FOM: RE: As to Aristotel's "Infinitum Actu Non Datur" Thesis

[Alexander Zenkin]

Dear friends,

Thanks very much for your kind, and very interesting responses and
remarks as to my request apropos of Aristotle's "Infinitum Actu Non
Datur" Thesis.

Samuel S. Kutler writes ([HM]: Thu, 11 Feb 1999 08:02:24):

>In the beginning of chapter 5, Aristotle uses (coins) the
>word adiexodon.  The x is a ksi,

>Let us dissect it beginning at the end:
>        Hodos = road or way
>        Exodos = way out
>        Dia-ex-hodos  = way out by going through, and finally the
negative
>        A Dia Ex Hodos = No way out by going through.

>The infinite is not addressed directly, but as something for which
there is no exit in an attempt to pass through it.

AZ:
    Thanks for that fine explanation of Aristotel's "A Dia Ex Hodos"
idea. I think that here once more thought is not addressed directly viz
the thought that a person who, nevertheless, tryes to pass through
(jumps over) the "A Dia Ex Hodos" makes the rough logic mistake "petitio
principi" (sorry for my "Latin") or "Jump to a conclusion". - Since, it
was too obvious and trivial for such the Great Logician as Aristotel.
Below, I will make some remarks apropos of that Aristotel point of view.

Julio Gonzalez Cabillon writes ([HM]: Wed, 17 Feb 1999 02:52:34):

Apropos of the Aristotle's famous dictum on infinity, I believe most
historians and philosophers will basically agree with Struik when he
states that

    "The scholastic writers of the Middle Ages, especially St. Thomas
Aquinas, accepted Aristotle's _infinitum actu non datur_ [= there is no
actual(ly) infinite], but considered every continuum as potentially
divisible ad infinitum.

<AZ: why a "but"? I think  the "and therefore" must be here >

Thus there was no smallest line. A point, therefore, was not a part of a
line, because it was indivisible: _ex indivisibilibus non potest
constare aliquod continuum_ [= a continuum cannot consist of
indivisibles]. ┘"
Is this the case?... Comments are welcome.

AZ:

    1. AN UNEXPECTED "CONNECTION" BETWEEN ARISTOTEL, St. THOMAS AQUINAS,
AND CANTOR.

    Let X be a set of "all" real numbers (points) of the segment [0,1].
Using binary number system (for a simplicity only), consider
CANTOR'S THEOREM. {Thesis A:} The set X is uncountable.
CANTOR'S PROOF by Reductio ad Absurdum (RAA) method.
Assume that {anti-Thesis not-A:} the set X is countable. Let

   x1, x2, x3, ┘ , xn, ┘ (1)

be an enumeration of ALL real number  x (- X, i.e.,

{B:} for any d: if d (- X then d (- (1).

Now, we construct a DIAGONAL real number (DRN), say,

   y = 0, y1 y2 y3 ┘ yn ┘   (2)

according to the well-known G.Cantor DIAGONAL rule:

    for any i > = 1: [if x_ii =0 then y_i :=1] & [if x_ii =1 then y_i
:=0]   (3)

Applying the DIAGONAL rule (3) to the enumeration (1), G.Cantor
construct DRN y and quite rightly states that the DRN y differs from
every element of  (1). Consequently:

{not-B:} the DRN y (- X, but  not-[y (- (1)].

   From the contradiction between B and not-B, G.Cantor concludes, by
RAA-method, that the assumption not-A is false, and, by the law of the
contradiction, the Thesis A is true. Q.E.D.

    Now, changing nothing in Cantor's proof, we add to Cantor's DIAGONAL
rule (3) a new DIAGONAL rule:

    For any i > = 1: [[if i=1 then z:=1] else [z := z+1]],   (4)

 and literally repeat the Cantor proof, using the both DIAGONAL rules
simultaneously at every step i > = 1.

