Representation of Infinite Numbers
Representation of Infinite Numbers John Einbu ABSTRACT. In the first section it is shown that Cantor’s diagonal method is not valid. In the following sections a symbol system is presented where each symbol possesses the properties of a unique infinite number and therefore can be considered a representation of such a number. Keywords: Set theory, Cantors diagonal method, Infinite numbers Spotlight on the Cantor diagonal method Cantor questioned if the real numbers could be mapped on the natural numbers, i. e. if the real numbers were countable. He let decimals between 0 and 1 represent the real numbers and found, using his famous diagonal method, that no matter how you organized these decimals in a column, it would always be at least one number that did not exist in this column. Cantor concluded therefore that the real numbers must be of a different order of magnitude than the natural numbers. The problem with the diagonal method, however, is that it is a process that moves from the finite to the infinite, i. e. from a domain where we have full control and to a more diffuse domain where we have insufficient control. And as we shall see, we are actually faced with two processes that both approaches infinity, but that ends up in an infinity of two different orders of magnitude. And with the result that Cantor’s diagonal method does not achieve its goal. To get a clearer picture of what the diagonal method is about, we first make a few thought experiments. Let us consider all decimal numbers between 0 and 1 with 100 digits (terminating zeros included). There are 10100 such numbers, and let us imagine that we arrange these numbers in a list in a certain way. Then we generate a number, d, following the pattern of the diagonal method, where the k-th digit in d is different from the k-th digit of decimal number k in the list. This number, d, will consist of 100 digits after the decimal point, and it will be different from the first 100 numbers in the specified list. But we have not thereby proved that this number, is also different from the 10100 - 100 remaining numbers in the list. Indeed, since we have assured ourselves that all decimal numbers with 100 digits are in the list, d must be among the 10100 -100 remaining numbers. Here we have a full overview of what is happening, and we see that the diagonal method only succeed in finding a decimal number that is different from the first 100 decimal numbers. So it then falls naturally to ask whether the same applies when we are dealing with the totality of decimals in the selected interval. Let us next assume that we have selected only 100 decimal numbers between 0 and 1, each with 100 digits after the decimal point, forming a list of them. Then we apply the diagonal method on this list to find a number, d, which is different from all of the 100 numbers. In this case it seems that the diagonal method is working. But why? It is because the list we have started with, takes the shape of a square. There are as many numbers in the list as there are digits in each number. Had it been 101 decimal numbers in the list, we could not be sure that decimal # 101 was different from d. In order to achieve that, we must have invented an optional rule in the formation of d, and nobody has suggested that. So Cantor’s diagonal method only works if the selected list of decimal numbers appear as a "square" as defined above. However, when we move from a finite number of decimals
and to all decimal numbers (in the selected interval), the number of digits goes to infinity with the number of decimals. One can then be misled into believing that in this case, the diagonal method encounters all decimals. But it is easy to see that this is not so. If we imagine a progression of lists of decimal numbers where each list has 10N numbers, each with N digits, and let N in this progression go to infinity, the lists will in this progression look less and less like a square after each step (the ratio between the number of digits and the total number of decimal goes quickly to zero), and this must also apply when the N reaches infinity. So one must conclude that Cantor’s diagonal method does not prove what Cantor claims, namely that the method shows that in any list of decimal numbers between 0 and 1, there is a number that is not in the list. A possible scheme Now it could still be the case that Cantor 's claim is correct, i.e. that no list of the relevant decimals can contain all decimals . But let 's make a third thought experiment in which we organize the decimals so that we initially insert all decimal numbers with one digit after the decimal point , followed by all decimal numbers with two digits behind the decimal point and further in the Nth step all decimal numbers with N digits are inserted (in each step no numbers with terminating zeroes are included). If we are assigning natural numbers to these decimals, a list that ensures that all both natural numbers and decimals are included may be 1 2
0.1 0.2 . .
9 10 11
0.9 0.01 0.11 . .
19 20
0.91 0.02 . .
