Uncountable setssowa <email@example.com>
Date: Mon, 24 May 93 08:34:27 EDT
From: sowa <firstname.lastname@example.org>
To: email@example.com, interlingua@ISI.EDU, firstname.lastname@example.org,
Subject: Uncountable sets
Since the discussion about uncountable sets on these lists has been
winding down, I hestiate to reopen it. But the following note from
John Nageley goes into further detail about the various doubts that
mathematicians have had concerning Cantor's diagonal proof. I have
already said more than I wanted to about this topic, and I am happy to
let someone else take up the agnostic side of this dispute. Please
put me on the cc list of any future discussions, but I would rather
stay on the sidelines.
Just to summarize my own position relative to KIF and CGs:
1. KIF and CGs should allow people to express any subject they like,
using whatever concepts or predicates they please with any axioms
they find useful. If they end up in contradictions, that is their
own tough luck -- it is not our responsibility to police or restrict
their use of the languages.
2. But at the same time, the doubts about uncountable sets indicate
that the foundations of mathematics in that area are not as solid
as in other parts. As Nageley points out, a number of highly
respected mathematicians have avoided topics that critically depend
on the assumptions that lead to uncountable sets. I believe that
KIF and CGs should also allow those people to use the languages
without being forced to make assumptions that they find problematical.
3. Therefore, I would like to see the base languages developed in such
a way that both sides can use them according to their own preferences.
Set theory, in Cantor's tradition, should be available as an
ontology expressed in KIF and/or CGs. But people who are not happy
with uncountable sets and the related problems should be able to
use the languages without depending on assumptions that lead to them.
4. This raises a question about the semantics of KIF, which is being
modelled using a version of VNBG (von Neumann, Bernays, Goedel set
theory), which does support uncountable sets. How can one use KIF
without presupposing its underlying semantics, which does seem to
lead to the dreaded "swamp of confusions" or "Cantor's paradise"?
5. I would answer that question in the following way:
a) For the present, I am willing to let the people who want to
establish the semantic foundations of KIF use VNBG if they like.
b) In the future, however, I would hope that someone might revisit
the foundational questions and develop a model theory for KIF
and/or CGs that does not involve any axioms that lead to
uncountable sets. Mereology, as I suggested in some earlier
notes, is an approach that I find congenial, but there are
probably other theories that would serve the purpose.
c) Meanwhile, computer science in general and AI in particular
are devoted to the study of algorithms and data structures
that can be implemented on digital computers. Even countably
infinite sets cannot be implemented, although many theorems
and proofs are simpler when they are stated without any finite
upper bounds. Therefore, I believe that talk of countable
infinities in foundational studies is OK, but that any
generalization to uncountable cardinalities is not only
unnecessary and irrelevant, but potentially dangerous.
d) Therefore, I am willing to use a language whose current model
theory happens to be based on VNBG, but I intend to restrict my
use only to the countable realm. If anyone else wants to use
the uncountable stuff, that's their business. But I hope that
while I'm happily working in the countable garden, some other
people will examine the walls and fences to make sure that the
uncountable swamp doesn't spill over into my tomato patch.
Following is the note from John Nageley.
John and Pat:
My approach to Cantor's work has been by way of logic and philosophy. And in
effect the following notes simply provide just one more appeal to authority
(more of the citation game that one of you mentioned) and provide evidence to
suggest that I may have done my homework. Included are quotes on the subject
>From well-known mathematicians and philosophers that did I not see in the
copies of your exchange that I received. (I see one of you referenced the
Encyclopedia of Philosophy and authors to read. It is an excellent resource:
If you haven't, you might also want to check it out for original writings of A.
A. Fraenkel, quotes from which are included below.)
My basic position is, and has been for numerous years, that though Cantor made
significant contribution to mathematics not all of it has been for the good. I
have been concerned particularly with the problem of the number line: simply
stated, the real number line is not a valid concept. I have written
extensively on the subject and hope eventually to publish. I gave a short
presentation on the subject at the 9th International Congress of Logic,
Methodology, and Philosophy of Science, August 1991, Uppsala, Sweden, entitled,
"The Doctrine of Analogy: Kantian and Hermetic Philosophy and Transfinite
Number Theory." Excerpts from that presentation are included in the notes
What seems clear to me now from my studies and your discussion on the subject
of Cantor's work is that no amount of debate or appeal to authority (i.e.,
again, the citation game) is going to resolve the issues or problems arising
>From that work. Instead, what is needed is a clear, convincing refutation of
Cantor's diagonal proof. And I think I have one, one that has been many years
in the making, one that I would like to publish soon.
NOTES ON CANTOR'S THEORY OF SETS
The major question that Kant asked was "What can we know?" He presented his
answer mainly in the Critique of Pure Reason but also in his Prolegomena to Any
Future Metaphysics. In partial answer to his question, Kant writes in the
first paragraph of the preface to the first edition of the Critique,
Human reason has this peculiar fate that in one species of its knowledge it is
burdened by questions which, as prescribed by the very nature of reason itself,
it is not able to ignore, but which, as transcending all its powers, it is also
not able to answer. (p. 7)
But as he also writes, the realization that we cannot answer some questions has
not prevented many from attempting to transcend all their powers in search of
answers. In their wake, they have left the many metaphysical schemes of which
he grew so weary. And the passing of time has not in any way diminished the
human propensity for creating ever more such large and grand schemes; to wit,
Cantor's theory of sets and transfinite numbers.