    Obviously, that if index i is finite, the Cantor DRN y is a finite
sequence of 0s and 1s and therefore it is not a real number. So, if
Cantor wish to preserve the MATHEMATICAL definition of the real number
notion as an INFINTE sequence of (in the binary number system) 0s and
1s, he MUST MAKE ACTUAL THE INFINITIES of the enumeration (1) and, as a
logic consequence, of a set of natural indexes of the elements of the
enumeration (1), i.e., he MUST also make actual the common series of
natural numbers: 1,2,3, ┘ So, if Cantor, by means of the application of
his DIAGONAL rule (3) to the infinite sequence elements x_i of (1), has
constructed his DIAGONAL real number y, then the simultaneous,
synchronous application of the DIAGONAL rule (4) to the indexes i of the
same elements x_i of (1) allows to construct an integer z which is
greater than any finite natural number, since until z is finite the
Cantor DRN y is not a real number.
    By Cantor's definition, the least integer which is greater than any
finite natural number is his famous least transfinite integer W
("omega"), i.e., after Cantor's proof, we also have z = W.
    REMARK. According to Cantor's methodology, the operation "+1"
applyed to integers can't leads us out the area of integers even if we
cross  over a "border" between finite and  non-finite (transfinite)
integers.
   I believe that the last is today the most rigorous PROOF of the
existence of the Cantor least transfinite integer "omega". Anyway, from
the point of view of that mathematics which declares valid the
traditional Cantor proof of the theorem on the uncountable sets
existence.   : -)
     But I wish to emphasize especially that it is the CONDITIONAL
proof: the Cantor  "omega" exists, IFF we wish to preserve the common
definitions of the notions of the common real number and of the common
geometrical point of a segment. So, the Cantor  "omega", the real
number, and the geometrical point have the same ontological status in
the sense that all objections against existence of one of these relate
equally to the existence of others.
    As is known, the original Cantor substantiation for the introduction
of the "omega" in mathematics is not a mathematical action, it is an act
of an ambitious belief (by the way, a lot of mathematicians did not
accept that belief up today), and it is beneath all criticism. Just the
last raised the distinctly negative relation of such the professional
classical German mathematician as L.Kronecker to Cantor's ideas. But I
take upon myself the liberty to assume that if Cantor guessed to add the
diagonal rule (4) to his uncountability theorem proof, L.Kronecker would
be stated before the difficalt choice: either to accept the Cantor
"omega", or to reject the Dedekind and Weierstrass conception of the
real number.
    However that may be, all said above contradicts cardinally to
Aristotel's and St. Thomas Aquinas'es point of view as to the potential
character of the infinity: the Cantor "omega", the real number, and the
geometrical point are ACTUALLY INFINITE (the first - in large, the
others - in small) objects.
     It is appropriate remember here the prophetic G.Cantor words:
"Transfinite Numbers themselves are, in a certain sense, new
irrationalities. Indeed, in my opinion, the method for the definition of
finite irrational numbers is quite analogous, I can say, is the same one
as my method for introducing  transfinite numbers. It can be certainly
said: transfinite numbers stand and fall together with finite irrational
numbers."

Luigi Borzacchini  (Date: Fri, 12 Feb 1999 11:46:17) writes:

These <AZ: Aristotel> texts are quite difficult to understand and not
actually convincing for modern readers. Strange enough, because here we
face theses that will <AZ: completely agree : -) > universally be held
true for two thousand years!
To understand the strength of Aristotle's arguments we must realize that
he gives the final setting to the crucial questions of "being" and
"becoming" in Greek and Middle Ages philosophy. In the earlier Greek
philosophers these themes were dealt with by principles ("archai") such
as water, air, fire, etc. and pair of 'physical' opposites such as
love/hate, cold/warm, wet/dry, etc.

AZ:  You are quite right. For modern mathematicians and logicians, the
Aristotel appeals to physics, psychology, and phylosophy (water, air,
fire, cold/warm, wet/dry, love/hate, principles "being" and "becoming",
etc), in such fundamental, but very abstract problem as whether the
Infinity is actual or potential, look like quite not cogent arguments
(of course, it must be added that there were not better arguments in
that time, and, in a framework of those possibilities, the Aristotel
argumentation is ingenious, even today). From that point of view, the
Cantor undoubted merit consists in that he went on from philosophical
discussions about the infinity to the practical usage of the infinity in
mathematics. But it was a quite dangerous step because henceforth all
consequences of such the infinity usage became, for the first time,
accessible to the rigorous mathematical and logical analysis..

