THE PROBLEM OF COUNABILITY OF
HIGHEST ORDINALS
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A.P. Kulaichev, 2007-2014
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Kulaichev A. P. The Problem of
Countability of Highest Ordinals. Pure and Applied Mathematics Journal.
2016. 5(4): 108-112. 
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Mathematics Subject Classification (2010) 03E10
Abstract.
In this study we use the alternate point of view on the structure of ordinals,
according to which each ordinal is the union of non-intersecting foregoing
segments of ordinals of equal exponentiation. Each ordinal wn=w1xw2x...xwn
is seen as wn=Uwji,
i=n-j+1
for any j=1,n-1 instead traditional union of foregoing intersecting
segments of ordinals of consistently increasing exponentiation
wn=Uwi,
i=1,n.
The first form corresponds to the geometric representation of ordinal wn
as an infinite n-dimensional matrix. According traditional
formulation ww=Uwi,
i=1,w,
thus ww is w-countable
union of countable ordinals so ww
is countable. According to alternate formulation ww=Uwni
, i=1,ww for any n,
thus ww is ww-union
of ordinals and the findings will be different.
These
findings are: 1) the proof of countability of countable union of countable
ordinals can not be directly or inductively transferred to its first limit
ww-union;
2) ww seems to be the first uncountable
ordinal with its power is equal to continuum; 3) the subsequent ascending
degrees of w-exponentiation of ww,
i.e. ww^w,
ww^(w^w),
... , correspond to consecutive À2,
À3,
À4,
... cardinals; 4) from here it also follows the direct justification of
continuum hypothesis.
Our study
shows that in the domain of transfinite sets different points of view and
its findings have the legal right to coexist as Nels Bohr's principle of
complementarity in physics.
Keywords:
set theory, ordinals, mathematical induction, absolute truth
1. INTRODUCTION
(1).
Ordinals or ordinal type sets except usual attributes, such as cardinality
and well-orderliness, also possess certain structure
as they consist of consecutive segments. And there can be different points
of view on structure of ordinals. For example according traditional interpretation
ordinal ww is defined as countable
sum or union of intersecting countable ordinals, therefore ww
is also countable, we quote [4]: «ww
is countable so far as ww = løü wi
= lim{w1,
w2,
w3,…}
it is the number, which owing to equalities 1+w
= w, 1+w+w2
= w(1+w) = w2,
etc. can be written down [by analytical transformations] as: ww
= 1+w+w2+w3+…
= Swn , n=1-w
»
[end of quote]. However, there is absolutely overlooked that the summation
of ordinals in its ascending order from left to right is identically equal
to the last on the right greatest ordinal due to the non-commutativity
of addition operation for ordinals. So the above expression for ww
is not the proof but only the tautology ww=ww.
Other proof
of ww countability is also based
on analytical transformation with decomposition of ordinals on other basis,
we quote [5]: «a = w2+w5+9
is the decomposition of a on w
basis. To spread out the same number on basis 2, it is enough to notice
that w = lim 2n = 2w.
Then w2 = (2w)2
= 2w2, w5
= 2w+2+2w,
from where we get a = 2w2+2w+2+2w+23+20.
By the same way we get ww = 2(w^2)
» [end of quote]. Taking into account w2
= 2w2 = 4w
= w we can continue ww
= 2(w^2) = 2(4^w)
= 2w = w.
The last view
of ww structure is significantly
different from the previous one and the both of them contradict the standard
description of ordinals in Cantor normal form [2]. Besides it is doubtful
that such formal transformations are applicable to any ordinals because
many other properties of operations with finite numbers aren't transferred
to ordinals, e.g.: 1+w ¹ w+1, 2w
¹ w2 and so on. Note also that such formal, inductive "proofs" can
be generated the next, for example: since 2w
= 4w = … = nw
= w and ww
= lim nw, then ww
= w.
(2).
Moreover, it is well known that in some sequence of ordinals {a,...,b,...,c}
some property P(b), which is inherent
to all ordinals b<c,
is not transferred to c if c
is the limit ordinal for all b<c.
The examples of such "intolerable" properties in {1, 2, 3, ..., w}
for its w-limit are: 1) Q(n) is
"n is the number represented by finite digits" but ØQ(w);
2) R(n) is "(n+1)/n>1" but ØR(w);
3) S(n) is "n+1=1+n" but ØS(w);
3) and so on. Below we also give the examples of not transferred properties
for ordinals of higher power. In such a situation the proof of any property
P(c)
has to be made by noninductive means. The same also concerns to the property
"ordinal c is countable".
(3). Let us look at the structure of ordinals with an alternative
point of view, where each ordinal is the union of non-intersecting sets.
