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  • Re: quantifier shift

    From joes@21:1/5 to All on Sat Oct 5 13:00:22 2024
    Am Sat, 05 Oct 2024 11:43:50 +0200 schrieb WM:
    On 05.10.2024 10:46, Moebius wrote:

    a quantifier shift is NOT reliable und wird daher in der Mathematik
    tunlichst vermieden (und nicht nur dort).
    I many cases it is correct. For instance if every definable natural
    number has ℵo natural successors, then there are ℵo natural numbers larger than all definable natural numbers. They are dark however and
    cannot be specified.
    Since it is logically invalid, you need to prove your deduction
    independently. In general those are two different propositions.

    --
    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 Sat Oct 5 20:49:07 2024
    On 05.10.2024 15:00, joes wrote:
    Am Sat, 05 Oct 2024 11:43:50 +0200 schrieb WM:
    On 05.10.2024 10:46, Moebius wrote:

    a quantifier shift is NOT reliable und wird daher in der Mathematik
    tunlichst vermieden (und nicht nur dort).
    I many cases it is correct. For instance if every definable natural
    number has ℵo natural successors, then there are ℵo natural numbers
    larger than all definable natural numbers. They are dark however and
    cannot be specified.
    Since it is logically invalid, you need to prove your deduction independently. In general those are two different propositions.

    If every definable number has ℵo-infinitely many successors, then no definable number is closer to ω. Then there is a infinite gap between definable numbers and ω.

    Regards, WM

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  • From Richard Damon@21:1/5 to All on Sat Oct 5 23:30:33 2024
    On 10/5/24 2:49 PM, WM wrote:
    On 05.10.2024 15:00, joes wrote:
    Am Sat, 05 Oct 2024 11:43:50 +0200 schrieb WM:
    On 05.10.2024 10:46, Moebius wrote:

    a quantifier shift is NOT reliable und wird daher in der Mathematik
    tunlichst vermieden (und nicht nur dort).
    I many cases it is correct. For instance if every definable natural
    number has ℵo natural successors, then there are ℵo natural numbers
    larger than all definable natural numbers. They are dark however and
    cannot be specified.
    Since it is logically invalid, you need to prove your deduction
    independently. In general those are two different propositions.

    If every definable number has ℵo-infinitely many successors, then no definable number is closer to ω. Then there is a infinite gap between definable numbers and ω.

    Regards, WM



    So?

    Would you expect less that an infinite gap between a finite number and infinity?

    Omega may be the next counting number pass the infinite set of the
    Natural Numbers, but that dpesn't mean the gap for that step is less
    than infinite. If it was only a finite step from the set, it would be
    part of the Natural Numbers, since that includes all the unit steps.

    Note, this also comes from the fact that there is no "last" Natural
    Number to be the one just before Omega, and that step between types of
    numbers has a bigger step then between the Natural Numbers, and bigger
    in a qualative way, a change of order of measurement, from finite to
    infinite.

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  • From joes@21:1/5 to All on Sun Oct 6 11:42:33 2024
    Am Sat, 05 Oct 2024 20:56:48 +0200 schrieb WM:
    On 05.10.2024 15:06, joes wrote:

    What about the gap between the last definable and the first dark UF?
    There is no last element in potential infinity - although it is finite.
    How useless. Every finite set is countable. Potential infinity cannot
    be finite.

    --
    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 Sun Oct 6 11:23:16 2024
    Am Sat, 05 Oct 2024 20:49:07 +0200 schrieb WM:
    On 05.10.2024 15:00, joes wrote:
    Am Sat, 05 Oct 2024 11:43:50 +0200 schrieb WM:
    On 05.10.2024 10:46, Moebius wrote:

    a quantifier shift is NOT reliable und wird daher in der Mathematik
    tunlichst vermieden (und nicht nur dort).
    I many cases it is correct. For instance if every definable natural
    number has ℵo natural successors, then there are ℵo natural numbers
    larger than all definable natural numbers. They are dark however and
    cannot be specified.
    Since it is logically invalid, you need to prove your deduction
    independently. In general those are two different propositions.
    If every definable number has ℵo-infinitely many successors, then no definable number is closer to ω. Then there is a infinite gap between definable numbers and ω.
    Of course. ω is infinite, and the naturals are finite.

    --
    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)