99 0.99 100 0.001 etc. If we then let N go to infinity, we have apparently achieved a complete list of all decimal numbers between 0 and 1 But can we now claim that we have found a one-to-one-correspondence between the selected decimal numbers and the natural numbers? Yes, to some extent we can, for example, the decimal number 0.1234 corresponds to the integer 4321, so the integer is thus a kind of mirror image of the decimal. But which integers correspond to the decimal numbers 0.3333... or 0.35091... in this list? Here the three dots signify an infinite sequence of known or unknown digits. This is a valid representation of decimal numbers between 0 and 1, 0.3333 ... can for instance represent the fraction 1/3. We see immediately that the integers corresponding to those two decimal numbers
must be infinite. But by admitting this, we face a serious weakness in the current number system, we have no representation of individual infinite integers. We have only one single symbol ∞, which represent any of the infinite integer number. We read sometimes about tribes in the Amazon jungle that counts 1, 2, 3, 4, many. In their arithmetic many + 1 = many and many + many = many. We consider this a primitive number system and have some compassion for these primitive people. But the natural number system of the rest of the world is not particularly more advanced: it contains on the one hand all integers that can be written with a finite number of digits and added to that is a single infinite integer, ∞ , such that ∞ + 1 = ∞ and ∞ + ∞ = ∞ . This is quite analogous to the system in the Amazon jungle. A new symbol system Now we introduce a symbol system where all symbols consist of a series of three dots and then one or more numeric digits. Two examples of these symbols are ...3333 and ...19053. We now see immediately that all decimal numbers between 0 and 1 with no final digit representation can be depicted uniquely into this symbol system. For example, we may have the one-to-onecorrespondence ...3333 ↔ 0.3333... and ...19053 ↔ 0.35091.... If we form the union of all finite integers and all possible symbols in this new symbol system, we see that we can depict all decimal numbers between 0 and 1 on this union in a one-to-one fashion. The question then becomes whether this new symbol system can represent the infinite natural numbers. There will of course be different opinions about this question. Some may reject this idea out of pure reflex. Others may refuse to accept this symbol system because it is not compatible with Cantors set theory. But to say that ...19053 cannot be a number is difficult to justify since 0.35091... is a legitimate number. By allowing booth these symbols to be numbers, we achieve symmetry in the number system, and symmetry is something desirable in both mathematics and science in general. That one can not place ...19053 on a number line cannot be decisive, ∞ cannot be placed there either. Relationships between these numbers cannot always be determined, for example which one is the largest of two arbitrary such numbers. But this would be an unreasonable demand, if we do not know how these two numbers are obtained. If we on the other hand know exactly how they are formed, we can certainly determine which is greatest. It may not be obvious how all the arithmetic computation rules can be applied to the new numbers, but considering that these numbers are after all different from the finite numbers in its structure, it may not be straight forward how all the calculation rules apply. But it is not inconceivable that creative mathematicians find good solutions to this problem. A positive consequence of the proposed expansion of the number system is that it provides new insights into this system and will lead to a much-needed revision of the current set theory. For by incorporating these symbols in the integer number system, we can conclude that there are as many integers as decimal numbers between 0 and 1, while the current set theory claims that the number of decimal numbers between 0 and 1 is of an entirely different order of magnitude than the number of integers. Personally, I perceive it as a benefit that we, the mathematicians, now no longer will be bullied for having a number system that is analogous to that employed in the jungle of the Amazon.
and to all decimal numbers (in the selected interval), the number of digits goes to infinity with the number of decimals. One can then be misled into believing that in this case, the diagonal method encounters all decimals. But it is easy to see that this is not so. If we imagine a progression of lists of decimal numbers where each list has 10N numbers, each with N digits, and let N in this progression go to infinity, the lists will in this progression look less and less like a square after each step (the ratio between the number of digits and the total number of decimal goes quickly to zero), and this must also apply when the N reaches infinity. So one must conclude that Cantor’s diagonal method does not prove what Cantor claims, namely that the method shows that in any list of decimal numbers between 0 and 1, there is a number that is not in the list. A possible scheme Now it could still be the case that Cantor 's claim is correct, i.e. that no list of the relevant decimals can contain all decimals . But let 's make a third thought experiment in which we organize the decimals so that we initially insert all decimal numbers with one digit after the decimal point , followed by all decimal numbers with two digits behind the decimal point and further in the Nth step all decimal numbers with N digits are inserted (in each step no numbers with terminating zeroes are included). If we are assigning natural numbers to these decimals, a list that ensures that all both natural numbers and decimals are included may be 1 2
0.1 0.2 . .