    2. MATHEMARICAL PROOF OF THE ARISTOTEL'S THESIS "INFINITUM ACTU NON
DATUR".

   The G.Cantor Uncountability Theorem and his proof of the Theorem
appeared in 1891 (please, correct the date if I am some wrong), but the
meta-mathematics and the mathematical logic (in their working versions,
but not in embryo) appeared about the half Century after, and added
nothing to that Theorem and to its Cantor's proof. So, that proof can't
be either a meta-mathematical one, or a mathematical logic one.
Moreover, I state that the G.Cantor proof is not also a mathematical
proof (see above), since it is quite hard to call mathematical a proof
which uses only three notions of the elementary mathematics, viz such
the notions as natural number, real number and a sequence of such the
numbers, that were well known in Aristotel's time. So, the G.Cantor
proof is a classical logic proof only! - But in such the case, the
Cantor Theorem might be proved in Aristotel's time!?

    Indeed, of course, not the Cantor Theorem itself, but its very close
analogue (similarity?) was really well known in Aristotel's time. Here
is it (though in a modern interpretation).

    Let N be the set of "all" finite natural numbers.
    The following "simple" statement is proved today as a classical
example of the "Reductio ad Absurdum" application.

THEOREM 1. {Thesis A:} The set N is an infinte set.
PROOF. Assume that {anti-Thesis not-A:} the set N is a finite set. Then
there is a maximal element, say, n in N. So, we have

{B:} n is a maximal element of N.

But if n is a natural number then n+1 is a natural number too, at that
n+1 > n. So, we have:

{not-B:} n is not a maximal element of N.

The obtained contradiction between B and not-B proves the Theorem.
Q.E.D.

REMARK. As I suspect, this Theorem was proved (and, unfortunately, by
just the Reduction ad Absurdum method) long before Aristotel's time. For
Ancient Grecian mathematicians it was obvious that for any n, if n+1 > n
then n+1 differs from every element of the sequence: 1,2,3,┘n. Therefore
they did not call their proof as the proof by means of a DIAGONAL
method. But the proof can be easy re-write as a proof  by means of just
a DIAGONAL method.

THEOREM 2. {Thesis A:} The set N is an infinte set.
PROOF. Assume that {anti-Thesis not-A:} the set N is a finite set.
Then there is an enumeration of all finite natural numbers in their
well-ordered natural form, say, in the such one:

1, 2, 3, ┘ n.     (5)

So, we have

{B:} for any mathematical object k, if k (- N then k (-(5).

Now we define a new DIAGONAL number, say, z by means of the following,
already known DIAGONAL  rule:

    for any i > = 1: [[if i=1 then z:=1] else [z := z+1]],   (4)

    Now we strictly follow G.Cantor: applying our DIAGONAL rule (4)
consequently to each element of the enumeration (5), we construct a new
mathematical object z=n+1, which, by Aristotel (not by Peano, since all
that was well known even to Pythagoras), is a finite natural number too,
and, by its construction, differes from all elements of the given
sequence (5). So, we have:

{not-B:} the mathematical object n+1 (- N, but not-[ n+1 (-(5) ].

    Thus, we have obtained the contradiction between the statements B
and [not-B]. Any meta-mathematicians, following to G.Cantor, claims
here: the contradiction proves that our assumption [not-A] was false,
and, consequently, by the classical Reduction ad Absurdum, the thesis A
is true.

