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:::The diagonalisation process requires a list in which there is a sequence of digits for every natural number. Your process has multiple digit sequences for each natural number; in fact an apparently infinite number. So there isn't a diagonalisation process until you define how to combine all your lists into one list. [[User:MartinPoulter|MartinPoulter]] ([[User talk:MartinPoulter|talk]]) 09:54, 21 April 2025 (UTC)
::::I have already answered each of your questions earlier. The question about the ___location of irrational numbers written in the decimal positional system is impossible. The matrix contains an infinite number of infinite sequences of rational numbers. If you write down in a sequence of numbers, the numbers that arise step by step during the creation of a new number, which the diagonal method offers, then a sequence of rational numbers will be created, i.e. such a sequence as are in the matrix. You are asking about creating a list, I have previously indicated a link on how to write down the numbers from the matrix into a single list. You can also do this; A1 B1 A2 C1 B2 A3 D1 C2 B3 A4 E1 D2 ... However, if you learn how a matrix is built, and how a new number is created in the diagonal method, you will understand that this method is pointless, i.e. it will not create a new number that is not in the matrix. Now a question for you; 1. Can the diagonal method create an irrational number or an infinite sequence of rational numbers? 2. If the list does not contain all the numbers, will the diagonal method find a new number that is not on the list? I apologize if something was misunderstood, maybe it's because of Google Translate. [[User:Krzysztof1137|Krzysztof1137]] ([[User talk:Krzysztof1137|talk]]) 17:21, 21 April 2025 (UTC)
::::I will describe it again to make everything clear. The matrix contains infinite sequences of rational numbers in the rows. The columns contain all possible numbers for any number of digits after the decimal point. Column "A" contains all numbers with one digit after the decimal point, column "B" contains all numbers with two digits after the decimal point, column "C" contains all numbers with three digits after the decimal point, etc... The number of columns is infinite. If you write the numbers in a list, e.g.; A1 B1 A2 C1 B2 A3 D1 C2 B3 A4 E1 D2 ... it turns out that the diagonal method will not create a number that is not on the list. The diagonal method only works on numbers written in the decimal positional system, in the next steps it creates subsequent rational numbers, if you write these numbers, an infinite sequence of rational numbers will be created. The diagonal method does not require the list to be complete, it does not check which number is missing, it creates a new number. -In the first step, we write down the first digit after the decimal point, the number that is in column "A" is created. Since column "A" contains all possible numbers with one digit after the decimal point, no new number is created that is not in column "A". -In the second step, we write down the second digit after the decimal point, the number that is in column "B" is created. Since column "B" contains all possible numbers with two digits after the decimal point, no new number is created that is not in column "B". -In the third step, we write down the third digit after the decimal point, the number that is in column "C" is created. Since column "C" contains all possible numbers with three digits after the decimal point, no new number is created that is not in column "C". -In the nth step, we write down the nth digit after the decimal point, the number that is in column "n" is created. Since column "n" contains all possible numbers with "n" digits after the decimal point, no new number is created that is not in column "n". I answered your questions, now you answer mine: - does the diagonal method create an irrational number or an infinite sequence of rational numbers? - if the list of numbers from the matrix is incomplete, why doesn't the diagonal method create a new number that is not on the list? [[User:Krzysztof1137|Krzysztof1137]] ([[User talk:Krzysztof1137|talk]]) 19:12, 25 April 2025 (UTC)
:::{{Re|Krzysztof1137}} ''"The matrix was created only to contradict the diagonal method."'' So it failed.<br>Apparently you confuse 'a possibility of expanding the decimal number infinitely' with 'having an infinite decimal number'. You certainly can find an arbitrarily long sequence of 3's after a decimal point in your table. But each such sequence is located in some specific row of the table and the number of the row determines how many threes there is in the sequence. You may have an infinite set of rows, but the ordinal number of each specific row is some natural number, which is certainly finite. As a result none of those sequences represents {{math|1/3}}. You definitely can find arbitrarily accurate aporoximations of one-third in your table, but not ''the'' {{math|1/3}} (let alone {{math|1/{{sqrt|3}}}}). --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 20:50, 21 April 2025 (UTC)
::::Now a question for you; 1. Can the diagonal method create an irrational number or an infinite sequence of rational numbers? 2. If the list does not contain all the numbers, will the diagonal method find a new number that is not on the list? [[User:Krzysztof1137|Krzysztof1137]] ([[User talk:Krzysztof1137|talk]]) 20:59, 21 April 2025 (UTC)
::::Please answer the questions. [[User:Krzysztof1137|Krzysztof1137]] ([[User talk:Krzysztof1137|talk]]) 20:40, 25 April 2025 (UTC)
:::::I am not CiaPan, but:
:::::If you are referring to specifically the method from the article page, then my answers
:::::are both No: That creates/finds a sequence of binary digits, not a real number.
