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SET THEORY
DEFINITIONS
Fraenkel defines Set Theory as "applied first order Predicate
Calculus" enlarged with the one specific primitive symbol,
the diadic "membership" predicate/operator "∈".
We shall read x∈y as "x is member of (set)y" or "(set)y
contains (member)x".
Set Theory being thus founded in Predicate Logic we could
- in the light of the previous chapter - dismiss it straight
off as another noumenal phantasm.
One cannot, however, exclude a priori the possibility of
some out-of-the-way peculiarities of the membership operator
doctoring the Set Theory and somehow making it rational.
We shall, therefore, examine the implications of the membership
operator, starting by asking what are the involved "set" and
"member" concepts.
Strangely enough, everybody seems to steer clear of defining
"set". Fraenkel and Bar-Hillel deal mainly with antinomies
and with innumerable axiomatic systems trying to avoid them
in different ways, but don't define "set". The closest to a
definition is Levy's following non-definition:
**By set we mean a completely structure-free set, and
therefore a set is determined solely by its members**
So, a set is a set. It's good to know, but it does not tell
us what it is, nor what is "member", nor what it means
"to have members".
The way out of the deadlock consists, as usually, in replacing
the definition with a symbolically expressed axiom. And so we
arrive at the first axiom of the set theory:
Axiom of Existentiality (Frege 1893)
∀x(x∈y←→x∈z)→y=z
(if y and z have the same members they are equal).
Next Levy's quote:
**The existence of sets: Now we face the question of finding
or constructing sets. We want any collection whatsoever of
objects, i.e. sets, to be a set. This is not a precise idea
and therefore we cannot translate it into our language.
We must therefore be satisfied with a somewhat weaker
stipulation. We shall require that every collection of sets
"specifiable" in our language is a set; i.e., for every
statement of our language this collection of all objects
which satisfy it is a set. We shall by no means assume that
it is necessarily true that all sets are specifiable;
moreover by introducing the Axiom of Choice we shall require
the existence of sets which are not necessarily specifiable.
The requirement that all specifiable collections are indeed
sets is the following one.
Axiom of Comprehension (Frege 1893)
∃y∀x(x∈y←→F(x))
(x is member of y IFF it satisfies the formula F(x))
No limitation specified on the formula F(x), an instance of
the axiom may be chosen taking
F(x)=x∉x
(x is not member of itself), leading to the contradiction
known as the Russell's Paradox:
Theorem: Russell's Paradox
¬∃y∀x(x∈y←→x∉x)
It's a theorem of predicate logic, since we don't use in the
proof any axioms of set theory.
Proof:
Suppose y is a set such that
∀x(x∈y←→x∉x)
Then, since what holds for every x, holds in particular for y
we have:
y∈y←→y∉y
which is a contradiction
(y is member of itself IFF y is not member of itself).
Let's resume: The very beginning of the set theory is
completely muddled. "Set" is "defined" as "structure-free
set" determined by its "members(?)". "We want any collection
whatsoever of objects, i.e. sets, to be a set". What are
"objects", what are their "collections"? Any "collection(?)"
of "sets(?)" is a "set(?)", but what, for heaven's sake,
is "set"? One takes with a grain of salt the requirement of
"sets" being "specifiable" in "our language" (Predicate
Logic plus membership operator). The foundation of mathematics
and science depending on possible arbitrary changes of "our
language" sounds rather thin.
And the axiom implementing this requirement leads right away
to an antinomy proving that something is rotten at the very
base of the Set Theory and of its founding Predicate Logic.
From this initial moment the Set Theory becomes rat hunting
for antinomies, just as its base, the Predicate Logic became
rat hunting for "reflexive fallacies". Fraenkel, prudently
dodging the definitions starts his elucidation of the Set
Theory by detecting and hunting rats such as antinomies of:
Russell, Cantor, Burali-Forti, Richard, Grelling and, of
course, the Liar, prudently dressed up in new but equally
fictitious clothes, leaving him naked as before.
FAKED AXIOMS
Antinomies are hunted by means of dogmatic decrees usurping
the name "axioms" in spite of lacking the principal feature
thereof, to wit the falsifiability. They simply embody the
Wishful Thinking Principle: -When X disturbs us, we issue
an "axiom" boiling down to "X is verboten"- usually dressed
up in impressing verbiage and followed by complex circular
procedure called "proof".
For instance, Zermelo disposes of the Russell paradox by
means of the "Axiom of Separation" and a theorem:
Axiom of Separation:"Whenever the propositional function -(x)
is definite for all elements of a set M, M possesses a subset
M' containing as elements precisely those elements x of M for
which -(x) is true".
The Theorem: "Every set M possesses at least one subset M0
that is not an element of M". Let M0 be the subset of M for
which, by Axiom of Separation, is separated out by the notion
"x is not member of x". Then M0 cannot be in M:
Now, that clearly boils down to the Wishful Thinking decree:
"Russell paradox is verboten". The following tedious "proof"
is a superfluous, circular tautology:
**
Proof
M0 cannot be in M:
1. If M0 is in M0, then M0 contains an element x for which
x is in x (i.e. M0 itself), which would contradict the
definition of M0.