    I think, Aristotel (if he were forced to consider that question)
should never have done such the hasty conclusion by the following three
reasons.
    1) Firstly, in that proof, there is an assumption, there is a
contrudiction, but there is not a classical "Reductio ad Absurdum" (RAA)
method. Because, that assumption is a formal, optional, i.e., removable,
element of the proof. In reality, we have here a proof of the following
DIRECT theorem: "any (arbitrary given) enumeration of elements of the
set N is not an enumeration, containing all natural numbers". In modern
time, S.C.Kleene approached to a true understanding of this non-trivial
fact. I remind his formulation of just the G.Cantor proof {S.C.Kleene,
"Introduction to Metamathematics", NY-Toronto, 1952, Part 1, Chapter 1,
paragraph 2}: "Assume now that

     x0, x1, x2, x3, ┘

is an INFINITE enumeration of SOME, BUT NOT NECESSARILY of ALL, real
numbers of the semi-interval (0,1]". As is easy to see, the Cantor
Theorem ({Thesis A:}"the set of all real number is uncountable") and the
Kleene assumption of the G.Cantor proof ("an INFINITE enumeration of
SOME, BUT NOT NECESSARILY of ALL, real numbers") are not contradictory
statements how that is strictly demanded by the classical RAA-method.
So, it would be, of course, a rough logic error, if here Kleene used
really that assumption for a "Reduction ad Absurdum" proof. But, in
reality, he proves the DIRECT theorem: "Any enumeration of real numbers
does not contain all real numbers". I think that here the outstanding
logician's intuition (right-hemispherical, geometrical thinking) was
victorious over his urge towards a logical punctuality
(left-hemispherical, rational-logical thinking). Though, alas, some
pages after, he calls explicitely another equivalent proof of the
G.Cantor Theorem (by means of an assumption on an 1-1-correspondence
between any set X and a set P(X) of all its subsets) by the "Reductio ad
Absurdum".

2) Secondly, in the modern propositional calculus it is proved the
famous theorem that anything follows from a(ny) contradiction (in
Russian: "iz protivorechija sleduet vsjo, chto ugodno"). That theorem
goes into any formal system that includes the propositional calculus (a
question: what formal systems do not include that calculus?). But in
Classical Aristotel Logic every contradiction has its own specific
reasons, its own specific structure, and  its own specific set of
consequences. Further, in Classical Aristotel Logic there are only two
method to PROVE the reliable FALSE of a statement, say, B: a) by the
laws of the negation and the excluded middle, from the proved truth of
not-B, and b) by the classical MODUS TOLLENS rule in a framework of the
RAA-method, from the proved false of a formal consequence, say, D of
that premis (assumption) B; at that the reliable false of the formal
consequence D itself is proved, by the laws of the negation and the
excluded middle, from the reliable truth of not-D; at that the reliable
truth of not-D itself is established independetly from and OUTSIDE of
the given formal deduction of the consequence D from the premis B (else
- the error "circle in a proof"). Just the last contradiction (between
these D and not-D) they mean in RAA-method saying the ritual words: "the
obtained contradiction proves the false of our assumption" {all these
and many other fine problems of the logical nature of the Reductio ad
Absurdum are considered in the paper: A.A.Zenkin, On Logic of Some
Quasi-Finite Reasonings of Set Theory and Meta-Mathematics. New Paradox
of Cantor's Set theory. - News of Artificial Intelligence, 1997, no.1,
64-98, 156-160. (in Russian)}.
    But what have we in the proofs of Cantor's Theorem and Theorems 1
and 2? - From a premise B, its formal concequence not-B is formally (or
constructively that is the same here) deduced, and then it is proved
that the concequence not-B is true. Then, by the law of the negation,
the false of the premise B is stated. So, here there is not the modus
tollens rule application that is the key stage of any Classical
RAA-method. What have we here? - We have here a very strange version of
a counter-example method: the single counter-example (not-B) itself is
FORMALLY DEDUCED from the UNRELIABLE common assumption or hypothesis (B)
which that counter-example (not-B) must disprove. It can be certainly
stated that the Classical Logic and Classical Mathematics do not know
such methods for reliable proofs.
    The fine moment. In original Cantor proof, there is a formal stage
of a conclusion from a proved false consequence B (not from the
contradiction between B and not-B) to the false of the assumption not-A,
almost by classical modus tollens rule. But as S.Kleene and I : -)
already said above, that assumpsion, not-A, is a formal, optional, fine
masking, unrelated to the proof appendix which can be painlessly
removed.

