Alexander Zenkin alexzen at
Mon Mar 22 13:47:49 EST 1999

Dear Colleagues,

First of all, thanks very much to all who were trying to answer the
following three my questions. The questions, of course, are not
rhetorical or pure historical ones as many persons already have guessed.

Subject:          L.Kronecker and G.Cantor
   Date:          Mon, 15 Mar 1999 10:45:27 +0300
   From:          Alexander Zenkin < alexzen at >
     To:          HISTORIA MATEMATICA <
historia-matematica at >
     CC:          fom at , philosop at

1) It is known that L.Kronecker had a very negative attitude toward
G.Cantor's set theory. What is known about mathematical argumentation of
L.Kronecker's objections?
2) There is a version (unfortunately, I can't recall a reference) that
just such the L.Kronecker attitude became the main reason of G.Cantor's
illness. But L.Kronecker passed away in 1891. So, it is quite difficult
to trust in the reliability of such the reason. What versions are there
as to the true reason of G.Cantor's illness?
3) Is anything known as to G.Cantor told or wrote apropos of his own set
theory after 1899?

   The context and the underlying theme of these questions, alas, are
not very new.
   As I emphasized earlier, many outstanding scientists, - such as
Aristotle, St. Thomas Aquinas,  Locke, Descartes, Spinoza, George
Berkeley, Leibniz, Gauss, Cauchy, Kronecker, Poincare,  Hilbert
(partly), Brouwer, Weyl, Heyting, Luzin, and many-many others, -
thought, supposed,  believed, stated, claimed and so on that ⌠Infinitum
Actu Non Datur■. - Of course, that were  intuition statements only. But
today, it can be said with the absolute distinctness that when such  the
scientists of genius who were knowing quite well what the Infinity is,
and who, in fact,  created the Science as a whole, unanimously maintain
the same paradigm during 2300 years (!),  it is too unlikely that they
all together do such the "so long" mistake. Some of them, beginning
from Kronecker, gave notice that it is impossible to erect a science
building on sands having in  their minds just the G.Cantor Theory.
   Not I, but I together with all these persons state that the G.Cantor
set Theory as a whole, i.e.,  from alpha to omega, is wrong because it
is based exclusively on the actualization of all infinite  sets , and
the grandiose building of the almost all modern meta-mathematics and
mathematical  logic states today on the G.Cantor "STUDY ON

    Apropos of the FIRST question. I think the main reason of
L.Kronecker's wild indignation  against G.Cantor's theory was a
psychological one, viz a feeling of intellectual impotence. I will  try
to explain that.
    G.Cantor's theory is not a single whole. Firstly, there is a set
theory language. It is a version of the universal Aristotle's
syllogistics language which was brilliantly visualized by Euler in his
known letters to a Deutsche Princess (by the way the famous Euler's
logical circles is one of the first example of just a cognitive
visualization of scientific abstractions. For more info see my
WEB-Site). The set theoretical version of Aristotle's language is
ideally adapted for a discription of attitudes between elements,
subsets, and sets, and just therefore it became a common language for
most areas of modern mathematics. Further, there are two absolutely
different parts of G.Cantor's set theory: the theory of transfinite
ORDINAL numbers and the theory of transfinite CARDINAL numbers. Lastly,
the fourth main part of G.Cantor's set theory is the CONTINUUM PROBLEM
(the Problem, but not one of a lot of concrete CP-hypotheses), i.e., the
deepest mathematical, philosophical, cognitive, scientific Problem about
the nature of the connection between "Discrete" and "Continuous". So,
the so-called G.Cantor's theory is a Trojan Horse whose very attractive
outward appearance hides a fantastical tangle of paradoxes,
contradictions, and ontological nonsences.
    As to the set theory language, - all is clear here: to raise an
objection against that ingenious G.Cantor's invention means to raise an
objection against universal language of Aristotle's Logic, i.e., against
the science language in general. I think nobody (including L.Kronecker)
and never did so.
    As to the G.Cantor theory of transfinite CARDINAL numbers. That
theory as a whole is based on the only G.Cantor theorem on the
uncountability of the real numbers set in the sense that were the
theorem absent, the idea itself about a distinguishing of INFINITE sets
by the quantity of their elements would be considered as an Alan Sokal's
type para-mathematical nonsense.
     I am sure that L.Kronecker could not present some convincing
mathematical arguments against G.Cantor's set theory because even modern
meta-mathematics does not understand that the "naive" and all modern
non-"naive" proofs of the famous G.Cantor's Theorem on the
Uncountability are not proofs by means of the "Reductio ad Absurdum"
method based on G.Cantor's Diagonal procedure. All such the proofs are a
PARA-mathematical version of the counter-example method in a framework
of which (of that version) a counter-example itself (G.Cantor's
"diagonal" real number which is differed from any real number of a given
enumeration of ALL real numbers) is formally (semi-constructively and
semi-algorithmically) DEDUCED from the common HYPOTHESIS (the assumption
of the proof itself: "a given enumeration comprises ALL real numbers")
which that counter-example must disprove.
    The popular explanation of the situation especially for
mathematicians (and mathematics philosophers) is such one.
    In the middle of the XVIII, Euler formulated his famous
Number-Theoretical Hypothesis.
    EULER'S HYPOTHESIS. For any power, r >= 3, the Diophantine equation
n^r = n1^ r + n2^ r + ... + ns^r has no solutions in integers, if s < r.