A brief review of the history of set theory, its introduction to and use in
mathematics and science, seems to confirm A.A. Fraenkel's belief that "set
theory has infiltrated into and largely transformed most branches of
mathematics." Though not everyone agrees on the value of set theory to
mathematics and logic, its application is found, for example, in analysis,
geometry and topology. The theory of functions is permeated by the methods and
results of the theory of sets of points. Then, also, almost every textbook
(even elementary school textbooks in some cases) has at least an introductory
chapter on sets in which the basic concepts and terms of the theory are
introduced (though we may question the value of such chapters).
Then too, many mathematicians seem to still agree even with Cantor's theory of
transfinite numbers and related or derivative concepts. As recently as 1962,
Felix Hausdorff reaffirmed Cantor's conviction that "actual" infinite exists.
Also, according to A. A. Fraenkel,
The majority of present-day mathematicians presumably agree with [David]
Hilbert's dictum that nobody will expel us from the paradise created by Cantor.
The following statement by Fraenkel seems to summarize the current status of
set theory in mathematics:
The overwhelming majority of mathematicians, even those who are theoretically
impressed by the critical arguments [against set theory], continue to apply the
methods of set theory. Very few, if any, have followed the example of Hermann
Weyl, who confessed that he avoided certain analytical research subjects
because they were immersed too deeply in set theory.
In short, as Cantor himself observed, mathematics has within it its own
corrective: if a theory is not useful--"fruitful"--it will soon be discarded.
And the history of his theory has shown that it can be fruitful.
Cantor's theory, and many of its parts, has however been severely criticized
since Cantor first introduced it over 100 years ago.A.A. Fraenkel, for example,
a mathematician who has made numerous, important contributions to set theory,
(he is the "F" in "ZFS" set theory) provides the following evaluation in his
book Set Theory and Logic:
Unfortunately...some of the results of set theory, in particular Cantor's
theorem and its proof, bring us close to a gulf which is difficult to bridge.
The matter is serious enough to have induced mathematicians to speak of
Cantor's work as a "pathological entanglement" which later generations will
look upon with bewilderment.
Also, though Fraenkel has labeled Cantor's fundamental theorem as "one of the
most important and applicable theorems in mathematics" still he adds, "it has
been attacked by philosophers, and also by some mathematicians and even
physicists, more than any other mathematical theorem, with the possible
exception of the well-ordering theorem."
Then, Paul Halmos in his introduction to Naive Set Theory states,
The student's task in learning set theory is to steep himself in unfamiliar but
essentially shallow generalities till they become so familiar that they can be
used with almost no conscious effort. In other words, general set theory is
pretty trivial stuff really, but, if you want to be a mathematician, you need
some, and here it is; read it, absorb it, and forget it.
In short, what is for some mathematicians a "paradise" is for others a possible
"pathological entanglement"--"pretty trivial stuff really."
To attack set theory, or any fundamental aspect of it, though, is to shoot at a
moving target. Witness, for example, the following statement by Hausdorff:
"In our opinion, it does not detract from the merit of Cantor's ideas that some
antinomies that arise from allowing excessively limitless construction of sets
still await complete elucidation and removal." There is no effective way to
address such an attitude. The approach is religious; not mathematical. It is
mathematics made religion. (Or it is mathematics made metaphysical.) [Please
note the pontifical "our" in Hausdorff's statement.]
But, regardless of whether the target will stand still, set theory must be
reexamined.Whether set theory is, indeed, either a "paradise" or possibly a
"pathological entanglement," "pretty trivial stuff really," may not be clear.
But what does seem clear is that the fundamentals of set theory need to be
As to an approach to such an examination, from his deep involvement in the
theory, Fraenkel believes that
the modern development of set theory seems to shatter mathematics altogether,
at least in its analytical parts. New axioms apparently need to be introduced,
corresponding to a deeper understanding of the primitive concepts underlying
logic and mathematics.
But, and he then takes the approach one step further, he declares, "nobody has
so far succeeded in discovering even a direction in which such axioms might be
Goedel has, however, suggested a direction: in his article, "What is Cantor's
Continuum Problem," he states, on "Restatement of the problem on the basis of
an analysis of the foundations of set theory and results obtained along these
this scarcity of results [from set theory], even as to the most fundamental
questions in this field, may be due to some extent to purely mathematical
difficulties; it seems, however, that there are also deeper reasons behind it
and that a complete solution of these problems can be obtained only by a more
profound analysis (than mathematics is accustomed to give) of the meanings of
the terms occurring in them (such as "set," "one-to-one correspondence," etc.)
and of the axioms underlying their use.