:::::(_That part_ is only to show that the set of
:::::infinite sequences of binary digits is uncountable.)
:::::If you are instead referring to
:::::" the real number such that its part before the decimal point is "0" and for all positions n after the decimal point, the digit in that position is the element of {1,2} with the opposite [[parity]] to the [[parity]] of the digit in position n after the decimal point of the [[Non-uniqueness of decimal representation and notational conventions|standard decimal representation]] of the n-th entry in the list "
:::::, then my answers are both yes: If you get different answers for this case, then:
:::::3. Do you think there is no real number
:::::"such that its part ... entry in the list"
:::::, like how there is no such integer for
:::::"the integer that is greater than all other integers" ?
:::::4. If you indeed think there is no real number
:::::"such that its part ... entry in the list"
:::::, then what about
:::::" the real number such that its part before the decimal point is "0" and for all positions n after the decimal point, the digit in that position is 3? "
:::::?
:::::[[User:JumpDiscont|JumpDiscont]] ([[User talk:JumpDiscont|talk]]) 23:06, 29 April 2025 (UTC)
::::::The diagonal method in the next steps creates the next numbers with a finite notation, if you write down all the possible numbers that can be created in the columns, then an infinite series of columns with numbers with a finite notation will be created. The columns contain all the possible numbers for any number of digits after the decimal point. Column "A" contains all the numbers with one digit after the decimal point. Column "B" contains all the numbers with two digits after the decimal point. Column "C" contains all the numbers with three digits after the decimal point. Column "n" contains all the numbers with "n" digits after the decimal point. The number of columns is infinite. If you write the numbers in a list, e.g.; A1 B1 A2 C1 B2 A3 D1 C2 B3 A4 E1 D2 ... etc. Only now can you use the diagonal method on the list, but we know that this list already contains all the numbers. There is a feedback loop that makes this list a trap for the diagonal method. There is no way to get beyond the numbers contained in the matrix. [[User:Krzysztof1137|Krzysztof1137]] ([[User talk:Krzysztof1137|talk]]) 01:40, 30 April 2025 (UTC)
:::::::For the orderings you gave, the column number from the matrix will be [[Big O Notation|Big O]] of the square root of the position within the list - since each column is first reached at a position within the list that approximately the corresponding [[triangular number]] - so for all sufficienctly large n, the n-th number in that list will have a 0 for its n-th digit after the decimal point, which means the diagonal number will have a non-zero digit for its n-th digit after the decimal point.
:::::::More generally, for all lists of numbers, if [for all sufficiently large n, the n-th number in the list has a non-zero n-th digit after the decimal place] then for all sufficiently large n, at most n numbers in the list have only n digits after the decimal point, even though there are 10^n numbers (not necessarily all in the list, and indeed for such n they can't all be in the list) with only n digits after the decimal point.
:::::::[[User:JumpDiscont|JumpDiscont]] ([[User talk:JumpDiscont|talk]]) 04:18, 30 April 2025 (UTC)
::::::::A thought experiment and an analogy to a matrix. A drum counter like the ones used to have on car odometers, i.e. each column of the matrix is a drum with digits from zero to nine. That is, after the decimal point there is an infinite number of drums, the digits on the drums arranged in a line create a number with an infinite decimal expansion. The first drum after the decimal point rotates changing the digit, e.g. every second, the second drum rotates ten times faster than the first drum, the third drum rotates ten times faster than the second drum, the fourth drum rotates ten times faster than the third drum, etc. each subsequent drum rotates ten times faster than the previous one. After a full rotation of the first drum, i.e. after ten seconds, the counter will create all possible combinations of digits in the line, which will ultimately create all possible numbers from zero to one written in the decimal positional system. [[User:Krzysztof1137|Krzysztof1137]] ([[User talk:Krzysztof1137|talk]]) 21:53, 29 June 2025 (UTC)
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