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  • Re: quantifier order

    From joes@21:1/5 to All on Mon Sep 9 19:55:41 2024
    Am Mon, 09 Sep 2024 21:03:24 +0200 schrieb WM:
    On 09.09.2024 17:49, joes wrote:
    Am Mon, 09 Sep 2024 17:32:11 +0200 schrieb WM:
    On 09.09.2024 17:15, joes wrote:
    Am Mon, 09 Sep 2024 16:53:32 +0200 schrieb WM:

    You claim that ℵo unit fractions are smaller than ANY x > 0.
    Yes. Not all the same ones of course.
    My question concerns same unit fractions only. Do ℵo unit fractions
    exist smaller than any x > 0? If not, how many same unit fractions
    exist smaller than any x > 0? How many are smalleror equal than all
    unit fractions?
    That is a different question
    No, that is THE question.
    The set of unit fractions smaller than all x is empty.
    However, the set of UFs smaller than one arbitrary x is infinite.

    Of course no unit fraction is smaller than every other unit fraction.
    --
    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 Sep 10 18:24:25 2024
    On 09.09.2024 21:55, joes wrote:
    Am Mon, 09 Sep 2024 21:03:24 +0200 schrieb WM:
    On 09.09.2024 17:49, joes wrote:

    No, that is THE question.
    The set of unit fractions smaller than all x is empty.

    Why is this so? Because all unit fractions are undercut by some x. The x undercutting the last unit fractions (you said that none remains because
    the set is empty, therefore all must be gone) do not satisfy your
    following claim:

    However, the set of UFs smaller than one arbitrary x is infinite.
    Regards, WM

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  • From Richard Damon@21:1/5 to All on Tue Sep 10 21:37:05 2024
    On 9/10/24 12:24 PM, WM wrote:
    On 09.09.2024 21:55, joes wrote:
    Am Mon, 09 Sep 2024 21:03:24 +0200 schrieb WM:
    On 09.09.2024 17:49, joes wrote:

    No, that is THE question.
    The set of unit fractions smaller than all x is empty.

    Why is this so? Because all unit fractions are undercut by some x. The x undercutting the last unit fractions (you said that none remains because
    the set is empty, therefore all must be gone) do not satisfy your
    following claim:

    However, the set of UFs smaller than one arbitrary x is infinite.
    Regards, WM


    Right, because any unit fraction you might want to try to nominate as
    the smallest always has at least one (actually Aleph_0) unit fractions
    smaller then it, and thus NO unit fractions are less than or equal to
    all unit fractions.

    You just don't understand that mathematic of the infinite set, which
    follow rules DIFFERENT from your normal rules for finite sets.

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