Fraenkel also points out that such mathematicians as Whitehead, Russell and
... do maintain that the logical foundation on which the [set] theory was
constructed at the turn of the century is insufficient and has to be
Later he states,
...More general doubts about the soundness of nondenumerable sets, transfinite
cardinals, higher number classes, etc., have spread and are at present by the
arguments of such important philosophers and logicians as Hao Wang and Paul
Though Goedel is concerned principally in this statement with Cantor's
"conjecture" or "fundamental theorem," still what he states seems generally
applicable to set theory: a disparity exists between "some well-determined
reality" that set theory is supposed to describe and the "concepts and axioms"
with which set theory can be applied. The disparity exists; therefore the
contradictions exist. The disparity, for example, between the fundamental
theorem and the world it seeks to describe cannot be fully reconciled. Too
many contradictions--the bane of set theory--remain unresolved. They make it
"unfruitful" for many applications.
The combined wisdom of Fraenkel, Goedel and Hilbert seems to be that we must
return to fundamentals: the very fundamentals of Cantor's theoretical work
need to be reexamined--to include even the concepts of a one-to-one
correspondence, set, nondenumerable sets, transfinite cardinals, higher number
>From my perspective, the fundamental problem lies with basic assumptions
underlying the diagonal proof and the resulting fundamental theorem and the
continuum hypothesis--but particularly the concept of the real numbers. As to
the importance of this hypothesis, the real numbers and such, according to
In 1900, David Hilbert, in a famous speech before an international conference
of mathematicians, summed up what he hoped would be accomplished in the
twentieth century by identifying 23 outstanding problems in mathematics. The
problem that he placed first in this list (because of its importance?) has two
parts: the first part is to determine the truth or falsity of Cantor's
continuum hypothesis. The second is to show, as was assumed in the proof, that
the real numbers are not countable, that there is a way to find the first
number for every set of real numbers.
>From a philosophical position, I see the foundation of Cantor's theory of
transfinite numbers as purely metaphysical; a noumenal theory, not a phenomenal
theory. Nowhere is the kind of thinking involved more vividly characterized
than in Kant's Critique of Pure Reason:
The light dove, cleaving the air in her free flight, and feeling its
resistance, might imagine that its flight would be still easier in empty space.
It was thus that Plato left the world of the senses, as setting too narrow
limits to the understanding, and ventured out beyond it on the wings of ideas,
in the empty space of pure understanding. He did not observe that with all his
efforts he made no advance--meeting no resistance that might, as it were, serve
as a support upon which he could take a stand, to which he could apply his
powers, and set his understanding in motion.
The alchemists, for example, may be said to have left the world of the senses
in search of fundamental principles of the universe; but they often returned.
Cantor, on the other hand, may be said to have left the world of the senses in
creating his theory--but never returned. Historically, Platonist have shown a
tendency to leave the world of the senses--and never want to return; witness
the Cantorian "paradise" in which some mathematician would prefer to dwell, a
never never land of pure speculation. Fraenkel is a self-confessed Platonist.
That humans will, and sometimes must, leave the world of the senses to explore
the outer regions of human cognition is an imperative laid on us by the very
nature of our minds. For, as Kant himself noted, "That the human mind will
ever give up metaphysical researches is as little to be expected as
that...we...should...give up breathing."
So be it. It is our "peculiar nature," according to Kant, to seek answers to
questions we can never hope to answer but question, nevertheless, that we
cannot ignore. Even though metaphysical speculation can often mitigate against
the soundest reason, we cannot give it up. Besides, what does it hurt that we
engage ourselves in, for example, playing with notions of transfinite numbers?
It can be fun to do so. For some, as we have heard, it can be even more than
simply fun: for them, the realm of transfinite number is a paradise.
Again, so be it. We should, however, be fully aware of when it is we are
engaged in such paradisiacal speculations and that we must then eventually
return to the realm of the phenomena--realizing that everything encountered in
paradise may not apply or be useful on our return: for example, transfinite
number theory. Current number line theory, a product of that theory, is an
excellent example of the absurdities that can be created when an attempt is
made to apply a transcendent (following Kant) notion to phenomena.
The most hopeful sign of a breakthrough in thinking relative to the concepts
discussed here on a more practical level is the conceptual basis for object-
oriented programming. Computer scientists seem to have managed to avoid or
ignore set theory (for whatever reason) in designing programming languages.
Their approach has, it seems, been practical; pragmatic. OOP, therefore, seems
to come closer to what I think a theory of sets should be than does Cantor's.
With its emphasis on objects and their attributes, for example, OOP bypasses an
historical hangup--that of the place of the concept of attribute in a theory of
sets, a concept that Quine once wrote that he abhorred. Set theory has always
had too many self-evident propositions--of which the notion of attribute is one-
-that have been self-evident only to those who have not wanted to deal with
If, however, OOP and other positive developments in modern thought are to prove
more fruitful (in following Cantor again) than have their predecessors, then
perhaps another of Kant's observations should be kept in mind:
It is, indeed, the common fate of human reason to complete its speculative
structures as speedily as may be, and only afterwards to enquire whether the
foundations are reliable.