For every line n of the Cantor-list we see that the antidiagonal differs from its entry. For every level of the Binary Tree we see that the number of paths that can be distinguished by nodes is finite.

Either both results can be extrapolated to the infinite limit, then uncountability is contradicted by the Binary Tree because everywhere only finitely many paths can be distinguished.

Or both results are not acceptable in the infinite limit, then the diagonal argument fails, and there is no proof of uncountability.

Regards, WM

On 1/11/2023 5:59 AM, WM wrote:
> For every line n of the Cantor-list we see that the antidiagonal differs from its entry. For every level of the Binary Tree we see that the number of paths that can be distinguished by nodes is finite.
>
> Either both results can be extrapolated to the infinite limit, then uncountability is contradicted by the Binary Tree because everywhere only finitely many paths can be distinguished.
>
> Or both results are not acceptable in the infinite limit, then the diagonal argument fails, and there is no proof of uncountability.
>
> Regards, WM

so now you replaced "Hilberts Hotel" with "WMs Binary Tree",

previously you had replaced your "Swapparoo Matrix" with "Hilberts Hotel",

before that you replaced "Framed Achilles the loser to turtles" w your "Swapadoodle Matrix"

Searching for a topic to replace good math practices is a fool's errand.

On 1/11/2023 3:59 AM, WM wrote:
> For every line n of the Cantor-list we see that the antidiagonal differs from its entry. For every level of the Binary Tree we see that the number of paths that can be distinguished by nodes is finite.

Huh? I am quite fond of trees, and an infinite binary tree has no
leaves. Do you know why?

>
> Either both results can be extrapolated to the infinite limit, then uncountability is contradicted by the Binary Tree because everywhere only finitely many paths can be distinguished.
>
> Or both results are not acceptable in the infinite limit, then the diagonal argument fails, and there is no proof of uncountability.
>
> Regards, WM

On 1/11/2023 7:35 AM, Sergi o wrote:
> On 1/11/2023 5:59 AM, WM wrote:
>> For every line n of the Cantor-list we see that the antidiagonal
>> differs from its entry. For every level of the Binary Tree we see that
>> the number of paths that can be distinguished by nodes is finite.
>>
>> Either both results can be extrapolated to the infinite limit, then
>> uncountability is contradicted by the Binary Tree because everywhere
>> only finitely many paths can be distinguished.
>>
>>   Or both results are not acceptable in the infinite limit, then the
>> diagonal argument fails, and there is no proof of uncountability.
>>
>> Regards, WM
>
>  so now you replaced "Hilberts Hotel" with "WMs Binary Tree",
>
>    previously you had replaced your "Swapparoo Matrix" with "Hilberts
> Hotel",
>
>       before that you replaced "Framed Achilles the loser to turtles" w
> your "Swapadoodle Matrix"
>
>
> Searching for a topic to replace good math practices is a fool's errand.

No shit. For some reason I don't think WM can even describe what an
infinite n-ary tree is.

"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:

> No shit. For some reason I don't think WM can even describe what an
> infinite n-ary tree is.

You have that right. In fact, there is a degree of deception going on
because what WM is talking about is not the tree of graph theory. He is
borrowing the term to mean some weird nonsense of his own. For example,
in a WMtree removing a node does not remove every path containing it.

--
Ben.

onsdag 11 januari 2023 kl. 12:59:10 UTC+1 skrev WM:
> For every line n of the Cantor-list we see that the antidiagonal differs from its entry. For every level of the Binary Tree we see that the number of paths that can be distinguished by nodes is finite.
>
> Either both results can be extrapolated to the infinite limit, then uncountability is contradicted by the Binary Tree because everywhere only finitely many paths can be distinguished.
>
> Or both results are not acceptable in the infinite limit, then the diagonal argument fails, and there is no proof of uncountability.
>
> Regards, WM

There is no contradiction and your "distinguished" is meaningless.

For village idiots of Germany-WM- with his slant cut of cone a ellipse when it truth it is a oval.

