The flaw in Cantor's Diagonalization Argument

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

  1. Seff

    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

    Seff

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    Can someone please tell me I'm not crazy and this makes sense?
     
    Seff, May 8, 2023
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  3. Seff

    Seff

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    Rephrased for readability:

    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
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