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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Wed, 2 Jul 2025 22:56:38 +0200
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On 02.07.2025 21:33, joes wrote:
> Am Wed, 02 Jul 2025 21:23:22 +0200 schrieb WM:
>> On 02.07.2025 21:05, joes wrote:
>>> Am Wed, 02 Jul 2025 15:51:01 +0200 schrieb WM:
>>>> On 02.07.2025 09:45, Mikko wrote:
>>>>> On 2025-06-30 18:21:09 +0000, WM said:
>>>>>> On 29.06.2025 12:25, Mikko wrote:
>>>
>>>>>> It means that no further element can be found later on.
>>>>> Whether an element is "found" has no mathematical meaning and in
>>>>> particular does not affect its being or not a member of some set.
>>>> "Numerals constitute a potential infinity. Given any numeral, we can
>>>> construct a new numeral by prefixing it with S." [E. Nelson:
>>>> "Hilbert's mistake" (2007) p. 3]
>>> Yeah, nothing about "finding" in there.
>> "Should we briefly characterize the new view of the infinite introduced
>> by Cantor, we could certainly say: In analysis we have to deal only with
>> the infinitely small and the infinitely large as a limit-notion, as
>> something becoming, emerging, produced, i.e., as we put it, with the
>> potential infinite. But this is not the proper infinite. That we have
>> for instance when we consider the entirety of the numbers 1, 2, 3, 4,
>> ... itself as a completed unit, or the points of a line as an entirety
>> of things which is completely available. That sort of infinity is named
>> actual infinite." [D. Hilbert: "Über das Unendliche", Mathematische
>> Annalen 95 (1925) p. 167]
> Nothing about "finding" in there either.

It is the contrary of "completely available".

> 
>>>>>> Then it cannot be. If it is that all natural numbers are subtracted
>>>>>> in their order, then it is that a last one is subtracted.
>>>>> Given two sets there is a set that is their difference. There is no
>>>>> opeartion of subtraction in order.
>>>> The set ℕ has an intrinsic order which can be used at any time.
>>>> Bijecting sets presupposes and requires order. Further the difference
>>>> of sets depends strongly on the order assumed.
>>> Bijections don't require order. Set difference has no order.
>> "thus we get the epitome (ω) of all real algebraic numbers [...] and
>> with respect to this order we can talk about the th algebraic number
>> where not a single one of this epitome () has been forgotten." [E.
>> Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und
>> philosophischen Inhalts", Springer, Berlin (1932) p. 116]
> Do you also have your own words to miss the topic?

Bijections don't require order.
with respect to this order we can talk about the th algebraic number

Regards, WM