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March 3

Larger infinity

N is used to call all the non-negative natural numbers (0; 1; 2; 3; 4;...). Why are there more numbers between 0 and 1 than in N? To check that [0;1] is equal to N, we should associate each number of [0;1] with N's :

N - [0;1]

0 - 0,403917364938...

1 - 0,740283758746...

2 - 0,748209488618...

3 - 0,284011847563...

...

According to Cantor, if we make another number from the list of [0;1] using the bold numbers and changing them (that is : 0,5591...), we obtain a number that isn't associate to any number of N. Why? If infinity is what it is, we can find all possible groups of numbers, so we could find 0,5591... somewhere in [0;1]! What didn't I understand? LLGH (talk) 19:45, 3 March 2019 (UTC)[reply]

Yes, 0.5591... is in the set [0,1], but the reason that it's impossible to find 0.5591... in your list above (and therefore your list can never contain all the numbers in the set [0,1]) is: 0.5591... isn't the first number in the list as the 1st number's 1st digit is 4, it isn't the second number as the 2nd number's 2nd digit it 4, it isn't the 3rd number as the 3rd number's 3rd digit is 8, etc. Therefore there must be numbers in [0,1] that cannot be in your list, so the size of [0,1] must be larger than the size of the integers. Iffy★Chat -- 20:02, 3 March 2019 (UTC)[reply]