    During about 200 years a lot of mathematicians tryed to prove that
Hypothesis. Why? - Not in the least turn , because the famous Fermate's
Great (Last) Theorem arises from the Euler Hypothesis by s=2.
    But today all we know well that all such attempts were beforehand
doomed to a failure, because, in 1967, Lander L.J., and Parkin T.R. { "A
counter example to Euler's sum of power conjecture." // Math.Comp., vol.
21, 101-103 (1967)} SEARCHED FOR the only COUNTER-EXAMPLE:
 144^5 = 27^5 + 84^5 + 110^5 + 133^5 ,
and the Euler's conjecture (the common statement) was disporved
absolutely rigorously by the CLASSICAL counter-example method.
    As is well known, Lander L.J., and Parkin T.R. were just SEARCHING
FOR a counter-example among the numbers 1,2,3, ┘, i.e., in the domain of
definition (or of possible realizations) of the Euler Hypothesis, using
a computer (and some simplifying number-theoretical and algorithmical
procedures). Why did they even not try to DEDUCE analytically their
counter-example directly from the Euler Hypothesis? Because they (and
all other mathematicians) know well that such kind of a "proof" is a
mathematical and logical non-sense.
    Shortly, if there is a mathematical  "proof" of the kind "IF A THEN
not-A", it can be always to prove that either such the "proof" breaks
the main deductive laws of Aristotle's Logic, or the premis A, as a
rule, IMPLICITLY already comprises  some contradictory statements, say B
and not-B. In the case of G.Cantor's Uncountability  Theorem, its premis
"Let (1) be an enumeration of ALL real numbers" (i.e., the proof
assumption of that, allegedly, "Reductio ad  Absurdum", i.e., a
hypothesis, i.e., a non-authentic common statement A)  already comprises
the contradiction: "The (1) is an ACTUAL, i.e., FINITE by its factual
usage, sequence" (not-B) and "The (1) is INFINITE, i.e., NON-FINITE,
sequence" (B). The G.Cantor's Diagonal method uses only not-B, and
allows to construct a "diagonal" real number differing from any real
number of the enumeration (1). If we use explicitly B, we prove that the
G.Cantor's proof contains a non-finite stage, i.e., proves nothing from
the mathematical point of view. All these my objections against the
G.Cantor Theorem of the Uncountability are proved absolutely rigorously
{see, for example, a lot of my messages to this [HM]- and FOM-discussion
lists, and references on my papers there, and in my WEB-Site}.
    In one word, all L.Kronecker's "mathematical" objections against
G.Cantor's set theory could be only of the same semi-philosophical kind
as the G.Cantor set theory itself.
    Now, some words about L.Kronecker's feeling of intellectual
impotence. I think that not G.Canotr's set Theory itself was the main
reason for such the feeling, but the quite inconceivable fact that he,
L.Kronecker, really a high professional, deep, true mathematician, could
not disprove, by means of all strict mathematical methods he well knew,
G.Cantor's quite doubtful argumentations. I think that the following
"passage" of G.Cantor, of his own pupil, revolted L.Kronecker most of
all. (Remark here that all L.Kronecker's energies to disprove G.Cantor's
Theory of transfinite ORDINAL numbers were doomed to a failure because
the last is the only part of G.Cantor's set Theory that is consistent.
As to a cost of that consistence see below).
     Indeed, G.Cantor states (almost word for word) the following: the
set of all finite natural numbers is infinite, and there does not exist
a <last> maximal number in the series 1,2,3, ┘ . Therefore, however it
would be CONTRADICTORY to speak about a maximal natural number, on the
other hand, there is not an ABSURDITY in that to imagine a new number -
call it by W ("Omega") ┘ They can even imagine that new-created NUMBER
as a LIMIT which the numbers 1,2,3,┘ are tending to, i.e., the W is the
first INTEGER which follows all the finite natural numbers, i.e., which
can be called greater than any finite natural number.
    So, without any Zigmund Freud's psychoanalysis, it is absolutely
clear that G.Cantor himself feels fine that his "DEFINITION" of the new
"INTEGER" W is a contradictory and absurd  one. But he disregards all
these contradictions and absurdities, and boldly goes further: applying
the common operation "+1", defined for finite integers, to the
transfinite "NUMBER" W, he obtains, by means of, supposedly, his "FIRST
generating principle" (i.e., by means of the common Peano's axiom
defined for the finite natural numbers: "IF n THEN n+1"), further
W, W+1, W+2 , ... , W+n , ... , 2W , ┘ , 3W , ┘ , nW , ┘ , W^2 , ┘, W^3
, ┘ , W^W , ┘ , W^W^W , ... , W^W^W^┘ (*)
   Thus he invents his famous series of the transfinite ordinal numbers.

