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  • The existence of dark numbers proved by the thinned out harmonic, serie

    From WM@21:1/5 to Alan Mackenzie on Tue Mar 18 16:06:21 2025
    On 18.03.2025 13:18, Alan Mackenzie wrote:
    WM <wolfgang.mueckenheim@tha.de> wrote:

    All elements of ℕ are there. That is the assumption. If no greatest can
    be identified, then the reason are dark numbers.

    No, the reason is that there is no greatest element.

    How can that be realizied? If all are there and all are smaller than
    omega, tzen there is a greatest one. If there is no greatest one, then
    the reason is either that new numbers are created (potential infinity)
    or that dark number become visible. There is no other alternative.

    "Dark numbers" do
    not exist, as has been proven in this thread.

    Dark numbers do exist as has been proven in this thread:
    The harmonic series diverges. Kempner has shown in 1914 that when all
    terms containing the digit 9 are removed, the series converges. Here is
    a simple derivation: https://www.hs-augsburg.de/~mueckenh/HI/ p. 15.

    That means that the terms containing 9 diverge. Same is true when all
    terms containing 8 are removed. That means all terms containing 8 and 9 simultaneously diverge.

    We can continue and remove all terms containing 1, 2, 3, 4, 5, 6, 7, 8,
    9 in the denominator without changing this. That means that only the
    terms containing all these digits together constitute the diverging series.

    But that's not the end! We can remove any number, like 2025, and the
    remaining series will converge. For proof use base 2026. This extends to
    every definable number. Therefore the diverging part of the harmonic
    series is constituted only by terms containing a digit sequence of all definable numbers.

    Note that here not only the first terms are cut off but that many
    following terms are excluded from the diverging remainder.

    This is a proof of the huge set of undefinable or dark numbers.

    Unable to comprehend?

    "Potential infinity" doesn't exist in modern mathematics.

    Therefore modern mathematics is rubbish. By the way this degenerated
    form of "mathematics" is the first kind which has turned out completely
    useless for any application. Small wonder. It is wrong.

    Regards, WM

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  • From joes@21:1/5 to All on Tue Mar 18 15:08:58 2025
    Am Tue, 18 Mar 2025 16:06:21 +0100 schrieb WM:
    On 18.03.2025 13:18, Alan Mackenzie wrote:
    WM <wolfgang.mueckenheim@tha.de> wrote:

    All elements of ℕ are there. That is the assumption. If no greatest
    can be identified, then the reason are dark numbers.
    No, the reason is that there is no greatest element.
    How can that be realizied? If all are there and all are smaller than
    omega, tzen there is a greatest one. If there is no greatest one, then
    the reason is either that new numbers are created (potential infinity)
    or that dark number become visible. There is no other alternative.
    No. There is no reason for a greatest to exist.

    "Potential infinity" doesn't exist in modern mathematics.
    Therefore modern mathematics is rubbish. By the way this degenerated
    form of "mathematics" is the first kind which has turned out completely useless for any application. Small wonder. It is wrong.
    Any examples?

    --
    Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
    It is not guaranteed that n+1 exists for every n.

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  • From WM@21:1/5 to joes on Tue Mar 18 16:39:42 2025
    On 18.03.2025 16:08, joes wrote:
    Am Tue, 18 Mar 2025 16:06:21 +0100 schrieb WM:
    On 18.03.2025 13:18, Alan Mackenzie wrote:
    WM <wolfgang.mueckenheim@tha.de> wrote:

    All elements of ℕ are there. That is the assumption. If no greatest
    can be identified, then the reason are dark numbers.
    No, the reason is that there is no greatest element.
    How can that be realizied? If all are there and all are smaller than
    omega, then there is a greatest one. If there is no greatest one, then
    the reason is either that new numbers are created (potential infinity)
    or that dark number become visible. There is no other alternative.
    No. There is no reason for a greatest to exist.

    Give a better explanation.

    "Potential infinity" doesn't exist in modern mathematics.
    Therefore modern mathematics is rubbish. By the way this degenerated
    form of "mathematics" is the first kind which has turned out completely
    useless for any application. Small wonder. It is wrong.
    Any examples?

    Sorry, there is not the slightest example for an application of
    transfinite set theory.

