Re: Simple enough for every reader?

On 2025-06-30 18:21:09 +0000, WM said:

On 29.06.2025 12:25, Mikko wrote:

On 2025-06-28 13:56:57 +0000, WM said:
 

On 28.06.2025 11:56, Mikko wrote:

On 2025-06-27 19:36:41 +0000, WM said:
 

On 27.06.2025 09:33, Mikko wrote:

On 2025-06-26 13:09:32 +0000, WM said:

 

 If we subtract in the order that is used for enumerating then a last one is necessary.

 No, there is no last one in an infinite enumeration.

 Then it is not finished or completed.

 No, but it can be continued.

 That is potential infinity. But Cantor claimed complete enumeration.

 There is no mathematical definiton of "complete enumeration"

 The definition of bijection requires completeness.

No, it doesn't. And it doesn't require enumeration.
However, that doesn't really matter as the distinction between complete
and incomplete is not mathematical.

so it is
possible that Cantor's enumeartion is "complete" is one sense and
"incomplete" in another.

 No. Cantor claims all, every and complete:
"The infinite sequence thus defined has the peculiar property to contain the positive rational numbers completely, and each of them only once at a determined place." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]
"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]
"such that every element of the set stands at a definite position of this sequence"

 

The notion set can only be applied to complete sets. i.e., sets which cannot be continued.

 Saying that every set is "complete" does not mean anything,

 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.

All are removed when all are removed.

 When done in natural order, then a last one is to be removed before all are removed. ℕ \ {1, 2, 3, ...} = { }.

 No. You said that every set is complete, so {1, 2, 3, ...}, which must
be a set in order to be valid for the context is complete and so is
ℕ \ {1, 2, 3, ...}, which is just another way to say { }-
 

This cannot be accomplished

 There is nothing to accomplish. What is is, that's all.

 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.

Being completed is not a mathematical concept. An infinite sequence just
is infinite.

     1.1 Cantor's original German terminology on infinite sets
 The reader fluent in German may be interested in the subtleties of Cantor's terminology on actual infinity the finer distinctions of which are not easy to express in English. While Cantor early used "vollständig" and "vollendet" to express "complete" and "finished", the term "fertig", expressing "finished" too but being also somewhat reminiscent of "ready", for the first time appeared in a letter to Hilbert of 26 Sep 1897, where all its appearances had later been added to the letter.
    But Cantor already knew that there are incomplete, i.e., potentially infinite sets like the set of all cardinal numbers. He called them "absolutely infinite". The details of this enigmatic notion are explained in section 1.2 (see also section 4.1. – Unfortunately it has turned out impossible to strictly separate Cantor's mathematical and religious arguments.)

 There is nothing religious in Cantor's arguments. The only traces of
his religious motivations are in the choice of his symbols, in paricular
aleph and omega.

 You are wrong. Here are only few pages of my Book Transfinity:
  4.1 Cantor on theology

Theology is not mathematics.
--
Mikko