Ordinal w is Un,
n=1,w,
w2=Uwn,
n=1,w,
w3
= Uw2n,
n=1,w
= Uwn,
n=1,w2
and so on. And finally,
ww=Uwni
, i=1,ww for any i=1,
2, 3, ... . We see that any wi, i=1,
2, 3, ... is at its maximum the w-union of wi-1.
But ww can not be at its maximum
the union of ww-i because ww-i=ww
and for any i=1, 2, 3, ... . Let us notice that this point of view
ascends to Cantor considering of w2
ordinal structure [1]. Thus, in a case of ww
we note the leap from all w-countable-unions
to ww-union. Hence the proof of countability
of w-countable-union of countable ordinals can
not be directly or inductively transferred to its first ww-union.
2. TWO METHODS OF RENUMBERING
OF ELEMENTS OF ORDINALS
The fist infinite
ordinal w designates the set of natural numbers
which is countable by definition.
Ordinal w2 is the
limit for the sequence 1, 2, 3, 4, ..., w, w+1,
w+2,
w+3,
... and the proof of its countability is made by direct reordering of its
elements therefore it is reduced to the countable sequence 1, w,
2, w+1, 3, w+2, 4,
w+3,
... . The same method is applicable for subsequent ordinals w3,
w4,
w5,
... .
For ordinal
w2
being a limit for the sequence
w, w2,
w3,
... the above mentioned method of direct reordering is inapplicable and
the proof of its countability (according its w2=Uwj,
j=1,w
representation)is
made by Cantor pairing function [1],
p(i,
j) = 0.5(i+j)( i+j+1)+j,
(1)
where: p(i, j)
– new natural order number of w2
element;
i – order number of element in wj
set;
j – order number of wj
set.
This method is equivalent to counterdiagonal
renumbering of two-dimensional and infinite in two directions matrix which
includes the elements of its non-intersecting subsets {1, 2, 3, ... },
{w, w+1, w+2,...
}, {w2, w2+1, w2+2,
... }, ... .
For ordinal w3
the corresponding set can be represented by a 3-dimensional and infinite
in three directions matrix. For other ordinals wn
its elements can be represented in n-dimesional infinite matrixes
wn=Uwjk...n,
j,k,…,n=1,w.
And for these ordinals it may be possible to use two various methods to
proof its countability.
Method 1
is based on counterdiagonal renumbering of wn
according its n-1-dimentional counterdiagonal planes. This procedure
is decribed by Cantor tuple function [6] according recursive equation:
p(n)(i,
j, k,...,n) = p(p(n-1)(
i,
j, k,...,n-1),n).
(2)
Method
2 is based upon step-by-step reducing of dimensionality of initial
matrix represented of wnset.
As ordinal w3 includes w
ordinals w2 then on the first step
we perform counterdiagonal renumbering of w
two-dimensional w2 matrixes, therefore
we obtain the single w2 matrix which
on the second step is reduced by counterdiagonal renumbering to single
natural set. To proof the countability of any wn
it demands to perform n-1 of similar steps to decrease dimensions
of initial matrix.
3. TWO OPPOSITE CONCLUSIONS CONCERNING
ww
COUNTABILITY
Using these two
methods it is possible to come to the opposite conclusions concerning ww
countability.
Proposition-1. ww
is not countable.
The proof is made directly by method
1.
(1). Because ww
is representable by infinite matrix with infinite dimensions, therefore
its first counterdiagonal hyperplane contains infinite number of elements
ww-1=ww,
so here the process of renumbering will be finished.
(2). For ww
as the limit of wn,n=w
we
have definition of Cantor tuple function (2):
p(w)(i,
j, k,...,w) = p(p(w-1)(i,
j, k,...,w-1), w)
= p(p(w)(i,
j, k,...,w), w),
(3)
So we get the equation which is unsolvable and the renumbering of elements
of ww
set is not possible.
Thereby the proposition-1
have been proved from both sides.
We can formulate objection and counter-objection against these reasoning.
Objection.
The number of elements mn,k in such counterdiagonal hyperplanes
of wn increase according to
the law of figurate numbers [3]:
mn,k
=(k+1)(k(n-2)+2)/2,
(4)
where k =0,1,2,... is the order number
of counterdiagonal hyperplane.
For any k-counterdiagonal-hyperplane
in ww, we have:
mw,k
= (wk +2) (k+1)/2 = (wk2+wk
+2k+2)/2 = wk2+wk
+k+1,
(5)
e.g.: mw,2=w5+3,
mw,3=w10+4,
mw,4=w17+5,...,
mw,w=w3+w2+w+1.
As in ww
we have only w counterdiagonal hyperplanes then
ww
is countable union of countable sets.
Counter-objections.
(1). According
to formula (5) ww is the union of
sets raised not more then to 3-th power. This goes against the nature of
ww
set and against the Cantor normal form [2].
(2).
Except counterdiagonal renumbering of elements of ww
set it is possible the alternate renumbering «on covers», e.g. for 2-dimentional
infinite w2 matrix:
1
2 5 10 ...