    Moreovere, the "inference" of the kind  B == > [not-B] is a half of
a paradox, which (the half) can be completed to the full classical
paradox: [B == > [not-B]] & [[not-B]  == > B]. But classical paradoxes
(the "Liar", Russel's ones and so on) have only a very far syntactical
relation to the Logic, and nothing more {for more info: A.A.Zenkin,
Automated Classification of Logical and Mathematical Paradoxes. On one
"Physical" Model of the "Liar" Paradox. - News of Artificial
Intelligence, 1997, no. 3, pp. 69-79. (in Russian); A.A.Zenkin,  The
Time-Sharing Principle and Analysis of One Class of Quasi-Finite
Reliable Reasonings (with G.Cantor's Theorem on the Uncountability as an
Example) - Doklady Mathematics, vol  56, No. 2, pp. 763-765 (1997).
Translated from Doklady Akademii Nauk, Vol 356, No. 6, pp. 733 -
735.(1997).}

3) Thirdly, the obtained above (see Theorem 2) contradiction between a
formal cosequence [not-B] and its premis B is easy solved without any
contradiction even with the assumption [not-A]. Following  Ancient
Grecian mathematicians, we construct a new finite enumaration

1, 2, 3, ┘ n, n+1.     (5.1)

and have

{B:} for any mathematical object k, if k (- N then k (- (5.1).

Now we can (more exactly - must) repeat that process up to (potentially)
infinity, and obtain already the known non-finite argumentation:

[not-A] == > B == > [not-B] == > B == > [not-B] == > B == > . . . (*)

And "there are neither logical nor mathematical arguments and reasons,
neither in heaven above nor in the earth benearth, nor in the water
under the earth, in order to stop (all the more, to jump over) this
infinite process (*) ever in the future", or, by Aristotel,

   So, some above, I formulated the Theorem 2: "The set N is infinite".
The fine peculiariry of its proof consists in that I shall not be able
ever to finish it  <according to Samuel S. Kutler 's and Aristotel's "A
Dia Ex Hodos = No way out by going through"> if, of course, I do not
wish to make the trivial logic mistake "jump to a conclusion", i.e.,
strictly speaking, I shall never be able to state Q.E.D. and to claim
that the set of "all" natural numbers is infinite. But in virtue of (*),
I can state (the true direct theorem) that none set of natural numbers
contains all natural numbers (of course, not because that the set is
uncountable, but) because the process of the construction of its
elements is not completed in principle, i.e., it is a potentially
infinite process, or, in other words, "Infinitum Actu Non Datur" in the
only rigorous sense: ANY set of natural numbers is a FINITE one, and
none set of natural numbers contains all natural numbers. The same
refers to the set of "all" points or of "all" real numbers of the
segment [0,1].

     Since "my" DIAGONAL method for natural numbers was known well to
Pythagoras and Aristotel who proved (in some other notation) the Theorem
2 on the Infinity of the series of finite Natural Numbers to his
students, I can't agree, for example, with S.C.Kleene, Hausdorff, and
many others,  that G.Cantor is the only author of the famous Diagonal
Method. :-)


    3) WHO DID INVENT THE SERIES OF TRANSFINITE ORDINALS?
    Meta-mathematicians  state that the famous series

1,2,3,┘,W,W+1,W+2,┘,W*2, W*2+1, W*2+2, ┘,
W^2,┘,W^3,┘,W^W,┘,W^W^W,┘,W^W^W^W^┘ (6)

where W is the G.Cantor Grecian "Omega", i.e., the least transfinite
ordinal number, was  invented by G.Cantor.
    According to "Abriss der Geschichte der Mathematik" von Dirk
J.Struik (Berlin, 1963. Chapter 4.3.), "The first known application {AZ:
in Europe} of the decimal positional number system refers to 595". It
means that Ancient Grecian mathematicians (including Aristotel) wrote
the common series of common finite natural numbers,

1,2,3, ┘ (7)

in the decimal NON-POSITIONAL system:

