• Re: Replacement of Cardinality --- infinitesimal number system

    From Moebius@21:1/5 to All on Wed Jul 31 01:54:59 2024
    XPost: sci.math

    Am 31.07.2024 um 01:46 schrieb olcott:
    On 7/30/2024 5:49 PM, Moebius wrote:
    Am 30.07.2024 um 20:37 schrieb Jim Burns:
    On 7/30/2024 1:30 PM, WM wrote:

    [...] between [0, 1] and (0, 1].

    Could you please explain to me the meaning of the phrase "between [0,
    1] and (0, 1]"?

    One can construe them as line segments that differ in length
    by a single geometric point on the number line. I came up
    with that a few years ago with my infinitesimal number system
    that <bla>

    Hint: IR does not comtain any "infinitesimal numbers" (except 0 that is).

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  • From Moebius@21:1/5 to All on Wed Jul 31 03:19:22 2024
    XPost: sci.math

    Am 31.07.2024 um 02:11 schrieb olcott:
    On 7/30/2024 6:54 PM, Moebius wrote:
    Am 31.07.2024 um 01:46 schrieb olcott:
    On 7/30/2024 5:49 PM, Moebius wrote:
    Am 30.07.2024 um 20:37 schrieb Jim Burns:
    On 7/30/2024 1:30 PM, WM wrote:

    [...] between [0, 1] and (0, 1].

    Could you please explain to me the meaning of the phrase "between
    [0, 1] and (0, 1]"?

    One can construe them as line segments that differ in length
    by a single geometric point on the number line. I came up
    with that a few years ago with my infinitesimal number system
    that <bla>

    Hint: IR does not comtain any "infinitesimal numbers" (except 0 that is).

    It is the case that the infinite set of points
    (0, 1] has exactly one point less than the infinite
    set of points [0, 1].

    Indeed! The point 0.

    [0, 1] \ (0, 1] = {0}.

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  • From Moebius@21:1/5 to All on Wed Jul 31 03:33:31 2024
    XPost: sci.math

    Am 31.07.2024 um 03:30 schrieb olcott:
    On 7/30/2024 8:19 PM, Moebius wrote:
    Am 31.07.2024 um 02:11 schrieb olcott:
    On 7/30/2024 6:54 PM, Moebius wrote:
    Am 31.07.2024 um 01:46 schrieb olcott:
    On 7/30/2024 5:49 PM, Moebius wrote:
    Am 30.07.2024 um 20:37 schrieb Jim Burns:
    On 7/30/2024 1:30 PM, WM wrote:

    [...] between [0, 1] and (0, 1].

    Could you please explain to me the meaning of the phrase "between
    [0, 1] and (0, 1]"? [...]

    One can construe them as line segments that differ in length
    by a single geometric point on the number line. I came up
    with that a few years ago with my infinitesimal number system
    that <bla>

    Hint: IR does not comtain any "infinitesimal numbers" (except 0 that
    is).

    It is the case that the infinite set of points (0, 1] has exactly one
    point less than the infinite
    set of points [0, 1].

    Indeed! The point 0.

    [0, 1] \ (0, 1] = {0}.

    Yes. That is the point.

    Remember that I originally wrote:

    "Could you please explain to me the meaning of the phrase "between [0,
    1] and (0, 1]"? What IS "between [0, 1] and (0, 1]". (Which real numbers
    are "between [0, 1] and (0, 1]"? 0? So why not just talk about 0 in this case?)"

    Thank you for your contribution. EOD.

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  • From Jim Burns@21:1/5 to olcott on Wed Jul 31 00:23:20 2024
    XPost: sci.math

    On 7/30/2024 7:46 PM, olcott wrote:
    On 7/30/2024 5:49 PM, Moebius wrote:
    Am 30.07.2024 um 20:37 schrieb Jim Burns:
    On 7/30/2024 1:30 PM, WM wrote:

    [...] between [0, 1] and (0, 1].

    Could you please explain to me
    the meaning of the phrase "between [0, 1] and (0, 1]"?

    The meaning of that phrase is {}.

    In WM's mind, {} proves darkᵂᴹ numbers exist by
    the application of (at least)
    shifting quantifiers,
    assuming what he intends to prove, and
    thinking that 'finite' and 'infinite' mean something they don't.

    One can construe them as line segments that
    differ in length by a single geometric point
    on the number line.
    I came up with that a few years ago with my
    infinitesimal number system that
    seems to map integers to a contiguous set of
    immediately adjacent geometric points
    on the number line.

    Consider a number line in which
    each split is situated.

    ⎛ In a situated split F′ ᵉᵃᶜʰ<ᵉᵃᶜʰ H′
    ⎜ a number.line.point x′ exists
    ⎝ either last.in.F′ or first.in.H′

    Consider the unit.fractions ⅟ℕ = {⅟j: finite j ∈ ℕ₁}

    The set F′ of lower.bounds of ⅟ℕ and
    the set H′ of not.lower.bounds of ⅟ℕ
    is a split of the number.line

    x′ situates F′ ᵉᵃᶜʰ<ᵉᵃᶜʰ H′

    0 is a lower.bound of ⅟ℕ
    Therefore, x′ isn't < 0

    x′ isn't > 0
    ⎛ Assume otherwise.
    ⎜ Assume x′ > 0
    ⎜ 2⋅x′ > x′ > ½⋅x′ > 0

    ⎜ ½⋅x′ < x′ is a lower.bound of ⅟ℕ

    ⎜ 2⋅x′ > x′ isn't a lower.bound of ⅟ℕ
    ⎜ finite unit.fraction ⅟k < 2⋅x′
    ⎜ finite unit.fraction ¼⋅⅟k < ¼⋅2⋅x′ = ½⋅x′
    ⎜ ½⋅x′ isn't a lower.bound of ⅟ℕ

    ⎝ Contradiction.

    Therefore, x′ isn't > 0
    x′ isn't < 0
    x′ = 0
    0 situates the split between
    lower.bounds and not.lower.bounds of ⅟ℕ

    No point δ > 0 is a lower.bound of ⅟ℕ
    Each point δ > 0 has a finite unit.fraction
    between δ and 0
    No point δ > 0 is an infinitesimal.

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