4 3
6 11 ...
9 8
7 12 ...
16 15 14 13 ...
... ... ...
... ...
Each k=0,1,2,3,.. cover in n-dimensional
wn
matrix includes en,k=(k+1)n-kn
elements, i.e. in ww matrix each
k-cover
includes ew,k=(k+1)w-kw=w-w=0
elements.
Summary.
Thus we see that depending on renumbering order of ww
elements we get different conclusions concerning ww.
But absolute mathematical truth should not depend on renumbering order
of ww matrix elements. It once again
shows that any formula or property derived for natural numbers, as well
as formula (5), can not be directly or inductively transferred to its w
limit, as it have been stated in the introduction. Therefore similar transfer
in the above-stated objection is incorrect, as well as the objection itself.
On the contrary,, the proof (2) of the proposition-1 clearly shows that
the rule of renumbering of previous wn
ordinals, which is going back to Cantor, can not be extended to their n=w-limit.
Proposition-2.ww
is countable.
The proof is made by method 2 of step-by-step
decrease of dimensions in infinite sequence w2,
w3,
w4,
..., ordinals enclosed each other up to ww.
We can
formulate four objections against these reasoning.
Objections.
(1).
This proof implicitly uses inductive procedure to transfer the property
of all elements of ordered set w2,
w3,
w4,
... to its unattainable ww limit.
That can lead to a wrong result as it noted in the introduction.
(2).
To prove that w3 is countable it
is previously necessary to renumber w copies
of w2 matrixes in w3
having received as a result a single w2
matrix which then is reduced to one natural row, that is to execute w+1
of counterdiagonal renumbering of w2
matrixes. For w4 it is necessary
in the same way to renumber w copies of w3
matrixes totally execute w2+w+1
renumbering of w2 matrixes. For wn
it is necessary to execute wn-2+wn-3+...+w
+1 of such renumbering of w2
matrixes. For ww it is necessary
to execute not less than ww-2 renumbering
of w2 matrixes, that is equal to
ww
renumbering. Thereby to prove that ww
is countable it is necessary to execute such number of renumbering which
countability we are going to prove. Therefore this proof, which implicitly
uses the still an unproven subject of the proof, is unconvincing.
(3). Let's
look at the problem from the other side. ww
set contains not less than ww-2 of
w2
sets. Thereby if we reduce each
w2
set to w set we obtain ww-2
set i.e. again initial ww set
as unsolvable "begging the question" arisen.
(4). Method
1 is based on Cantor tuple function (3) which allows us unambiguously
to calculate for each element of wn
its position in a resultant natural sequence. On the contrary the inverse
Cantor tuple function allows us uniquely to decompose natural numbers set
to wn set. Method 2
for any ordinal wn does exactly
the same, i.e. it formulates another tuple function with the same properties.
But proof (2) of proposition-1 states that for ordinal ww
the tuple function and its inversion does not exist.
On the adduced
four arguments we did not find any counter-objections.
Summary.
We considered two antithetics about countability of ww
ordinal and their proof. On ratios of possible objections and counter-objections
the proposition-1 is more convincing. For a final choice from these
two alternatives it is desirable to find such property of ww
ordinal, proceeding from the used paradigm of its matrix representation,
which proof can't be disproved neither by method 1, nor by method
2.
4. THE CARDINALITY OF ww
Theorem.ww
cardinality is equal to cardinality of powerset of natural numbers or continuum
2À0=ñ.
Proof.
Powerset of natural numbers includes: all subsets of one number, all subsets
of two numbers and so on up to all subsets of infinite natural numbers.
On the other hand, all subsets of two numbers are representable in a two-dimensional
infinite w2 matrix and the first
line of this matrix contains subsets of one element.
{1}
{2} {3} {4} {5} ...
{1,2} {1,3} {1,4} {1,5}
{1,6} ...
{2,3} {2,4} {2,5} {2,6}
{2,7} ...
{3,4} {3,5} {3,6} {3,7}
{3,8} ...
{4,5} {4,6} {4,7} {4,8}
{4,9} ...
...
... ... ...
... ...
All subsets of
three numbers are representable in 3-dimensional infinite w3
matrix at which the first plane contains subsets of one and two elements.
And so on to all subsets of natural numbers which is representable by infinite
matrix with infinite number of dimensions and this matrix is corresponded
to ww set.
Thereby this
theorem have been proved.
Consequence
1. Continuum hypothesis follows from this proof because in the
ascending and continuous sequence of all countable transfinite ordinals
w,...,
w2,...,
w3,...,
wn,...
ordinal ww is the first noncountable
one and it is equal to cardinality of powerset of natural numbers 2À0=ñ.
Therefore its cardinality is the first one after À0
i.e. À1.
