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The First Mathematical Problem of David Hilbert: Continuum Hypothesis Continuum Hypothesis, also known as the first Hilbert's problem is one of the most interesting mathematical problems. It is often spoken within the context of the basic mathematical tasks and multitude theory. Continuum Hypothesis is closely related to such simple and natural questions like how much? , More or less? , How many? , etc. Even the smallest child is able to understand the formulation of this problem. Nevertheless, we have to provide the reader with some additional information to formulate the essence of this mathematical problem. So, lets get started.
Equivalence of multitude Lets dwell on the following example. There is a dance party in the school. How can we answer this question: How many boys and girls are dancing within the context more or less? Certainly, we can calculate the exact quantity of boys and girls and then to compare numbers received.
At the same time we can answer this question easily if the orchestra is playing and the children are dancing. In such a way, there will be pairs of boys and girls dancing. If all the children present are dancing, it will mean that the quantity of boys and girls is equal. If there are boys standing aside, it means that there are more boys than girls. Sometimes such principle is more natural than direct calculation. It is also called the principle of division into pairs or the principle of mutually univocal correspondence.
Now lets examine the totality of objects of derived nature the multitude. The objects creating multitude are called its elements. If element x forms part of multitude X, it is described as follows: x X. If the multitude X 1 forms part of multitude X 2 (all elements of the multitude X 1 are the elements of multitude X 2), they say that X 1 sub-multitude X 22. The multitude is finite if the quantity of elements is finite. Multitude can be finite (for example, multitude of students in the class) and infinite (for example, the multitude of natural numbers 1, 2, 3, ).
The multitudes that contain the elements consisting of numbers are called numerical. Let X and Y be two multitudes. It is accepted that there is a mutually univocal correspondence between them, if all the elements of these two multitudes are divided into pairs (x, y) where x X, y Y. Each element from X and each element from Y multitude takes part only in one pair. The example, where all boys and girls are divided into dancing pairs is the example of mutually univocal correspondence between the multitude of boys and the multitude of girls.
If we are able to establish mutually univocal correspondence between the multitudes, we call them equivalent or uni-equal. Two finite multitudes are equivalent when and only when they contain the same quantity of elements. In such a way, it is natural to suppose that if one infinite multitude is equivalent to the other, it contains the same quantity of elements. At the same time, when we accept such definition of equivalency, we can receive quite sudden and unexpected properties of infinite multitudes. Infinite Multitudes Lets examine any finite multitude and any sub-multitude of its own (non-empty and not-coincident with itself). In such a way, the number of elements in sub-multitude is less than the number of elements in the multitude (K.
Kunen and J. Vaughan 112). Naturally, the part is always less than the whole. Do infinite multitudes possess such property?
Is it correct to say that one infinite multitude has fewer elements than another multitude that is also infinite? Speaking about two infinite multitudes we can only say whether they are equivalent or not. Do non-equivalent infinite multitudes exist? Lets try to answer all these questions correspondingly.
Lets start with a short funny fantastic story. The action takes place in the future, when the inhabitants of different galaxies can meet each other. There is an enormously huge hotel for such space travelers crossing several galaxies. This hotel has infinite quantity of rooms. As it should be, each room has its number. Any natural number n corresponds to the room with such number.
Once the meeting of cosmo zoologists took place in this hotel. The representatives of all galaxies took part at this meeting. As the quantity of galaxies equals infinite multitude, all rooms in the hotel were occupied. At the same time the friend of hotels owner asked the owner to place his friend in one of the rooms.
After several minutes of thinking the owner of hotel asked the administrator: Lets give him the room # 1 Where Ill place the person who lives in the room # 1? asked the administrator of hotel Lets move him to room # 2. The person who lives in room # 2 will move to the root. In such a way, the person who lives in room #k will move to room #k+ 1 as it is shown here: Each guest will have his own room and the room # 1 will be free. In such a way, the new guest will be able to receive his own room because there is infinite multitude of rooms in this hotel.
Primarily the participants of the meeting occupied all rooms of hotel. Therefore, there were established mutually univocal correspondence between the multitude of cosmo zoologists and the multitude X. Each cosmo zoologist received his unique number. The corresponding natural number was written at the door of each room. It is natural to consider that the quantity of participants was equal to quantity of natural numbers.
