Evgueny Kadissov
A disproof of the continuum cardinality existence diagonal "proof" removes an obstacle to reunification of mathematicians
Abstract
Mathematicians of all attitudes unite!
The known Georg Cantor's diagonal proof (DP) with his continuum cardinality (CC) and continuum hypothesys (CH) splitted community of mathematicians.
Mathematicians not accepting CH began to propose some different from classical variants of grounds: finitism, intuitionism and constructivism.
On the other hand we do not know any attempts to see what is the number that will be a result of Georg Cantor's diagonal procedure. It turns out that in this way lays a simple disproof both of Georg Cantor's diagonal proof and his continuum hypothesis. So long as a result of a diagonal process is related to the enumeration from which it is constructed in the same manner as the limit of a sequence is related to the sequence.
As attempts to prove or disprove DP, CC and CH have no success up to now a proposed disproof must at last give mathematicians a ground to unite.
Key words.
Georg Canor's set theory, countable sets, uncountable sets, continuum hypothesis, cardinality of the continuum, real, rational, irrational and transfinite numbers.
UDC 510.22.
Forword.
The known Georg Cantor's diagonal proof (DP), his cardinality of the continuum (CC) and his continuum-hypothesis (CH) splitted community of matematicians.
The matter is that Georg Cantor's proof of uncountability of reals is based on a rather simple and in appearance convincing procedure. Let us suppose an opposite (a set of reals is countable) and arrange all reals one under another in the order by which we counted them (further - counted reals - CR). Then we will compose a number taking from these numbers digits laying on a diagonal and changing them. It is evident that this new number differs from each previous number (further - numbers differing from previous numbers - NDFPN). From this G. Cantor draws an appearantly natural conclusion that the set of reals contains too many elements to make a bijection of reals and naturals.
But if we accept CC we must accept all contradictions related with CH. Let us consider only some of them.
On one hand it is stated that there are sets (named uncountables) consisting of such a quantity of elements that one can not count them not because they contain infinitely many elements. E.g. there are infinite quantity of naturals but a set of naturals is considered countable.
On the other hand in spite of the fact that one can not count elements in such sets some special numbers (called transfinite numbers) are established to be equivalent to the quantities of elements in such sets.
More than that one must take it in consideration that there exists some hierarchy of these transfinite numbers.
But from the times of antiquity it is known that there is no such a natural number that one can not find a still bigger one. How to agree this fact with existence of transfinite numbers?
Many mathematicians have not accepted CH. CH was not accepted by such prominant mathematicians as e.g. Kronecker and Poincaré.
Mathematicians not accepting CH began to propose various differing from classical grounds of Mathematics that led to e.g. finitism, intuitionism and constructivism.
A question if CH is true was put by Hilbert at 1900 as first in his list of unsolved problems of mathematics.
One of various interpretations of CH states that there is no set whose cardinality is strictly between that of the integers and that of the real numbers. In other words a countable set is infinitely small as compared with any uncountable one.
For examle a set of irrationals on a segment as far as it is uncountable has incomparably more elements than a set of rationals on the same segment.
Attempts to prove or disprove CH were undertaken and earlier. Kurt Gödel (1940) has proved that a negation of CH cannot be proved in Zermelo-Fraenkel set of axioms with an axiom of choice (ZFC). Cohen (1963) has proved that CH cannot be proved in ZFC.
Hence ZF and CH are independent of each other. So in accordance with one point of view CH (in spite of all contradictions linked with it) one must add to the system of axioms.
On the contrary it seems to us that another point of view is more adequate to a mathematical reality. The point of view that the first Hilbert's problem is not solved. Hence a solution of the question whether CH is true one must look for in other fields.
A proof of an inconsistency of CH one can see in a work of prof. A. Zenkin (Russian А.А.Зенкин) (2000). It is evident to us that prof. Zenkin's proof was insufficiently convincing. And here we are going to remove this defect.
But before we will proceed we will make one remark. We must chose a numeral system. If we will chose decimal system with the help of Cantor's diagonal procedure we can construct many NDFPN. If we chose binary system we can construct only one NDFPN. This fact replenishes the number of contradictions mentioned above.
A question if the diagonal procedure is consistent must not depend upon a choice of numeral system. So we will chose the binary system because it helps to solve the ptoblem. In above mentioned work A. Zenkin also took binary system but his proof led to unlimited number of adding NDFPN that made his proof insufficiently convicing.
