Re: Simple enough for every reader?

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