Another man arrived; he was located at the same hotel and quantity of inhabitants increased for 1 person. However, the quantity remained the same and remained equal to the quantity of natural numbers as all of guests were still located in the same hotel. If we label the quantity of cosmo zoologists as X 0, we will receive identity: X 0 = X 0 + 1 However, it is not valid for any finite multitude X 0. In such a way we came to an interesting conclusion: We have a multitude that is equivalent to X. If we add one more element we receive the multitude that will be still equivalent to X. However, it is evident that the participants of the meeting represent one part of the multitude of people located at the hotel after the arrival of a new guest.
It means that the part of multitude in this case is not smaller than the totality, but is equal to it. In such a way, taking into account the definition of equivalency (which, however, doesnt give us any problems speaking about finite multitudes), we come to conclusion that one part of infinite multitude can be equivalent to the whole multitude. Probably, well-known mathematician Bolcano, who tried to use the principle of mutually univocal correspondences during his work, was afraid of such unusual effects and, therefore, didnt try to develop this theory. He considered is to be an absurd. However, Georg Cantor during the second half of the XIX century was interested in this principle and started to examine it. He created the theory of multitudes.
This theory became the important part of basic mathematics. Lets continue our story about the space hotel. The new guest wasnt surprised when the administrator proposed him to move into the room # 1, 000, 000 the next morning after his arrival. The explanation was that the cosmo zoologists from the galaxy BCK- 3472 arrived and the administrator had to locate another 999, 999 guests. Further on, due to some mistake in calculations, the philatelists arrived at the same hotel for their meeting. The quantity of philatelists was also equal to an infinite multitude: one representative from each galaxy.
How the administrator was able to locate them all? This mathematical task was quite difficult. However, the administrator found the way out. First of all, the administrator ordered to move the guest from # 1 to the room # 2. The guest from room # 2 should be moved into the room # 4, from the room # 3 to the room # 6. In other words, the guest from the room #n should be moved into the room # 2 n.
Now, his plan is quite clear. In such a way he made the infinite multitude of odd numbers free and was able to place there the philatelists. In result of this operation the multitude of even rooms were occupied by cosmo zoologists, and the odd numbers were occupied by the philatelists. We came across with another interesting effect. If we unite two multitudes, each of which is equivalent to X, we receive the multitude that is also equivalent to X. It means that even if we double the multitude, we receive the multitude that is equivalent to the first one.
Further lets examine only the numerical multitudes the sub multitudes of the numerical straight line. The multitude of all numbers belonging to this numerical straight line is the multitude of real numbers. Countable and Uncountable Sets Lets examine the chain where there is a multitude of whole quantities and the multitude of rational quantities: Where the multitude of numbers like p / q , where p and q are whole quantities and q is more than 0 (Maddy 482). All multitudes are infinite. Lets dwell on the question of their equivalency. Lets establish mutually univocal correspondence between then and create pairs (n, 2 n) and (-n, 2 n+ 1), n and pair (0, 1).
There is another way to establish such mutually univocal correspondence. For example, we can write down on a piece of paper all whole quantities in table, and further, to attach to each whole quantity a definite number. In such a way we will calculate all whole quantities: each number Z will correspond to a certain whole quantity (number) and each number will correspond to a certain whole quantity. In such a way we will confirm that Z is equivalent to N. Any multitude equivalent to the multitude of whole numbers is called countable.
We can count the numbers in this multitude: to attach a natural number to each of its elements. You can suppose that the quantity of rational numbers at the straight line is more than quantity of whole numbers. The element density seems to be quite high: there is an infinite quantity of these numbers at any small interval of the straight line. From the other side, such multitude is still countable. Lets prove the supposition that such multitude (multitude of all positive rational numbers) is still countable.
Lets write down all such elements into a table, where the first line will contain all numbers with denominator 1 (e. g. whole numbers) and the second line will contain all numbers with denominator 2, etc. Each positive rational number will inevitably be written in this table and even not once) Now lets recalculate these numbers: following the arrows we will attach each number its own number (or omit the number if we have already met it before in the other record).