    I state that modern meta-mathematics added nothing of novelty to
that G.Cantor's "definition" of the least transfinite ordinal number W,
and to his (or Peano's) FIRST and SECOND principles of the series (*)
generation. I think any modern mathematician assesses such the
definition of, supposedly, mathematical notion as a not too well prank
in Alan Sokal's kind (if overwhelming majority of mathematicians were
not avoiding some "dangerous", by H.Weil, areas of their science, it
would be understood much earlier). The G.Cantor's definition of W is the
next Trojan Horse of the G.Cantor's set theory. Indeed, if you TAKE
ANYTHING (either chairs, tables, and beer mugs, by D.Hilbert, or an
actual, by G.Cantor, or even potential, by Pythagoras and Aristotle,
infinity), DENOTE that ANYTHING by ANY SYMBOL, say alpha, betta, gamma,
┘, omega, CALL that SYMBOL by an INTEGER, - it's all! Further the
Peano's inductive axiom (or even more precisely, the primordial,
genetic, Pythagoras definition of the natural number notion itself) "IF
? is an integer THEN ?+1 is an integer too" works only. No mathematician
will object against Peano or Pythagoras, but just therefore some
essential details pass by a critical mathematical conciousness.
    Usually, meta-mathematical experts state: infinite things are very
different from finite things, because the first possess of very unusual
(for our normal mathematuical intuition) properties which the last can
not possess (e.g., an infinite set is equivalent to its own subset), and
therefore it is impossible to attribute properties of finite sets to
infinite sets. G.Cantor himself breaks that commandments. Indeed, from a
childhood we know well that every finite thing "?" possesses the
property "IF ? is a finite natural number THEN ?+1 is a finite natural
number too". And conversely, "IF a thing ?+1 is a finite natural number
> 1 THEN a thing ? is finite natural number too". It is easy to see that
G.Cantor's invention, W, W+1, W+2, ┘, is based upon the attribution of
the main, fundamental property of finite sets to infinite sets. I think,
to call a SYMBOL W an INTEGER is not a sufficient reason for such the
attribution. All the more that nobody, not for a while yet, has proved
that the application of the strictly FINITE operation "+1" to INFINITE
(even TRANSFINITE) "integers" is a logically legitimate action.
    The next shocking moment. G.Cantor underlined that the philosophical
value of his Set Theory  is even greater than its mathematical value. I
believe that he had in his mind those well-known enormous efforts which
he made in oder to "prove" (to persuade others to trust in) that all
infinite sets are ACTUAL INFINITE sets. According to G.Cantor, the
potential infinity is a non-definite, variable, non-constant quantity
which can not be called a (integer) number. In oder to call the infinite
series 1,2,3,┘, by the least transfinite integer NUMBER W, he was forced
to make (or, rather, to declare) ACTUAL that series. But it occures that
all G.Cantor's Theory of ordinal numbers does not depend absolutely on
whether the INFINITE series 1,2,3,┘, denoted by the symbol W, is assumed
to be ACTUAL or POTENTIAL one. From the point of view of the paradigmal,
mathematical and philosophical, foundations of G.Cantor's set Theory as
a whole it is one of the most unexpected and scandalous fact.


