Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

Liste des GroupesRevenir à s math 
Sujet : Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
De : james.g.burns (at) *nospam* att.net (Jim Burns)
Groupes : sci.math
Date : 14. Dec 2024, 05:54:47
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <4327ba4a-e810-4565-83e8-b9572018ac35@att.net>
References : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
User-Agent : Mozilla Thunderbird
On 12/13/2024 2:31 PM, WM wrote:
On 13.12.2024 20:00, Jim Burns wrote:
On 12/13/2024 6:25 AM, WM wrote:

Ignoring that Cantor's claim requires to
empty the endsegments from all natural numbers
in order to use them as indices in mappings
>
Each finite.cardinal
⎛ is first in an end.segment.
⎝ is used as a index of an end.segment.
>
Each finite.cardinal
⎛ is absent from an end segment.
⎝ is emptied from the end.segments.
>
Require at will, sir.
>
If endegments were defined as
E(n) = {n+1, n+2, ...}:
>
E(0) = {1, 2, 3, ...}
E(1) = {2, 3, 4, ...}
E(2) = {3, 4, 5, ...}
...
E(ω-1} = { }.
Then this change from content to index
would even be more obvious.
The index of Eᑉ(2) is content of Eᑉ(1), etc.
The number doesn't change.
Which after.segment changes.
----
One problem which
Eᑉ(ω-1) = {}
has  is that
'finite' is NOT defined the way in which
you (WM) think 'finite' should be,
which means
ω is NOT defined the way in which
you (WM) think ω should be.
ω is the set of finite.ordinals.
k ∈ ⟦0,ω⦆  :⇔  finite.ordinal k
Part of what we mean by 'finite k' is that
the priors ⟦0,k+1⦆ of k+1 cannot map to
the priors ⟦0,k⦆ of k
¬∃¹ᵗᵒ¹f:⟦0,k+1⦆⇉⟦0,k⦆
It is a theorem[1] that
⎛ if ¬∃¹ᵗᵒ¹f:⟦0,k+1⦆⇉⟦0,k⦆
⎝ then ¬∃¹ᵗᵒ¹g:⟦0,k+2⦆⇉⟦0,k+1⦆
Which means
⎛ if k is finite
⎝ then k+1 is finite
[1]
⎛ If ∃¹ᵗᵒ¹g:⟦0,k+2⦆⇉⟦0,k+1⦆
⎜ then ∃¹ᵗᵒ¹f:⟦0,k+1⦆⇉⟦0,k⦆
⎜⎛ Define
⎜⎜ f(g⁻¹(k+1)) = g(k+2)
⎜⎝ otherwise f(j) = g(j)

⎜ Contrapositively,
⎜ if ¬∃¹ᵗᵒ¹f:⟦0,k+1⦆⇉⟦0,k⦆
⎝ then ¬∃¹ᵗᵒ¹g:⟦0,k+2⦆⇉⟦0,k+1⦆
Therefore,
k ∈ ⟦0,ω⦆  ⇒  k+1 ∈ ⟦0,ω⦆
Consider the after.segments of the finite.ordinals.
Define
Eᑉ(n) = {n+1, n+2, ...} ⊆ ⟦0,ω⦆
Eᑉ(0) = {1, 2, 3, ...} ⊆ ⟦0,ω⦆
Eᑉ(1) = {2, 3, 4, ...} ⊆ ⟦0,ω⦆
Eᑉ(2) = {3, 4, 5, ...} ⊆ ⟦0,ω⦆
...
⎛ Assume Eᑉ(ω-1) = {}

⎜ ∀k ∈ N: Eᑉ(n) = {n+1}∪Eᑉ(n+1)
⎜ ω-1 ∈ Eᑉ(ω-2)

⎜ k ∈ ⟦0,ω⦆  ⇒  k+1 ∈ ⟦0,ω⦆
⎜ ω-1 ∈ Eᑉ(ω-2)  ⇒
⎜ (ω-1)+1 ∈ Eᑉ(ω-2)  ⇒
⎜ (ω-1)+2 ∈ Eᑉ(ω-2)  ⇒
⎜ (ω-1)+3 ∈ Eᑉ(ω-2)  ⇒
⎜ ...

