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 Aristotle'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, tries 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 a Great Logician as Aristotle.
Below, I will make some remarks apropos of that Aristotle's 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 actually 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 ARISTOTLE, 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 in X, i.e.,
{B:} for any d: if d in X then d in (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 wishes to preserve the MATHEMATICAL definition of the real number
notion as an INFINITE 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"
applied to integers can't lead 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 to 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
have been stated before the difficult choice: either to accept Cantor's
"omega", or to reject Dedekind's and Weierstrass's conception of the
real number.
However that may be, all said above contradicts cardinally to
Aristotle's and St. Thomas Aquinas's 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 to 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 ([HM] Fri, 12 Feb 1999 11:46:17) writes:
> These < AZ: Aristotle's > 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
Aristotle's appeals to physics, psychology, and philosophy (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 Aristotle's
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. MATHEMATICAL PROOF OF THE ARISTOTLE'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 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 the
notions such as natural number, real number, and a sequence of such
numbers, that were well known in Aristotle's time. So, the G.Cantor
proof is a classical logic proof only! - But in such a case, Cantor's
Theorem might be proved in Aristotle's time!?
Indeed, of course, not Cantor's Theorem itself, but its very close
analogue (similarity?) was really well known in Aristotle'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 infinite 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, such 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 Aristotle's time. For
ancient Greek 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 easily rewritten as a proof by
means of just a DIAGONAL method.
THEOREM 2. {Thesis A:} The set N is an infinite 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:
1, 2, 3, ... n. (5)
So, we have
{B:} for any mathematical object k, if k in N, then k in (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 Aristotle (not by Peano, since all
that was well known even to Pythagoras), is a finite natural number too,
and, by its construction, differs from all elements of the given
sequence (5). So, we have:
{not-B:} the mathematical object (n+1) in N, but not-[ (n+1) in (5) ].
Thus, we have obtained the contradiction between the statements B
and [not-B]. Any meta-mathematicians, following G.Cantor, would claim
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, Aristotle (if he were forced to consider that question)
should never have done such a hasty conclusion by the following three
reasons.
1) Firstly, in that proof, there is an assumption, there is a
contradiction, 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's
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 explicitly 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 Aristotle's Logic every contradiction has its own specific
reasons, its own specific structure, and its own specific set of
consequences. Further, in Classical Aristotle's 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 premise (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 independently from and OUTSIDE of
the given formal deduction of the consequence D from the premise 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 consequence not-B is formally (or
constructively that is the same here) deduced, and then it is proved
that the consequence 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's 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 assumption, not-A, is a formal, optional, fine
masking, unrelated to the proof appendix which can be painlessly
removed.
Moreover, 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", Russell'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 consequence [not-B] and its premise B is easy solved without any
contradiction even with the assumption [not-A]. Following ancient
Greek mathematicians, we construct a new finite enumeration
1, 2, 3, ..., n, n+1. (5.1)
and have
{B:} for any mathematical object k, if k in N, then k in (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 beneath, 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 Aristotle,
So, above I formulated Theorem 2: "The set N is infinite".
The fine peculiarity of its proof consists in that I shall not be able
ever to finish it <according to Samuel S. Kutler's and Aristotle'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 no 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 well known to
Pythagoras and Aristotle 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^3, ..., W^W, ..., W^W^W, ..., W^W^W^W^... (6)
where W is Cantor's Greek "Omega", i.e., the least transfinite ordinal
number, 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 the year 595". It means
that ancient Greek mathematicians (including Aristotle) 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 shortened forms (e.g., 10+10+10 = 10*3, 10*10 =10^2, and
so on), ancient Greek mathematicians could re-write 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 Latin ABC,
they could represent the notation (9), or 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 to 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 Aristotle'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 Greek Mathematics. Because, all that was so
natural in that far Aristotle's time.
As is known, the ontological status (essence) of the finite natural
numbers has its roots in the real world,
[see, for example, a posting at
http://www.math.psu.edu/simpson/fom/postings/9801.45
From: Jon Barwise <barwise@phil.indiana.edu>
To: Foundations of Mathematics <fom@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, 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 transform 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,
which 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 refers 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-correspondence between all elements of the G.Cantor series (6) and
the series (7) under the order preservation in the last.
Further, if somebody wish to continue a construction of the G.Cantor's
series (6), for example, in such a 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 denoted 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
Aristotle Logic point of view, for any consistent system.
4) ON THE INCOMPLETENESS OF THE CLASSICAL ARISTOTLE'S 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 Aristotle's Logic
sounds so.
POSTULATE 1. Under the correct using of the Aristotle's 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 broken never,
long before Aristotle 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 TRUE. 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 Aristotle'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 desirable FALSE consequence and
therefore we shall never finish our proof. However, why, though that
fatal possibility, the RAA-method is widely applied 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 premise B.
All these (and many other) reason allows to formulate the "new" (or
old, since the RAA-method was known long before the Aristotle's time)
axiom of Classical Logic.
POSTULATE 2. If a premise A in the formal inference (**) is FALSE,
then any its consequence B is necessarily FALSE too, or shortly:
FALSE ==> FALSE (only!). IFF the Aristotle's Logic deductive rules are
applied correctly.
Why did Great Aristotle's not formulate explicitly such obvious
Postulate 2? I think the reason is purely psychological: like many
centuries later the Great Gauss was afraid of "Biothiers cry" (in Russian
"Krik Biotiytsev") apropos of his non-Euclidean Geometry, Aristotle 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 Greek (some historians
from the Moscow University gave their view that primarily Aristotle was
creating his Logic not for a pure science, but just for a judicial
defence of his nationals.).
It is evident that the Postulate 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 includes 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 Aristotle's Logic, which was, is,
and will always be 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@com2com.ru
WEB-Site http://www.com2com.ru/alexzen
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"Infinitum Actu Non Datur" - Aristotle