I. AXIOM OF DESIGNATION. Any THING can be designed by any SYMBOL.
II. AXIOM OF DENOMINATION. Any SYMBOL can be denominated as an INTEGER.
III. AXIOM OF GLUING (CONCATENATION). If a1,a2,a3,┘, and b1,b2,b3,┘, are
two things then a1,a2,a3,┘,b1,b2,b3,┘, is a thing too.
arbitrary symbol, and n be any finite natural number (possibly different
one in different occurences). Then the following purely symbolic

   W, nW, nW^n, nW^n + nW, nW^(nW^n + nW +n) + nW^n + nW, nW^(nW^(nW^n +
nW +n) + nW^n + nW +n) + nW^n + nW, and so on,

are NAMES (complex symbols) of the "second kind" by G.Cantor.

REMARK. It is the most ingenious invention by G.Cantor (his next Trojan
Horse): using the idea of the common polinomial notation, he writes
summands of such the common polinomial in an additive NON-POSITIONAL
form with the symbol W (or "Omega" that is the same) as a radix, and
generates an infinite sequence of quasi-mathematical NAMES (complex

EXAMPLE. It is well-known that the number of elementary particals, say,
N in the physical Universe (even taking into account the virtual
particals) is much less then, say, N=10^1999. Following G.Cantor's
additive, non-positional number system notation, that number N must be
written in the following form: N=10^(10^3+9x10^2+9x10+9).


DRAW (by PEANO): 1,2,3,┘, and designate that thing as  W (by CANTOR, and
DRAW (by PEANO): W, W+1,W+2,W+3,┘, and designate that thing as W2 (by
DRAW (by PEANO): W2, W2+1,W2+2,W2+3,┘,  and designate that thing as W3
DRAW (by PEANO):  . . . , and designate all that things as W^2 (by
DRAW (by PEANO): W^2, W^2+1,W^2+2,W^2+3,┘, and designate that thing as
And so on: W^W, ┘, W^W^W, ┘ , W^W^W^┘, and designate all that things as
e0 .
Further you can repeat the same with the names: e0,e1,e2,e3, ┘ e_W (by
CANTOR, and AXIOMS I,II,IV), and so on.
GLUE all things drawn above in one thing according to AXIOM IV:

1,2,3,┘,W, W+1,W+2,W+3,┘,W2, W2+1,W2+2,W2+3,┘, W^2, W^2+1,W^2+2,W^2+3,┘,
W^W, ┘, W^W^W, ┘, W^W^W^┘ (= e0), e0+1, e0+2,e0+3,┘,e0 2, e0 2+1,┘,e0
3,┘,e0^e0,┘ ┘ ┘

 Sorry, I forgot to formulate the main axiom of G.Cantor's transfinite
"arithmetic". I will give it in the most common form.

V AXIOM. If  $  is any radix, and n is any natural number which is
strictly less than the radix  $ , then
   n + $ = $ =/= $+n.