⎜ Eᑉ(ω-2) = {ω-1, (ω-1)+1, (ω-1)+2, ...}
⎜ Eᑉ(ω-1) = {(ω-1)+1, (ω-1)+2, ...} ≠ {}
⎝ Contradiction.
No last finite ordinal ω-1 exists.
Its existence gives us the contradictions.
No end.segment E(ω-1) exists.
Its existence gives us the contradictions.

Each set has
its emptier.by.one and fuller.by.one counterparts.
>
If all natnumbers can be used for mappings as indices
then every natnumber has to leave the sequence of endsegments.
Do you agree?
Yes.

Therefore their intersection is empty.
Do you agree?
Yes.

Emptying by one only implies
finite endsegment intersetcions.
Do you agree?
No.

If not describe the process according to your opinion.
Each end.segment ⟦k,ω⦆ of ⟦0,ω⦆ contains,
for each finite.cardinal j
a subset ⟦k,k+j⟧ holding more.than.j.many.
That contradicts |⟦k,ω⦆| being finite.
No _finite_ cardinal j = |⟦k,ω⦆| which doesn't
give us the contradictions  exists.

Date Sujet#  Auteur
27 Nov 24 * Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)873WM
27 Nov 24 +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)2joes
27 Nov 24 i`- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1WM
28 Nov 24 `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)870Jim Burns
28 Nov 24  +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)868WM
28 Nov 24  i+* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)860joes
28 Nov 24  ii`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)859WM
28 Nov 24  ii +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)4joes
28 Nov 24  ii i`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)3WM
28 Nov 24  ii i `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)2joes
28 Nov 24  ii i  `- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1WM
28 Nov 24  ii `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)854FromTheRafters
28 Nov 24  ii  `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)853WM
28 Nov 24  ii   +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)2joes
29 Nov 24  ii   i`- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1WM
29 Nov 24  ii   `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)850FromTheRafters
29 Nov 24  ii    +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)848WM
29 Nov 24  ii    i`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)847FromTheRafters
29 Nov 24  ii    i `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)846WM
29 Nov 24  ii    i  `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)845FromTheRafters
30 Nov 24  ii    i   `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)844WM
30 Nov 24  ii    i    `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)843FromTheRafters
30 Nov 24  ii    i     `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)842WM
30 Nov 24  ii    i      +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)10FromTheRafters
30 Nov 24  ii    i      i+* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)7WM
30 Nov 24  ii    i      ii`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)6joes
30 Nov 24  ii    i      ii `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)5WM
30 Nov 24  ii    i      ii  `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)4joes
30 Nov 24  ii    i      ii   `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)3WM
1 Dec 24  ii    i      ii    `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)2joes
1 Dec 24  ii    i      ii     `- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1WM
30 Nov 24  ii    i      i`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary, infinite-middle)2Ross Finlayson
2 Dec 24  ii    i      i `- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary, infinite-middle)1Chris M. Thomasson
2 Dec 24  ii    i      `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)831Chris M. Thomasson
2 Dec 24  ii    i       +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)90Moebius
3 Dec 24  ii    i       i`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)89Chris M. Thomasson
3 Dec 24  ii    i       i +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)84Moebius
3 Dec 24  ii    i       i i+- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1Chris M. Thomasson
3 Dec 24  ii    i       i i`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)82Chris M. Thomasson
3 Dec 24  ii    i       i i `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)81Moebius
3 Dec 24  ii    i       i i  `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)80Chris M. Thomasson
3 Dec 24  ii    i       i i   `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)79Chris M. Thomasson
3 Dec 24  ii    i       i i    +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)17Moebius
3 Dec 24  ii    i       i i    i`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)16Chris M. Thomasson
3 Dec 24  ii    i       i i    i +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)3Chris M. Thomasson
3 Dec 24  ii    i       i i    i i+- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1Chris M. Thomasson
3 Dec 24  ii    i       i i    i i`- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1Moebius
3 Dec 24  ii    i       i i    i +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)2Moebius
3 Dec 24  ii    i       i i    i i`- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1WM
3 Dec 24  ii    i       i i    i +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)9Chris M. Thomasson
3 Dec 24  ii    i       i i    i i+* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)7Chris M. Thomasson
3 Dec 24  ii    i       i i    i ii`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)6Chris M. Thomasson
3 Dec 24  ii    i       i i    i ii +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)4Moebius
3 Dec 24  ii    i       i i    i ii i`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)3Moebius
3 Dec 24  ii    i       i i    i ii i `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)2Chris M. Thomasson
3 Dec 24  ii    i       i i    i ii i  `- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1Moebius
3 Dec 24  ii    i       i i    i ii `- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1Chris M. Thomasson
