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NNTP-Posting-Date: Tue, 17 Sep 2024 02:55:23 +0000
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (repleteness)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Mon, 16 Sep 2024 19:55:30 -0700
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On 09/16/2024 11:24 AM, Jim Burns wrote:
> On 9/16/2024 2:13 PM, Jim Burns wrote:
>> On 9/15/2024 9:31 PM, Ross Finlayson wrote:
>>> On 09/15/2024 03:07 PM, FromTheRafters wrote:
>>>> on 9/15/2024, Ross Finlayson supposed :
>>>>> On 09/15/2024 11:03 AM, FromTheRafters wrote:
>>>>>> After serious thinking Ross Finlayson wrote :
>>
>>>>>>> "What, no witty rejoinder?"
>>>>>>
>>>>>> What you said has no relation to
>>>>>> the 'nextness' of elements in discrete sets.
>>>>>> What is 'next' to Pi+2 in the reals?
>>>>>
>>>>> In the, "hyper-reals", it's its neighbors,
>>>>> in the line-reals, put's previous and next,
>>>>> in the field-reals, there's none,
>>>>> and in the signal-reals, there's nothing.
>>
>>>> What is the successor function on the reals?
>>>> Give me that, and maybe we can find the
>>>> 'next' number greater than Pi.
>>>
>>> Ah, good sir, then I'd like you to consider
>>> a representation of real numbers as
>>> with an integer part and a non-integer part,
>>> the integer part of the integers, and
>>> the non-integer part a value in [0,1],
>>> where the values in [0,1], are as of
>>> this model of (a finite segment of a) continuous domain,
>>> these iota-values, line-reals,
>>> as so established as according to the properties of
>>> extent, density, completeness, and measure,
>>> fulfilling implementing the Intermediate Value Theorem,
>>> thus for
>>> if not being the complete-ordered-field the field-reals,
>>> yet being these iota-values a continuous domain [0,1]
>>> these line-reals.
>>
>> As n → ∞, (ι=⅟n), ⟨0,ι,2⋅ι,...,n⋅ι⟩ → ℚ∩[0,1]
>>
>> ℚ∩[0,1] is not complete.
>> ℚ∩[0,1] has one connected component,
>
> Sorry. I meant the opposite of that.
> ℚ∩[0,1] is NOT connected, NOT "continuous".
>
> For each irrational x ∈ (ℝ\ℚ)∩[0,1]
> ℚ∩[0,x) and ℚ∩(x,1] are open in ℚ∩[0,1]
>
>> being what you (RF) call "continuous".
>> ℚ∩[0,1] has no points next to each other.
>>
>>>>> I wonder what you think of something like Hilbert's
>>>>> "postulate of continuity" for geometry, as with
>>>>> regards to that in the course-of-passage of
>>>>> the growth of a continuous quantity, it encounters,
>>>>> in order, each of the points in the line.
>>
>> That sounds like the Intermediate Value Theorem,
>> where "encounters each" == 'no skips".
>>
>> The Intermediate Value Theorem
>> is equivalent to
>> Dedekind completeness.
>>
>>
>

It's shewn that [0,1] has no points not in ran(f).

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