Talk:Cantor's diagonal argument/Arguments: Difference between revisions

Revision as of 06:55, 5 March 2024

Revision as of 07:02, 5 March 2024

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:::: The answer is that it does not. There is a key difference that separates this from Cantor's diagonal argument, which is that there are more rows in the list then there are columns. In Cantor's argument, the diagonal drawn through the list meets each row one time. This is key to the argument, because the reason the number obtained from the diagonal is not on the list is because it differs from the 1st number on the list at the 1st decimal place, it differs from the 2nd number on the list at the 2nd decimal place, etc. However, in this example, the diagonal misses some rows altogether. '''111''' is indeed different from the 1st listed string (011) because it differs in the 1st character, and different from the 2nd listed string at the 2nd character, and the 3rd listed string... but then, there is no reason that '''111''' should different from the 4th, 5th, etc. strings on the list, since we have run out of characters to compare. So it is fine for there to be a matching from the three-element strings from {0,1} with numbers from 1 through 8, since the diagonal argument needs the diagonal to meet each row once before Cantor's proof by contradiction can kick in.

:::: However, this problemloophole never happens in Cantor's argument about real numbers. Given any proposed matching from the natural numbers to the real numbers, once you get the number from the diagonal and change each digit, even the trillionth number on the list must differ from this one since they both have trillionth decimal places, and they are different. There is no "running out of digits to compare" that happens with real numbers. [[User:C7XWiki|C7XWiki]] ([[User talk:C7XWiki|talk]]) 06:55, 5 March 2024 (UTC)

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