3 Dec 24  ii    i       i i    i i`- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1Moebius
3 Dec 24  ii    i       i i    i `- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1Moebius
3 Dec 24  ii    i       i i    `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)61Ben Bacarisse
3 Dec 24  ii    i       i i     +- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1WM
3 Dec 24  ii    i       i i     `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)59Chris M. Thomasson
3 Dec 24  ii    i       i i      +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)55Moebius
3 Dec 24  ii    i       i i      i`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)54Moebius
4 Dec 24  ii    i       i i      i +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)52Chris M. Thomasson
4 Dec 24  ii    i       i i      i i`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)51Moebius
4 Dec 24  ii    i       i i      i i `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)50Moebius
4 Dec 24  ii    i       i i      i i  `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)49FromTheRafters
4 Dec 24  ii    i       i i      i i   `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)48Ben Bacarisse
4 Dec 24  ii    i       i i      i i    `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)47Moebius
4 Dec 24  ii    i       i i      i i     +- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1FromTheRafters
4 Dec 24  ii    i       i i      i i     `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)45Ben Bacarisse
4 Dec 24  ii    i       i i      i i      +- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1FromTheRafters
4 Dec 24  ii    i       i i      i i      `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)43Chris M. Thomasson
4 Dec 24  ii    i       i i      i i       +- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1Ben Bacarisse
5 Dec 24  ii    i       i i      i i       `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)41WM
5 Dec 24  ii    i       i i      i i        +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)19joes
5 Dec 24  ii    i       i i      i i        i`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)18WM
5 Dec 24  ii    i       i i      i i        i +- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1Richard Damon
5 Dec 24  ii    i       i i      i i        i +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)8joes
5 Dec 24  ii    i       i i      i i        i i`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)7WM
6 Dec 24  ii    i       i i      i i        i i `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)6joes
7 Dec 24  ii    i       i i      i i        i i  `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)5WM
7 Dec 24  ii    i       i i      i i        i i   +- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1joes
7 Dec 24  ii    i       i i      i i        i i   `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)3Richard Damon
7 Dec 24  ii    i       i i      i i        i i    `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)2Chris M. Thomasson
7 Dec 24  ii    i       i i      i i        i i     `- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1Richard Damon
5 Dec 24  ii    i       i i      i i        i `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)8FromTheRafters
5 Dec 24  ii    i       i i      i i        i  +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)5WM
5 Dec 24  ii    i       i i      i i        i  i`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)4FromTheRafters
5 Dec 24  ii    i       i i      i i        i  i `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)3WM
5 Dec 24  ii    i       i i      i i        i  i  `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)2FromTheRafters
6 Dec 24  ii    i       i i      i i        i  i   `- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1WM
6 Dec 24  ii    i       i i      i i        i  `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)2Chris M. Thomasson
6 Dec 24  ii    i       i i      i i        i   `- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1Moebius
5 Dec 24  ii    i       i i      i i        +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)20Richard Damon
5 Dec 24  ii    i       i i      i i        i+* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)18WM
6 Dec 24  ii    i       i i      i i        ii+* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)8Richard Damon
6 Dec 24  ii    i       i i      i i        iii`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)7WM
6 Dec 24  ii    i       i i      i i        iii +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)5joes
6 Dec 24  ii    i       i i      i i        iii i`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)4WM
6 Dec 24  ii    i       i i      i i        iii `- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1Richard Damon
6 Dec 24  ii    i       i i      i i        ii`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)9Chris M. Thomasson
6 Dec 24  ii    i       i i      i i        i`- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1Chris M. Thomasson
6 Dec 24  ii    i       i i      i i        `- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1Chris M. Thomasson
4 Dec 24  ii    i       i i      i `- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1WM
4 Dec 24  ii    i       i i      `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)3Ben Bacarisse
3 Dec 24  ii    i       i `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)4Jim Burns
2 Dec 24  ii    i       +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)2Moebius
2 Dec 24  ii    i       +* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)2Moebius
2 Dec 24  ii    i       `* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)736FromTheRafters
29 Nov 24  ii    `- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1Ross Finlayson
29 Nov 24  i`* Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)7Jim Burns
28 Nov 24  `- Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)1Ross Finlayson

Haut de la page

Les messages affichés proviennent d'usenet.

NewsPortal