2. If M0 is not in M0, M0 is an element of M that satisfies
the definition "x is not member of x", and so is in M0, which
is a contradiction.
So M0 cannot be in M, hence not all objects of the universal
domain B can be elements of one and the same set. "This
disposes of the Russell antinomy as far as we are concerned".
**
However, not all of us seem convinced. One tends to find the
Wishful Thinking procedure void of sense and "the domain B"
coming out of the blue mysterious and unrelated to the issue.
One may therefore prefer the bold and simple procedure
inspired by cutting the Gordian knot and consider Russell's
paradox as the proof that Russell's collection is not a set!
Using this approach we must legalize some collections of
sets that are not sets, or, as says Levy, some collections
of objects not being objects themselves. In order to denote
with a general term all set- and non-set collections one
introduces the term "Class", postulating that all specifiable
collections are classes, which may or may not be sets.
We still don't know what are "collections" and "objects"
and get an additional puzzle - which decree commands the
"collections of objects" to be or not to be "objects"
themselves.
Most axioms added hereafter to the ST seem to embody the
Wishful Thinking Principle and to fall into three principal
types:
1.Axioms banning culprits of antinomies from the community
of sets, e.g. the dealing with Russell's antinomy described
above.
2.Hijacking of mathematical constructs and principles and
presenting them post factum as founded in the Set Theory.
We may mention here neighborhood and limit, which worked
rigorously and efficiently for years, but suddenly got
contested unless founded in the axiom of choice.
3.Dealing with infinity and continuity. It intersects with
the preceding type, because most non-trivial mathematical
problems deal with infinity and continuum. However, there
is a particular set-theoretical problem of constructing
continuum from discrete collections, which deals with
transfinite numbers defined by Cantor as:
**cardinal numbers or ordinal numbers that are larger than
all finite numbers, yet not necessarily absolutely infinite.**
Calling A0 and A1 respectively the transfinite cardinals of
sets of natural and real numbers, Cantor proved that A1 is
greater than A0 and enonced the Continuum Hypothesis stating
that there is no intermediate cardinal number between A0
and A1. Wishful Thinking elevated the hypothesis to an axiom
and transfinity was saved, having nevertheless induced shock
waves of unusual magnitude.
Poincare condemned the theory of transfinite numbers as a
"disease" from which he was certain mathematics would someday
be cured and Kronecker even attacked Cantor personally,
calling him a "scientific charlatan", a "renegade" and
a "corrupter of youth".
One cannot help feeling uneasy in face of transfinity and
of the awkward dealing with "collections" in the stage of
opening definitions of the set theory. It seems to be due
to the current foundational crisis of mathematics, the
third one, according to Abraham A. Fraenkel and Yehoshua
Bar-Hillel, as quoted in the following paragraph.
THE THREE CRISES
The twentieth century is not the first period in which
mathematics underwent a foundational crisis. It might add to
the perspective in which contemporary antinomies should be
looked upon if prior crises are, if only briefly, sketched.
In the fifth century B.C., only a short time after mankind
obtained one of the most brilliant achievements in its
history, viz. the development of geometry as a rigorous
deductive science, two discoveries were made that were
extremely paradoxical: The first was that not all geometrical
entities of the same kind are commensurable with each other,
so that, for instance, the diagonal of a given square could
not be measured by an aliquot part of its side (in modern
terms, that the square root of 2 is not a rational number);
the other were the paradoxes of the Eleatic school (Zenon
and his circle) developing with many variations the theme
of the non-constructibility of finite magnitudes out of
infinitly small parts.
This crisis shocked the Greek mathematicians into obtaining
two more brilliant achievements: the Theory of Proportions,
as contained in books 5 and 10 of Euclid's Elements, and
the Method of Exhaustion, as invented by Archimedes, that
was nothing less than a strict, though not sufficiently
general forerunner of modern theories of integration. Their
Theory of Proportions should have enabled the Greeks to
define irrational number and develop, accordingly, an
arithmetical theory of the Continuum; somehow they did not
quite make it.
The Greek Theory of Proportions was soon forgotten - so much
so that when rigorous arithmetical theories of irrational
numbers were constructed in the second half of the 19th
century, one was not at first aware of the fact that these
methods were not in principle much different from those
already in the possession of the Greek mathematicians two
thousand years earlier. Before that, in the 17th and 18th
centuries, the great power and fruitfulness of the newly
invented calculus led most mathematicians of those times
into feverish applications of the new ideas without caring
much for the solidity of the basis upon which the calculus
was founded. However, the shakiness of this basis became
clear at the beginning of the 19th century, constituting
the Second Crisis in the foundations of mathematics.
In order to overcome this crisis, Cauchy, in the eighteen
thirties showed how to replace the irresponsible use of
infinitesimals by a careful use of limit, while Weierstrauss
and others, in the sixties and seventies, demonstrated how
all of analysis and function theory could be "arithmetized".
