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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 21:23:22 +0200
Organization: A noiseless patient Spider
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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]

>>>> 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]

Regards, WM