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  • Re: density of the reals

    From joes@21:1/5 to All on Wed Sep 11 06:05:03 2024
    Am Tue, 10 Sep 2024 21:13:57 +0200 schrieb WM:
    On 10.09.2024 20:38, Chris M. Thomasson wrote:
    It seems that any non-zero gap can have unit fractions small enough to
    fit in it...
    A gap of countably many points is not sufficient but is a subgap of
    every gap between unit fractions.
    Why is it not sufficient? Since when are the reals countable?

    --
    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 Wed Sep 11 16:33:14 2024
    On 11.09.2024 08:05, joes wrote:
    Am Tue, 10 Sep 2024 21:13:57 +0200 schrieb WM:
    On 10.09.2024 20:38, Chris M. Thomasson wrote:
    It seems that any non-zero gap can have unit fractions small enough to
    fit in it...
    A gap of countably many points is not sufficient but is a subgap of
    every gap between unit fractions.
    Why is it not sufficient?

    Because between any two unit fractions there are uncountably many points.

    Since when are the reals countable?

    Every uncountable set has a countable subset, in fact uncountably many.
    The countable set (0, a) is a subinterval of the uncountable set (0, d) comprising ℵo unit fractions. No x of (0, a) can have smaller unit
    fractions. Well, perhaps one.

    Regards, WM


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