As far as we know up to now no one looked what is a number that we would have as a result of G. Cantor's diagonal procedure. It turns out that in this way lays a simple disproof both of Georg Cantor's diagonal proof and his continuum hypothesis.
Formulation of the problem.
Let us prove that G. Cantor's diagonal procedure is inconsistent and hence a set of reals in the interval [0-1] is countable. The set of the reals consists of two sets of countable rationals and uncountable irrationals. By this way we will prove that a countable set of reals is bigger than an uncountable set of iirationals. This is obviously a contradiction. This contradiction is solved only if CH is wrong and a set of uncountables is not such that it contains too many elements to be countable it is ill-ordered to be countable.
By this way we will show that Georg Cantors' procedure is not sutable to prove that there are sets containing too many elements to be counted.
The proof.
We will use binary system. But we will number reals so to more clearly show inconsistency of Georg Cantor's diagonal proof.
To proof uncountability of reals G. Cantor used interval (0, 1). To facilitate our problem we will expand our interval including to it its left boudary and so we will have a [0, 1) interval. So we will proceed our proof.
Any real number in the interval [0, 1) can be represented as a binary fraction with an infinite row of binary digits and a zero to the left of a point.
A = 0.a1a2a3a4a5...ai... (1)
Between these numbers there are such that have to the right from some other digit an infinite row of 0 or 1. These numbers can be naturally written with finite quantity of digits and not obligatorily added to the right of them a zero in brackets standing for an infinite row of zeroes.
A = 0.a1a2a3a4a5...an(0) (2)
where binary digits a1...an-1 can be 0 or 1, and a digit an only 1.
A real number zero we will pair with a number zero in a set of natural numbers.
Bijection of other numbers of two sets we will begin with a number that is represented with only one digit to the right of the binary point. This is a number: 0.1(0) or 0.1, which we will pair to a nanural number 1.
Now we will proceed with numbers that are represented with 2 binary digits to the right of the point. These are 2: 0.01 and 0.11. These will be paired to natural numbers 2 and 3.
Analogically we will number reals with 3 digits to the right of the point. These are 4: 0.001, 0.011, 0.101, 0.111. They will get numbers from 4 to 7.
Numbers with 4 digits are 8: 0.0001, 0.0011, 0.0101, 0.0111, 0.1001, 0.1011, 0.1101, 0.1111. They will get numbers from 8 to 15. After that we will number reals with 5, 6, 7 and so on digits to the right of the point.
So a number of real A = 0.a1a2a3a4a5...ai... is defined in such a way:
N = Σ∞i=1 ai*2i-1 (3)
Let us turn our attention upon a sequence or reals with numbers 2n-1. These reals have right after the point n ones in succession. So this sequence has as a limit a real equal to 0.1111111111... или 0.(1).
For our goal we will show several first numbers in our enumeration.
0 - 0.000000000000000000
1 - 0.100000000000000000
2 - 0.010000000000000000
3 - 0.110000000000000000
4 - 0.001000000000000000
5 - 0.101000000000000000
6 - 0.011000000000000000
7 - 0.111000000000000000
8 - 0.000100000000000000
9 - 0.100100000000000000
10 0.010100000000000000
11 0.110100000000000000
12 0.001100000000000000
13 0.101100000000000000
14 0.011100000000000000
15 0.111100000000000000
In our list we skip no one real with given number of digits. Hense in a limit when our list will be infinite we will not skip no one real with infinite number of digits.
Now we will show that as a result of a Georg Cantor's diagonal procedure in spite of his opposite statement we will get a real that is in our list.
In the above numbers especially marked out digits standing on a diagonal. These are the digits that must be changed in a process of Georg Cantor's diagonal method to get a number that is absent in our enumeration.
As far as on the diagonal there are only zeroes we wil get as a result of Georg Cantors's diagonal method a real number in which to the right of the point will be infinite row of ones: 0.1111111111... or 0.(1).
But this number is the same that is the limit of above mentioned sequence. So in our list no real is omitted and the real that we get as a result of diagonal procedure is in our list.
That is what to be proved.
The discussion.
So diagonal procedure with the help of wich G. Cantor proves that the set of reals is more that that of naturals cannot be used for this aim.
