The flaw in Cantor's Diagonalization Argument

Discussion in 'Number Theory' started by Seff, May 8, 2023.

1. Seff

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Diagonaliztion as a process involves constructing a number that cannot possibly exist in an infinite list of numbers of a set such as the reals, then because that list was assumed to have a bijection with the naturals it concludes that a bijection is impossible. This conclusion however is flawed in that it is never tests if diagonalization will also create a new natural number not in the list of natural numbers that we can then use to continue the bijection.

Say we have a list of all natural numbers:
3948593...
1085483...
7688312...
...
we can add one to the first digit of the first number, the second of the second number, and so on diagonally in order to construct a natural number not in the list: 4290337...
Because diagonalization can always create a new number in the naturals we can continue the bijection that Cantor abandoned and show that there is no contradiction.

Seff, May 8, 2023
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2. Seff

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Can someone please tell me I'm not crazy and this makes sense?

Seff, May 8, 2023
3. Seff

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Cantor assumes a bijection between the reals and the naturals is possible.
Cantor shows a surjection from the reals to the naturals is impossible using diagonalization.
Cantor concludes his assumption leads to a contradiction and must be false.

I assume a bijection between the reals and the naturals is possible.
I show a surjection from the naturals to the reals is impossible using diagonalization.
I conclude Cantor's assumption of a contradiction leads to a contradiction itself because if two sets are not surjective into each other then both must be strictly larger than each other which is impossible.

I conclude there cannot be a contradiction in Cantor's argument and so there must be a bijection between the reals and the naturals.

Seff, May 8, 2023
4. TorusField

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I might be wrong, but wouldn't there be an issue if you get to a natural number in the list which has less digits than you need? For example,

3948593...
1085483...
7688312...
2...
...

How could we create a new natural number with the fourth digit as one greater than the fourth digit of 2?

I also think that a number with infinite digits is not a natural number, and therefore one could not create a new natural number by infinitely adding one to each diagonal digit in the list.

TorusField, Jun 19, 2023
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5. Tommiy

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Hi, How to find the sum of the elements of a series (from 1 to 1000 ) for example, Is it possible to find the sum of the elements, 1+ (n^2)/(n^3)-((m+1)^2)/((m+1)^3 )+((n+2)^2)/((n+2)^3)-?. (from 1 - to - 1000.) n=m. ciao. Привет, как найти сумму элементов ряда (от 1 до 1000 ), например, можно ли найти сумму элементов, 1+ (n ^ 2)/(n ^ 3)-((m+ 1)^2)/((т+1)^3)+((п+2)^2)/((п+2)^3)-?. n=m. пока..

Tommiy, Jul 27, 2023
6. HallsofIvy

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If m= n why use different letters? Also, n^/n^3= 1/n, (n+ 1)^2/(n+ 1)^3= 1/(n+ 1), and (n+ 2)/(n+ 2)^3= 1/(n+ 2). So you are asking for $1+ \sum_{n=1}^{1000} 1/n+ 1/(n+ 1)+ 1/(n+ 2)$.

HallsofIvy, Aug 4, 2023

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