9 10 11
0.9 0.01 0.11 . .
19 20
0.91 0.02 . .
99 0.99 100 0.001 etc. If we then let N go to infinity, we have apparently achieved a complete list of all decimal numbers between 0 and 1 But can we now claim that we have found a one-to-one-correspondence between the selected decimal numbers and the natural numbers? Yes, to some extent we can, for example, the decimal number 0.1234 corresponds to the integer 4321, so the integer is thus a kind of mirror image of the decimal. But which integers correspond to the decimal numbers 0.3333... or 0.35091... in this list? Here the three dots signify an infinite sequence of known or unknown digits. This is a valid representation of decimal numbers between 0 and 1, 0.3333 ... can for instance represent the fraction 1/3. We see immediately that the integers corresponding to those two decimal numbers
must be infinite. But by admitting this, we face a serious weakness in the current number system, we have no representation of individual infinite integers. We have only one single symbol ∞, which represent any of the infinite integer number. We read sometimes about tribes in the Amazon jungle that counts 1, 2, 3, 4, many. In their arithmetic many + 1 = many and many + many = many. We consider this a primitive number system and have some compassion for these primitive people. But the natural number system of the rest of the world is not particularly more advanced: it contains on the one hand all integers that can be written with a finite number of digits and added to that is a single infinite integer, ∞ , such that ∞ + 1 = ∞ and ∞ + ∞ = ∞ . This is quite analogous to the system in the Amazon jungle. A new symbol system Now we introduce a symbol system where all symbols consist of a series of three dots and then one or more numeric digits. Two examples of these symbols are ...3333 and ...19053. We now see immediately that all decimal numbers between 0 and 1 with no final digit representation can be depicted uniquely into this symbol system. For example, we may have the one-to-onecorrespondence ...3333 ↔ 0.3333... and ...19053 ↔ 0.35091.... If we form the union of all finite integers and all possible symbols in this new symbol system, we see that we can depict all decimal numbers between 0 and 1 on this union in a one-to-one fashion. The question then becomes whether this new symbol system can represent the infinite natural numbers. There will of course be different opinions about this question. Some may reject this idea out of pure reflex. Others may refuse to accept this symbol system because it is not compatible with Cantors set theory. But to say that ...19053 cannot be a number is difficult to justify since 0.35091... is a legitimate number. By allowing booth these symbols to be numbers, we achieve symmetry in the number system, and symmetry is something desirable in both mathematics and science in general. That one can not place ...19053 on a number line cannot be decisive, ∞ cannot be placed there either. Relationships between these numbers cannot always be determined, for example which one is the largest of two arbitrary such numbers. But this would be an unreasonable demand, if we do not know how these two numbers are obtained. If we on the other hand know exactly how they are formed, we can certainly determine which is greatest. It may not be obvious how all the arithmetic computation rules can be applied to the new numbers, but considering that these numbers are after all different from the finite numbers in its structure, it may not be straight forward how all the calculation rules apply. But it is not inconceivable that creative mathematicians find good solutions to this problem. A positive consequence of the proposed expansion of the number system is that it provides new insights into this system and will lead to a much-needed revision of the current set theory. For by incorporating these symbols in the integer number system, we can conclude that there are as many integers as decimal numbers between 0 and 1, while the current set theory claims that the number of decimal numbers between 0 and 1 is of an entirely different order of magnitude than the number of integers. Personally, I perceive it as a benefit that we, the mathematicians, now no longer will be bullied for having a number system that is analogous to that employed in the jungle of the Amazon.