But what shall we do with irrationals? It looks as if the same diagonal procedure but applyed to a set of irrationals cannot be disproved.
Because for this goal one have to "count irrationals" (to set a bijection with the set of naturals).
It seems that for this aim it is sufficient to mark out rational numbers from set of reals and to count the rest. But it is impossible. Impossible for several reasons.
In the system of recording that is suitable to count rationals there is no space to irrationals.
In those scales of notation in which one can count reals it is practically impossible to distinguish rational and irrational numbers.
For this aim one have to look in an infinite chain of digits that represents a real for some sequence repeated infinite times.
To look for a repeated sequence of digits one must know its maximal length. Repeated sequence can have any length.
So to test a real for belonging to rationals or irrationals one needs an infinite time. To test onother real there shall be no time.
So it is impossible to disprove G. Cantor's diagonal procedure applyed to irrationals by the method that we used to disprove it applyed for reals.
But it is very easy to show contradictions in its theory. As far as uncoutable set of irrationals is a subset of a countable set of reals an uncounable set is not obligatorily bigger than a countable one.
Hence a continuum cardinality and transfinite numbers must be moved to a museum hall of scientific delusions where are already phlogiston and aether.
Advocates of continuum hypothesys and transfinite numbers may dispute this proof proceeding from the assumption that we can give numbers only to those reals that have in their binary presentations finite number of ones. Because to reals that have infinite number of ones in their binary presentation we will get infinite numbers.
On this we will say in the following way. Our opponents when actual infinity helps them they take it into account when it hinders they reject it. When G. Cantor in his diagonal proof gives numbers to reals with infinite number of digits he uses the actual infinity. When our opponents dispute our proof because of infinite numbers they in fact reject the actual infinity.
Yes. Infinitum actu non datur. The actual infinity is not given. It is not to be used in such exact sciences as the methematics. The is no such practical spheres where an actual infinity is used. So in no practics there are iirrational and transcedent numbers. And the diagonal procedure itself is inconceivable if there is no actual inginity.
To show contradictoriness of a notion of inginity it is interesting to strengthen Galileo's paradox. Instead of Galileo's sequence 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 .... n2, we will take another sequence : 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000 ... nn. These numbers according to G. Cantor there are exactky so many as there are natural numbers.
This contradiction is solved if one remembers G. Cator's definition of a real number as a limit of a Cauchy sequence or a fundamental sequence.
It is easy to see that for any real number in our interval [0;1), one can find a sequence of reals in our list converging to the given real.
On the other hand one can show some properties of functions N = f(R) and R = g(N) defining bijection between sets of reals and natural numbers. E.g. we want to know a real that is next after such a real as π/4. It will be a real equal to π/4-0.5. For the real π/8 next after it will be a real equal to π/8+0.5.
So we have shown that Cantor's diagonal procedure is inconsistent. It is not adequate to prove that a set of reals is "bigger" than a set of natural numbers. A set of reals is countable.
Conclusions.
The continuum hypothesis is disproved. The diagonal procedure can not be used to prove uncountability of reals. An uncountable set does not contain too many elements it is only ill ordered to be countable. Cardinality of continuum does not differ from cardinality of integers. Also there are no grounds for transfinite numbers and hierarchy of ones. Hense there are no grounds to splitting on mathematical issues. On the contrary there is a ground to mathematicians of all attitudes to unite. And instead of a system of axioms ZFC mathematicians by their common efforts must elaborate a system more adequate to a mathematical reality.
Acknowledgments
Author sincerely thanks doctors Alexandre Maslov, Valdis Egle and Maksim Makarov, my group mate Yulia Kostyliova and my wife Margarita Kadissova for useful discussions of this work.
References.
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Кантор Г. Труды по теории множеств. М., Наука, 1985.
Пол Дж. Коэн Теория множеств и континуум-гипотеза. — М.: Мир, 1969. — С. 347.
Gödel, K. (1940). The Consistency of the Continuum-Hypothesis. Princeton University Press.
Cohen, Paul J. (December 15, 1963). "The Independence of the Continuum Hypothesis". Proceedings of the National Academy of Sciences of the United States of America. 50(6).
А.А.Зенкин, "Ошибка Георга Кантора". Вопросы философии, 2000, No. 2, 165-168.