    As it is easy to see, the axiom V in such the common form can be
applyed even to PA (Peano's Arithmetic): for example by radix 10 we
shall have: 7+10 = 10 =/= 10+7. I suspect that [HM]-experts possibly
know some examples of such non-commutative PA applications in the
ancient Greek mathematics.
    So, as it is easy to see, all G.Cantor Theory of the so-called
transfinite ordinal "numbers" consists in the construction of the same
Pythagoras series of the common natural numbers,
       0, 0+1, 0+2, 0+3, 0+4, ┘, 0+n, ┘
based on a recursive procedure of renaming (the quasi-mathematical NAME
of a current series transforms into a new quasi-mathematical NAME for
the "ZERO" in the next series), and a successive gluing of constructed
series. Modern meta-mathematics supposes that such the "process" can't
be ever ended, i.e. that the "process" is POTENTIALLY infinite. Of
course, it contradicts roughly to G.Cantor's Theory Spirit itself, but
else they had to abandon from a splendid dream to arrive at even though
the least inaccessible transfinite ordinal number ever.
     Anyway such a tautological process, every step of which satisfys to
the Peano induction axiom, can't lead to a contradiction. Consequently,
the G.Cantor Theory of transfinite ordinal numbers is (relatively)
consistent, of course, iff the Peano induction axiom "n == > n+1" is
consistent. That explains completely why the outstanding German
mathematician, L.Kronecker, could not disprove the G.Cantor Theory of
transfinite ordinal numbers, the only "mathematics" of which is Peano
induction axiom, and was very sad by that reason for a long time. - It
is impossible to win such the "Trojan Horse" by means of mathematical

   So, my second question was:
   2) There is a version (unfortunately, I can't recall a reference)
that just such the L.Kronecker  attitude became the main reason of
G.Cantor's illness. But L.Kronecker passed away in 1891.  So, it is
quite difficult to trust in the reliability of such the reason. What
versions are there as to the true reason of G.Cantor's illness?

    As to the SECOND question. I have serious suspicions and a quite
plausible hypothesis that during 1895-1905 G.Cantor was understanding
that all his set Theory is wrong. Taking into account the fact that 1900
was the year of the world-wide triumph for G.Cantor himself (he was only
55 years old!) and for his Set Theory, one can only conjecture what
terrible tragedy was occuring in his soul during that time.

3) Is anything known as to G.Cantor told or wrote apropos of his own set
theory after 1899?

     I believe that there is some (maybe, very private) information
about that period of G.Cator's life which could confirm that the Great
German Mathematician, G.Cantor, was the first scientist in the XX
Century who understood that "there is no permanent place in the
mathematics" for the actual infinity, and that ALL INFINITE SETS IN

    Thus, all said above is only rigorous logical and mathematical
consequences of the following absolute, MATHEMATICAL statement which is,
in its turn, the direct consequence of the famous G.Cantor Theorem on
the Uncountability.

 ARITOTLE'S THEOREM. "Infinitum Actu Non Datur".

   So, I propose the Open List for persons who believe firmly that:

⌠The World is Number ┘ and Harmony of Singing Heaven Spheres■, by
"Die ganzen Zahlen hat der liebe Gott gemacht,
alles andere ist Menschenwerk", by Leopold Kronecker, and
"Mathematics is the Queen of Sciences -
and Number Theory is the Queen of Mathematics-XXI", <almost> by Carl
Friedrich Gauss.


HILBERT (partly),
. . .

The List is open . . .

"He that hath an ear, let him hear what the Spirit saith ┘"

Best regards,


P.S. Today, I have a complete, unque series of lectures about the NEW
Foundations of Mathematics-XXI and the NEW Philosophy-XXI of Infinity.
The course comprises the complete resolution of the I, II, and III Great
Historical Crises in the Foundations of Mathematics, the complete
solutions of Continuum Problem, Paradoxes Problem (including the famous
"Liar"), and so on.

> ############################################
> Prof. Alexander A. Zenkin,
> Doctor of Physical and Mathematical Sciences,
> Leading Research Scientist of the Computer Center
> of the Russian Academy of Sciences.
> e-mail: alexzen at
> WEB-Site
> ############################################
> "Infinitum Actu Non Datur" - Aristotle.

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