As we move by diagonals, earlier or later we will finish the whole table (e. g. earlier or later we will recalculate any of the numbers) (Godel 517). So, we found the method to number all the numbers. We proved that the multitude (multitude of all positive rational numbers) is also countable. Note that such method of numeration doesnt preserve the order: if we have two rational numbers, the greater number can be met earlier or later.
What about the negative rational numbers and zero? Lets do the same thing as we did with cosmo zoologists and philatelists in the infinite hotel. Lets numerate the multitude by only even numbers (giving them numbers not 1, 2, 3 but 2, 4, 6, etc). Lets attach the zero number 1, and all negative rational numbers lets attach the odd numbers starting from 3 using the same scheme as we used for positive rational numbers. Now all rational numbers are numerated as natural. Therefore, the multitude is countable.
We have a natural question. Probably, all endless multitudes are countable. Remember the fact that the multitude of all dots at the numerical straight line is uncountable. This result, received by Cantor made great impression on the mathematicians. Lets prove this fact as it was proved by Cantor: using the method of diagonal process.
As we already know, each real number x can be written in form of fractional number: where A is a whole number (not obligatory positive), and 1, 2, ... , n, ... represent numbers from 0 to 9. However, this assumption is ambiguous: for example According to one variant, the numbers after 0, 5 equal zeroes, whereas in the second variant the numbers after 0, 4 contain 9 -s. We will choose the first variant for convenience of our explanations. In such a way, each number will correspond to its exact fractional number. Lets suppose that we managed to calculate all real numbers.
In such a way, we are able to locate them using the following order: To come to contradiction we need to create such number y which is uncountable (the number that is not located in the table). For any number we define the number as follows: 1, if a 1; 2 if a = 1 Lets suppose. For example if In such a way, we received the real number y using the method of diagonal process. This number y doesnt coincide with any of the numbers presented in the table, sa y differs from each. We supposed that it is possible to calculate all real numbers. We came to contradiction when we found the number impossible to calculate.
Therefore, multitude R is uncountable. The multitudes R and N are not equivalent, and. Thats why the quantity of all real numbers is a bit more than the quantity of all natural numbers to some extent. There is a supposition that capacity of multitude R (capacity of continuum) is more than the capacity of multitude N.
Continuum Hypothesis Finally, we have all necessary information to formulate the well-known first Hilbert's problem: There is no set whose size is strictly between that of the integers and that of the real numbers. There exist only two types of infinite multitudes: countable multitude and continuum accurate within the equivalency (Zenkin 156). In other words, we need to find out whether the multitude of intermediate capacity, e. g.
the multitude that is not equivalent either to N or R. This problem was examined by many mathematicians. Even Cantor many times tried to prove this hypothesis, but each time he was mistaken. However, it was found out that the hypothesis of Hilbert possesses an interesting solution.
Paul Korean proved that the hypothesis can be neither proved nor refuted. It means that is we will take a standard system of axioms by Cermelo-Frenkel (ZF) and will add to it the hypothesis of continuum as axiom we receive non-contradictory system of assertions. At the same time is we will add to ZF the denial of hypothesis of continuum (the opposite assertion) we receive the same non-contradictory system of assertions. In such a way, neither continuum hypothesis, nor its refutation can be taken out from the standard system of axioms. Bibliography Godel, The Consistency of the Continuum Hypothesis, Ann. Math.
Studies no. 3, Princeton University Press, 1940. What is Cantor's continuum problem? Amer. Math.
Monthly 54 (1947) Kunen, Set Theory: An Introduction to Independence Proofs, North-Holland, 1980 Kunen and J. Vaughan, eds. Handbook of Set-Theoretic Topology, North-Holland, 1984. Maddy, Believing the axioms. I and Believing the axioms. II, J.
Symbolic Logic 53 (1988) Zenkin A. A. , Cognitive Visualization Of The Continuum Problem And Mirror Symmetric Proofs In Transfinite Numbers Mathematics. - ISIS-Symmetry Congress and Exhibition. Abstracts. Haifa, Israel
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