1,2,3,┘,10,10+1,10+2,┘,10+10, 10+10+1, 10+10+2,┘, 10+10+10, 10+10+10+1,
10+10+10+2, ┘ (8)

     Using modern shortenings (e.g., 10+10+10 = 10*3, 10*10 =10^2, and
so on), Ancient Grecian mathematicians could re-write the series (8)
(i.e., the same (7)) in the following much more compact form:

1,2,3,┘, 10,10+1,10+2,┘,10*2, 10*2+1, 10*2+2,┘,10^2,
┘10^3,┘,10^10,┘,10^10^10,┘,10^10^10^10^┘ (9)

Using any arbitrary radix, say, the last letter, Z, of the Latine ABC,
they could represent the notation (9), or, that is the same, the
initial  natural series (7) in the form:

1,2,3,┘,Z, Z +1, Z +2,┘, Z*2, Z*2+1, Z*2+2, ┘, Z^2,┘, Z^3,┘, Z^Z,┘,
Z^Z^Z,┘, Z^Z^Z^Z^┘ (10)

    It is easy to see that the notation (10) of the series (7) of common
finite natural numbers is in the symbol-by-symbol, 1-1-correspondence
with the G.Cantor series (6) of transfinite ordinal numbers up to the
famous transfinite ordinal, e0 ("epsilon-zero"). Of course, it is well
known that e0 is an countable ordinal, and there is nothing of a
surprising in a 1-1-corresponding between two series (6) and (7) of
equal cardinality, but it is appropriate mention here that the
constructed above 1-1-correspondence between notations (10) and (6)
preserves the elements order in (7), and such kind of the
1-1-correspondence between the series (6) and (7) is stated firstly
(after Aristotel's time, of course). However, I can be wrong: maybe some
natural series notations like the forms (9) or (10) are known in the
History of the Ancient Grecian Mathematics. Because, all that was so
natural in that far Aristotel's time.
    As is known, the ontological status (essense) of the finite natural
numbers has its roots in the real world (see, for example, From: Jon
Barwise <barwise at phil.indiana.edu>, To: Foundations of Mathematics
<fom at math.psu.edu>, Subject: FOM: Feferman's 10 theses; Date: Mon, 5 Jan
1998 16:57:37), the ontological status of the transfinite integers is
exhausted by their position in a notation of the form (6) in the sense
that there is nothing in the real world what the transfinite integers
might be associated with. In other words, the transfinite ordinal
integers have no sense outside of a notation (6). Maybe, G.Cantor's
transfinite ordinals will raise their ontological status thanks to the
similarity of (6) to (7)?
    Of course, there are a lot of different concrete notations of the
series of G.Cantor's transfinite integers that is depending only from an
endurance of a writer. So, for example, my notation (6) has a
"dimension" of about 1 x 10 cm^2, S.Kleene's notation in his
"Introduction to Metamathematics" (the Russian edition, p. 421) has a
"dimension" of about 5 x 10 cm^2, P.S.Alexandrov's notation in his
"Introduction to the Common Theory of Sets and Functions" (the Russian
edition, p. 78) has a "dimension" of about 9 x 10 cm^2, and so on. But
at any case, the following trivial  theorem takes place: for any given
notation of the series of G.Cantor's transfinite integers, such the
finite radix Z can be taken that the corresponding notation (10) of the
series (7) of the common finite natural numbers will be
1-1-corresponding, order-isomorphic, similar to the given notation of
the G.Cantor series of the transfinite ordinals. {For more info see:
A.A.Zenkin, Whether the Lord exists in G.Cantor▓s Transfinite Paradise?
√ News of Artificial Intelligence, 1997, No. 1, стр. 156-160.}

    One my opponent who distinguishes badly a formal system of objects
(for example, by S.C.Kleene) from different possible interpretations of
such the system, was repeating quite a long time: "But the symbol "┘" in
(10) denotes a finite quantity of numbers, whereas the same symbol "┘"
in (6) denotes an infinite quantity of numbers ┘?" - Especially for him,
I suggested some another method to construct a 1-1-correspondence
between (6) and (7) under the simultaneous preservation of the finite
numbers order in (7), and the quantity of all transfinite numbers in (6)

