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From: joes <noreply@example.org>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Wed, 2 Jul 2025 19:33:16 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
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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.

>>>>> 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?

-- 
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.