This solidification of the foundations was so successful
that Poincare, in an address delivered in 1900 before the
Second International Congress of Mathematicians on the role
of intuition and logic in mathematics, could proudly claim
that mathematics had by then acquired a completely solid
and sound basis. In his own words:
"Today there remain in analysis only integers ... Mathematics
... has been arithmetized ... We may say today that absolute
rigor has been obtained."
Ironically enough, at the very same time that Poincare made
his proud claim, it had already turned out that the theory
of the "infinite system of integers" - nothing else but a
part of the set theory - was very far from having obtained
absolute security of foundations. More than the mere
appearance of antinomies at the basis of the set theory, and
thereby of analysis, it's the fact that the various attempts
to overcome these antinomies ... revealed a ... surprising
divergence of opinions and conceptions on the most fundamental
mathematical notions, such as set and number themselves,
which induces us to speak of the Third Fundational Crisis
that mathematics is still undergoing.
CONTINUITY AND DISCRETENESS
The Third Fundational Crisis brought up by Fraenkel and
Bar-Hillel is also known as the Fin-de-Siecle Crisis.
19th Century Physics and Logic were dominated by reaction of
Dogmatism against Rationality of the First Enlightenment.
Physics was founded in the dogmatic delusion of Aether.
Logical systems culminated in Predicate Logic (PL).
This noumenal, ill founded Logic was assumed to be the
universal, absolute foundation of transcendental "Reality"
and of human knowledge thereof. All areas of Science were
assumed to be recursively founded in PL, although Mathematics
stayed for some time unfounded. Finally, in 1873 Cantor
conceived the Set Theory which supplied the missing link and
completed the "Cantorian Paradise", which represented the
edifice of Science solid and stable as Cheops pyramid,
bestowing upon scientists the gratifying conviction that Logic,
THEIR Logic, explains and determines the "Real Universe".
At the end of the 19th century two breaches appeared in the
pyramid setting off the Fin-de-Siecle Crisis of Thought which
triggered the Second Scientific Revolution:
1.In Physics the Michelson-Morley experiment showing invariance
of the speed of light and thus falsifying the mechanistic
Dogma.
2.Russell's Paradoxes calling in question the foundations
and sense of PL and founded in it Set Theory.
Question arose, if apparently local breaches in otherwise
solid pyramid may be patched locally, or should lead to
global reconsideration of the whole structure.
In Physics Einstein has chosen the global approach, revised
entire Physics and laid the cornerstone of the Second
Enlightenment with its new Rationality. Without generating
any "surprising divergences of opinions and conceptions",
the new physics encompasses four or five principal models,
complementary, giving uniform satisfaction and sharing the
same foundations.
Logicians did not notice that Russell's Paradoxes did not
wreck Logic as such, but only the PL and the set theory, and
try in vain, till our days, to patch them rather than to
scrap the wrecks and to conceive in their place new rational
foundations. As to "divergences of opinions and conceptions",
we may count over 40 TYPES of theories formalized, intuitive,
logistic, (not)-axiomatizable, (un)-decidable, elementary,
first- second- ... nth-order, formally (in)-consistent,
(in)-complete, (in)-completable, recursively incompletable,
semantically (in)-consistent, semantically (in)-complete,
(non)-categorical, relatively categorical, etc. Each type
encompassing several theories, a lifetime would not suffice
to learn them all, due on the one hand to their obscurity
and, on the other hand to new ones popping up while you
struggle with one. And after 100 years of this proliferation
none can tell us - to believe Fraenkel and Bar-Hillel -
what's "set" and what's "number".
In face of this unbridled chaos one is tempted to look at
the stable and robust foundations of physics and to see if
and how they could assist those - hopelessly confused -
of mathematics.
Our inquiry of time and events (Part "A_FOUNDATIONS")
brought to light the CD (Continuum/Discreteness) Polarity
as the basic element of the human universe of discourse and
the foundational primacy of its continuous term. Following
this primacy, physics is founded in the continuum of
SpaceTime-Field, discretizing it to singularity areas and
particles.
Set theories inverse this natural primacy and are ill founded
in discrete "collections" ascending to continuum via
phantasmic transfinities. As long as this primacy violation
persists, mathematics will lack proper foundations. In other
terms arithmetics should be founded in geometry and not the
other way round as the establishment pretends.
Another confirmation of our view may be found in the
ultimate Frege's disappointment. In 1923 he came to the
conclusion that the aim he had set himself throughout most
of his career, namely to found arithmetic in (predicate)
logic, was wrong. He decided instead (like ourselves) that
one had to base the whole of mathematics on geometry.
He began to work on these ideas but had not progressed far
by the time of his death
The proper foundations of mathematics will have to be
conceived from the scratch, based upon continuum, locally
discretized whenever necessary.
References:
1.Abraham A. Fraenkel and Yehoshua Bar-Hillel "Foundations
of Set Theory".
2.Azriel Levy "Basic Set Theory".
