Instance Rule for an arbitrary set s without
considering its extent. g, the set of all Frenchmen is finite. What
about infinite sets, such as z, the set of all sets?
By definition z contains itself, what we can write by virtue of
the Instance Rule (ABBB1): z(t-) <- z(t).
This statement is rigorous IFF the domain of z(t) and, consequently,
that of z(t-) is cognizable. The Foundation Rule restricts the domain
of cognition to the past. This condition is necessary and satisfied by
situating the element "z(t-)" in the past. It is, however, sufficient?
Is a collection of entities extending over an infinite domain
cognizable?
The answer is clearly negative. Indeed, any conceivable scrutiny of
an infinite domain would last for infinity, so that situating it
in the past is not sufficient to make it cognizable. One may object
that finite, but large and vaguely documented domains such as "all
Frenchmen" are too practically noncognizable. It is true enough,
but the lack of knowledge results here from practical problems, not from
essential impossibility. We may admit that the domain "all Frenchmen"
is vaguely known, or, at the limit, entirely unknown. These propositions
are still rigorous, as they are answers to the rigorous question:
'how precisely is the domain 'all Frenchmen' known'. In no way they
contest the essential cognizability of the domain.
On the contrary, the very question about knowing the domaine
of "all sets" is in UH meaningless and any attempt to answer it
is a nonsens.
At the first glance we seem to face an unsolvable dilemma. On the one
hand infinite domains appear to be incompatibe with the scientific
context and, on the other hand, eliminating them would mean throwing the
baby out with the bath water, as most, if not all non-trivial problems
of mathematics deal with infinity. However, a second glance alows a more
optimistic view. Indeed, we have established the illegitimacy of sets
extending over infinite domains, but not of the domains themselves.
While entities extending over an infinite domain are illegitimate,
the domain itself is perfectly acceptable as virtuality, as support of
entities extending over its finite parts. Such virtual domain may be
called "class". Rather than an entity, a class is a rule of creating
entities. It could therefore be called "function", or "predicate", which
would stress its constructive aspect. For reasons appearing later in
this study and concerning modeling we prefer the term "class" which
stresses the collecting aspect of the concept.
ABBB. SET-THEORETICAL ANTINOMIES 4
Let us compare some essential characteristics of set and class and
review the involved terminology.
A set is an existing entity. It may contain sets as elements. It may
be contained in a set or in a class as an element. It may be included
in a set or in a class as a subset.
A class is virtuality. It is a rule alowing to verify if some existing
entity is its element. It may be included in a set or in a class as
a subclass. It may not be contained in a set nor in a class as an
element.
We have formulated above (ABBB3) the Subset Instance Rule:
[RG] : s(t-) << s(t).
It has a time restriction due to the objectivity being restricted to
the past in the UH. This restriction applies to sets, which are
"existing objects", but not to classes, which are domains of potential
objects, indifferent to their actual existence. Consequently, a class
is invariable with respect to time and the Class Recursivity Rule
may be formulated:
ABBB5. [RG] : C << C.
A class is subclass of itself.
Let us now examine the Russel's Antinomy in the light of the Foundation
Rule and the Instance Rule. Its usual verbal formulation is:
Set of all sets not being elements of themselves is element of itself
IFF is not element of itself.
"Set of all sets ..." is illegitimate and has to be replaced by "class
of all classes ...". A class may not be element of a class. It may only
be a subclass of a class, i.e. be included in a class.
The Antinomy refers now to Class of all classes not included in
themselves. Let us call classes included in themselves "recursive".
The reformulated Antinomy becomes:
The class of all non-recursive classes is a contradiction.
However, by virtue of the Class Recursivity Rule (ABBB5): C << C
every class is recursive so that the term "non-recursive class"
is meaningless. Consequently, the Antinomy itself becomes meaningless
and vanishes.
ABBB. SET-THEORETICAL ANTINOMIES 5
The Foundation Rule has once more proved its efficacity in dealing
with antinomies. The concept of class derived from it is, however,
by no means new. It goes back to von Neumann's aggregates and functions
and to Bernay's distinction between sets and classes. It is to day
part of the axiomatic of all set theories. However, it appears in
those axiomatics arbitrarily, without other guide than subjective
intuition of exceptionally gifted thinkers and without other
justification than the consecutive disappearing of antinomies.
The Foundation Rule, on the contrary, endeavours to provide an objective
guide for establishing rigorous epistemological foundations of logical
systems.
Let us have a quick look at two other set-theoretical antinomies.
Cantor's Antinomy:
By virtue of Cantor's Theorem the set c(s) of all subsets of s
(the power set of s) has a greater cardinality than s.
Then: u being the set of all sets has the cardinality greater then
itself which is paradoxical, as u is the most inclusive of sets.
We see immediately that in the light of the foundation rule Cantor's
Antinomy refers to the meaningless concept "set of all sets" and,
consequently is meaningless itself. After replacing u with the class U
we may talk about its extent, but not about its cardinality and the
problem vanishes.
Burali-Forti's Antinomy:
Well-ordered set w of all ordinals has an ordinal greater than any
element of w, hence greater than any ordinal.
Again, the proposition is meaningless and when the illegitimate set w
is replaced with the class W, the problem disappears.