Consequence
2. The subsequent ordinals of ascending w-exponentiation
of ww represent powerset of ordinals
of previous degree of w-exponentiation
of ww thereby w(w^w),
w(w^(w^w)),
... correspond to cardinals À2, À3,
À4,
… . Thus, in a case of cardinal À2=2À1
we should apply the proof of theorem to transfinite matrix of ww-length
instead of w-length and with ww
number of dimensions instead w number
of dimensions. This matrix represents ordinal w(w^w).
And so on.
5. DISCUSSION
At the turn of
XX century in the history of mathematics it has been occurred the significant
but not noticed event. Prior to that mathematics revealed or took from
some unknown storage the absolute truths such as multiplication table,
squaring the circle, Fourier transform, etc. But in the last third of XIX
century in mathematics there were began to appear theories and hypotheses.
But any theory represents only some set of initial intuitively or substantially
reasonable provisions intended by means of certain rules and conclusions
logically convincingly and consistently to explain some properties and
phenomena of external or mental world. Thus for the similar explanation
may be used several different theories and the adoption of one or another
theory is not a consequence of its absolute truth, but a matter of public
agreement. One or other social community might prefer one or another theory
and follows it in accordance with the freedom of its choice. The same situation
has occurred in respect of numerous set theories [7].
Moreover the
degree of public recognition of any theory in natural sciences area is
determined by predictive adequacy of concrete theory, since these predictions
are usually available for further experimental testing. Mathematics is
not a science with possibility of experimental verification of its theoretical
provisions in area of set theories. Instead of experimental validation
of mathematical findings the only criterion of its truth remains the consistency
of its conclusions in the framework of the theory itself, but does
not the presence of contradictions with the conclusions of other theories.
Further, any
result in any theory can not be wrong iff it is not consistent with other
results in other theory obtained from a different point of view and using
a different proof. Results can be wrong iff the point of view is wrong
or the proof includes errors. Indeed it is impossible to disprove the theorems
in Euclidean geometry on the base of results in spherical geometry. In
such a situation, at the degree of public acceptance of one or another
set theory and its results there begin to influence professional consents,
political considerations and teleological grounds, which is not always
compatible with the dispassionate scientific approach to the evidences.
Numerous subtleties
of such public consents in most accepted version of set theory force some
professors and lecturers clearly to warn their audience about these subtleties
in their textbooks [8], we quote: "In fact, we have already reached a dangerous
border where visual representation of sets lead to a contradiction", and
also: "Here again we come to the dangerous border of paradoxes and so we
have to be expressed evasively".
6. CONCLUSUON
As we above have
seen the question of countability or uncountability of some ordinals also
depends on a particular viewpoint on the structure of such ordinals. So
in most popular set theory according to the axiom of exponentiation of
ordinals it is considered that ordinal wn
is the union or sum of intersecting segments of ordinals of previous sequence
of w-exponentiation, i.e. wn=Uwi,
i=1,n
or wn=Swi,
i=1,n
[4] that ascends to Cantor natural form [2]. According to that ww=Uwi,
i=1,w
and ww seems like w-countable
union of countable intersecting ordinals, so ww
is also countable. But it is possible the alternate point of view: wn=Uwji,
i=n-j+1
for any j=1,n-1, thus ww=Uwni,
i=1,ww
for any n, so ww is ww-union
of foregoing non-intersecting segments of ordinals of equal exponentiation,
that leads to the conclusion that ww
is uncountable.
Our study shows
that in the domain of transfinite sets the different points of view and
its findings have the legal right to coexist as Nels Bohr's principle of
complementarity in physics. Acceptance or rejection of a particular viewpoint
in this area is not the subject of absolute truth but the subject of a
particular public agreements.
REFERENCES
[1] Georg Cantor, 1878, Ein beitrag zur
mannigfaltigkeitslehre. Journal fr die reine und angewandte Mathematik,
84, 242-258.
[2] Georg Cantor, 1883, Grundlagen
einer allge.meinen Mannigfaltigkeitslehre: Ein mathematisch-philo-sophischer
Versuch in der Lehre des Unendlichen. Leipzig: Teubner.
[3] Deza, E., Deza, M., 2011, Figurate
Numbers. World Scientific.
[4] Felix Hausdorff, 1914, Grundzuge
der Mengenlehre. Veit and Company, Leipzig.
[5] Kuratowski K., Mostowski A., 1967,
Set theory. Amsterdam.
[6] Lisi M., 2007, Some remarks on
the Cantor pairing function. Le Matematiche, 62(1), 55-65.
[7] Maydim A. Malkov, 2014, Stairs
of Natural Set Theories, Pure and Applied Mathematics Journal. 3(3), 49-65.
[8] Vereshchagin N.K., Shen A., 2002,
Lectures on mathematical logic and theory of algorithms. Moscow, MCNME.