    The method, - shortly called the transfinite cavitation method {see:
A.A.Zenkin, Transfinite Cavitation in the Ranks of G.Cantor's Ordinals.
- News of Artificial Intelligence, 1997, no. 3, pp. 131-137.}, - is
based on the well-known real G.Cantor invention (S.C.Kleene calls it by
"a matrix method") that allows to transformate any WxW-type array (for
example, the set of all positive rational numbers) into an alone W-type
sequence like (7).
    So, consider the initial semi-interval [1, W^2) of the Cantor series
(6). It can be represented as the matrix (11):

   0,         1,         2,         3, ┘
 W,    W+1,   W+2,   W+3, ┘
W2, W2+1, W2+2, W2+3, ┘
W3, W3+1, W3+2, W3+3, ┘
. . .

   Using G.Cantor's "matrix method" we have the following W-type
sequence like (7):

0, W, 1, 2, W+1, W2, W3, W2+1, W+2, 3, and so on      (12)

   So, the initial semi-interval [1, W^2) of the W^2-type is cavitated
into a one W-type sequence (12) under the preservation of the quantity
of all transfinite numbers in [1, W^2). Obviously, that all similar
semi-intervals [ nW^2, (n+1)W^2), n=1,2,3,┘, of the series (6) can be
cavitated by the same way into similar W-type sequences like (12), or,
that is the same, like (7). So, now the initial semi-interval [1,W^3) of
(6) is a countable set of countable subsets of the W-type, i.e., it is a
matrix like (11). Using the G.Cantor matrix method, we cavitate the
initial semi-interval [1,W^3) into a W-type sequence like (12), or like
(7). The same referes to all intervals [nW^3, (n+1)W^3). Continuing that
process, we shall cavitate all the G.Cantor series (6) into an only
W-type sequence, which will contain all transfinite numbers of the
initial series (6) up to the famous transfinite ordinal number
"epsilon-zero", of course, by an other order. So, we obtain an
1-1-correspodence between all elements of the G.Cantor setries (6) and
the series (7) under the order preservation in the last.
    Further, if somebody will wish to continuate a construction of the
G.Cantor series (6), for example, in such the manner:

e0, e0+1, e0+2, ┘, e0*2, ┘, e0^2, ┘, e0^e0,┘,e0^e0^e0,┘, e0^e0^e0^┘ ,
(6.1)

where the last countable ordinal e0^e0^e0^┘ might be denote as, say, e1
("epsilon-one"), the transfinite cavitation method allows to cavitate
the countable series (6.1) into an alone W-type sequence like (7).
    In one word, going along the series like (6.1) for
e1,e2,e3,┘,f0,f1,f2,┘,g0,g1,g2,┘, and so on, the transfinite cavitation
method allows to cavitate the series of ALL COUNTABLE ORDINALS into a
one W-type sequence like (7), say, such one:

0, $1, $2, $3, ┘, W1   (13)

where the W1 ("Omega-one") is a transfinite ordinal that follows ALL
COUNTABLE transfinite ordinals, i.e., by G.Cantor's definition, that is
the least uncountable transfinite ordinal and its cardinal number is the
"aleph-1". But on the other hand, the cardinal number of the W-type
sequence 0,$1,$2,$3, ┘ is equal to the "aleph-0". So, we obtain some
quite strange equality:

"aleph-0" = "aleph-1".

As it is easy to see, the last result is too powerful for any
contradiction-free system, or, that is the same from the Classical
Aristotel Logic point of view, for any consistent system.


    4) ON THE INCOMPLETENESS OF THE CLASSICAL ARISTOTEL LOGIC.

    Consider a deductive inference ( = a logical proof) of a consequence
B from a premise A, or shortly:
       A == > B          (**)
As is known, the main epistemological paradigm of the Aristotel Logic
sounds so.

POSTULATE 1. Under the correct using of the Aristotel Logic deductive
rules, IF A is TRUE, THEN B is necessarily TRUE too, or shortly: TRUTH
== > TRUTH (only!).