    Regards, WM


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  • From WM@21:1/5 to FromTheRafters on Tue Mar 18 16:41:41 2025
    On 18.03.2025 16:14, FromTheRafters wrote:
    After serious thinking WM wrote :
    On 18.03.2025 13:18, Alan Mackenzie wrote:
    WM <wolfgang.mueckenheim@tha.de> wrote:

    All elements of ℕ are there. That is the assumption. If no greatest can >>>> be identified, then the reason are dark numbers.

    No, the reason is that there is no greatest element.

    How can that be realizied? If all are there and all are smaller than
    omega, then there is a greatest one

    That is just your intuition making you think that. Try mathematical
    thought.

    Apply it and explain how a fixed set can be well-ordered striving for
    omega but not reaching it.

    Regards, WM

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  • From joes@21:1/5 to All on Tue Mar 18 20:45:47 2025
    Am Tue, 18 Mar 2025 16:39:42 +0100 schrieb WM:
    On 18.03.2025 16:08, joes wrote:
    Am Tue, 18 Mar 2025 16:06:21 +0100 schrieb WM:
    On 18.03.2025 13:18, Alan Mackenzie wrote:
    WM <wolfgang.mueckenheim@tha.de> wrote:

    All elements of ℕ are there. That is the assumption. If no greatest >>>>> can be identified, then the reason are dark numbers.
    No, the reason is that there is no greatest element.
    How can that be realizied? If all are there and all are smaller than
    omega, then there is a greatest one. If there is no greatest one, then
    the reason is either that new numbers are created (potential infinity)
    or that dark number become visible. There is no other alternative.
    No. There is no reason for a greatest to exist.
    Give a better explanation.
    Lol. Why does the FSM not exist? (Sacrilege, I know.)

    "Potential infinity" doesn't exist in modern mathematics.
    Therefore modern mathematics is rubbish. By the way this degenerated
    form of "mathematics" is the first kind which has turned out
    completely useless for any application. Small wonder. It is wrong.
    Any examples?
    Sorry, there is not the slightest example for an application of
    transfinite set theory.
    (Modern) mathematics isn't all transfinite set theory. But I meant
    examples where applied mathematics is useless.

    --
    Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
    It is not guaranteed that n+1 exists for every n.

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From joes@21:1/5 to All on Tue Mar 18 20:46:28 2025
    Am Tue, 18 Mar 2025 16:41:41 +0100 schrieb WM:
    On 18.03.2025 16:14, FromTheRafters wrote:
    After serious thinking WM wrote :
    On 18.03.2025 13:18, Alan Mackenzie wrote:
    WM <wolfgang.mueckenheim@tha.de> wrote:

    All elements of ℕ are there. That is the assumption. If no greatest >>>>> can be identified, then the reason are dark numbers.
    No, the reason is that there is no greatest element.
    How can that be realizied? If all are there and all are smaller than
    omega, then there is a greatest one
    That is just your intuition making you think that. Try mathematical
    thought.
    Apply it and explain how a fixed set can be well-ordered striving for
    omega but not reaching it.
    Why shouldn't it? It's infinite.

    --
    Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
    It is not guaranteed that n+1 exists for every n.

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From WM@21:1/5 to joes on Wed Mar 19 12:51:18 2025
    On 18.03.2025 21:46, joes wrote:
    Am Tue, 18 Mar 2025 16:41:41 +0100 schrieb WM:

    Apply it and explain how a fixed set can be well-ordered striving for
    omega but not reaching it.
    Why shouldn't it? It's infinite.

    The dark set does reach omega, but that is denied.

    Regards, WM

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  • From WM@21:1/5 to joes on Wed Mar 19 12:48:31 2025
    On 18.03.2025 21:45, joes wrote:
    Am Tue, 18 Mar 2025 16:39:42 +0100 schrieb WM:
    On 18.03.2025 16:08, joes wrote:

    "Potential infinity" doesn't exist in modern mathematics.
    Therefore modern mathematics is rubbish. By the way this degenerated
    form of "mathematics" is the first kind which has turned out
    completely useless for any application. Small wonder. It is wrong.
    Any examples?
    Sorry, there is not the slightest example for an application of
    transfinite set theory.
    (Modern) mathematics isn't all transfinite set theory.

    But matheologians claim that it was.

    But I meant
    examples where applied mathematics is useless.

    I said that transfinite set theory is useless.

    Regards, WM

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