    Remind, that the Postulate 1 is not a proved theorem, it is a very
plausible empirical statement (axiom), only, that was breaked never,
long before Aristotel and about 2300 years after him.
    What does the Classical Logic state when A in (**) is FALSE?
    It states very cautious thing: IF A=FALSE then its logical
consequence B is UNRELIABLE. What does it mean? It means that the
corresponding inference (**) proves nothing, and B can turn out to be
FALSE as well as TRURE. When B is FALSE, all is obvious and not
interesting here (but see below). But what is the case when A=FALSE, but
B is TRUE? The huge scientific experience during the same time period
showed that the last, i.e., FALSE == > TRUE,  can be in the following
two cases only:
    a) when A is a non-essential (i.e., removable) "premise" for B, and
in such the case the "premise" A must be, simply and smoothly, removed
from the "proof". So, we have here the trivial gross logical error "It
does not follow" (in Russian "Ne sleduet");
    b) in the proof process, the other (usually, much more fine) logical
error takes place - the error "substitution of notions (terms) " (in
Russian "Podmena ponjatii").
    It allows to state that the "inference" ("proof") of the kind  FALSE
== > TRUE is possible in Classical Logic or in Classical Mathematics,
based upon that Logic, only if such the "inference" ("proof") breaks
grossly the main deductive inference rules of Aristotel's Logic.
    Further, what is the most terrible case for the RAA-method
applications? - The most terrible case for the RAA-method applications
is a case when the (unreliable!) assumption of the RAA-proof will occur
a true statement! - Why? - Because in such the case, by virtue of the
Postulate 1, we shall never obtain a disirable FALSE consequence and
therefore we shall never finish our proof. However, why, though that
fatal possibility, the RAA-method is widely applyed in Classical Logic
and in Classical Mathematics? Because the 2500 years experience of the
successful RAA-method usage in logical and mathematical proofs convinced
us of that if the RAA-assumption, say, B is FALSE then, deducing its
FORMAL sequences, we will obtain, sooner or later, such a formal
sequence, say, D which negation not-D is (or can be proved that is) a
reliable Truth. The last will allow us to prove the reliable FALSE of D,
by the laws of the negation and the excluded middle, and then, by modus
tollens rule, to prove the reliable FALSE of premis B.
    All these (and many other) reason allows to formulate the "new" (or
old, since the RAA-method was known long before the Aristotel time)
axiom of Classical Logic.

    POSTULATE 2. If a premis A in the formal inference (**) is FALSE,
then any its consequence B is necessarily FALSE too, or shortly: FALSE
== > FALSE (only!). IFF the Aristotel Logic deductive rules are applyed
correctly.

    Why did Great Aristotel not formulate explicitely such the obvious
Postulate 2? I think the reason is purely psychological: like many
centuries later Great Gauss was afraid of "Biothiers cry" (in Russian
"Krik Biotiytsev") apropos of his non-Euclidean Geometry, Aristotel was
afraid of a cry of his own Biothiers, i.e., sophists, who,
professionally using just the "rule" FALSE == > TRUE, robed shamelessly
purses of respectable citizens  in law-courts of the Ancient Greece
(some of [HM]-people of Moscow University give their view that primarily
Aristotel was creating his Logic not for a pure science, but just for a
judicial defence of his nationals.).
    It is evident that the Postulat 2 contradicts flatly to the
definition itself of the implication notion of the propositional
calculus which admits the case FALSE  == > TRUE as a true formal
inference. Since, as said above, any formal system of modern
meta-mathematics includs the propositional calculus axioms system as a
subsystem of its own axioms system, it can be said that all modern
formal systems theory (meta-mathematics) based on the case FALSE  == >
TRUE. In such the case, modern meta-mathematics and Classical Aristotel
Logic, which was, is, and will be always the main basis of the really
working and really verified, experimental and theoretical Science, are
quite different things.

Best regards,

A.Z.

############################################
 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 com2com.ru
 WEB-Site   http://www.com2com.ru/alexzen
 ############################################
 "Infinitum Actu Non Datur" - Aristotel.