William,Astrid Pundt,Arnulf Quadt,Karl-Henning Rehren,WM,Wolfgang M. Weyand,Brigitta Wolff,Metin Tolan,Anja Karliczek,Susanne Schneider,Steffen Schumann, Cynthia A. Volkert are they forever going to be mindless idiots of math--slant cut of cone is oval, never ellipse geometry.
> On Tuesday, January 10, 2023 at 3:08:14 PM UTC-6, Dan Christensen wrote:
> > STUDENTS BEWARE: Don't be a victim
> > Time for another spanking,
>
> > And here you can plainly see the side EC is smaller than CF, while a ellipse requires them to be equal
> > ......./\E
> > ...../ c.\
> > F / .......\
> >
> > entrance E of planar cut, and "c" the center axis, and F the exit of planar cut, cF is far larger than Ec.
> >
> > Are they trying to turn sci.math into a gay pick up bar rather than legitimate math with endless day after day spam?

William 96 "Who recognizes.. dark numbers"
Fritz Feldhase 95 "Who recognizes...dark numbers"

> 
> > WM 30 "Who recognizes... dark numbers"
> 
> > Sergi_o 68, "Questions on...."
> > Wolfgang Mueckenheim 324 "Three proofs of dark numbers..."
> > of the fine Bacarisse cheese
> > Ben Bacarisse 323 "Three proofs of dark numbers..."
> > FromTheRafters (more like gutters) 221 "Two similar..."
> > Sergi_o 161 "Two similar..."
> > Ben Bacarisse 133, "Two similar properties..."
> >
> >
> >
> > WM turning Gottingen Germany into geometry failures.
> > Kibo chasing corpses in WM gay pick up bar, for Kibo sure does not know the difference between Oval and ellipse as seen in his latest stupidity-- a revolving axis as Kibo the moron and BWR describes it--
> >
> > Let's try again with your little diagram, fixed.
> > > > >>> .......A
> > > > >>> ....../.\E
> > > > >>> ...../.C.\
> > > > >>> ..../B....\
> > > > >>> .../.......\
> > > > >>> .F/....G....\
> > > >
> > > > You chose point C to be the intersection of line segment EF and the axis
> > > > of the cone AG. As you say segment EC is smaller than CF. Not in
> > > > dispute. But that doesn't show that the intersecting curve isn't
> > > > symmetric around EF. C is not the center of EF. To see if the
> > > > intersection figure is symmetric around EF, you obviously have to start
> > > > at the center, the halfway point of EF. Here I called it B, and the
> > > > length BE = length BF. Now you need to show that the curve is or is not
> > > > symmetric around B. That the width at B+d is or is not = the width at
> > > > B-d, as bwr stated. Not quite as simple.
> > >
> >
> > On Monday, December 5, 2022 at 1:48:16 PM UTC-6, Michael Moroney wrote:
> > > On 12/5/2022 12:22 AM, Chris M. Thomasson wrote:
> > > > On 12/4/2022 8:19 PM, Michael Moroney wrote:
> > > >> On 12/4/2022 7:43 PM, Mathin3D wrote:
> > > >>> Serious Question:
> >
> >
> > Kibo Parry M on Harvard's Dr. Hau.
> > Kibo on > I want to fuck her corpse
> > On Sunday, September 11, 2022 at 12:07:15 AM UTC-5, Michael Moroney wrote:
> > > I want to fuck her corpse
> > Why Kibo??? Because she refuses to finish her experiment and see all the light vanishes simultaneously, or is it because she can not admit slant cut of cone is Oval, never ellipse???
> >
> > On Wednesday, November 2, 2022 at 12:21:30 AM UTC-5, Michael Moroney wrote:
> > >"Imp of Math"
> > On Monday, January 18, 2021 at 2:06:24 PM UTC-6, Michael Moroney wrote:
> > > fails at math and science:
> >
> > Re: Drs.Larry Summers, Sheldon Glashow, Lisa Randall of Harvard, teach percentages correctly??-- Moroney//never realizing the Real Electron = muon, proton=840MeV, .5MeV = Dirac's monopole (1)
> > By Michael Moroney 1/23/18, 44 posts 461 views
> >
> > On Sunday, September 11, 2022 at 12:07:15 AM UTC-5, Michael Moroney wrote:
> > > I want to fuck her corpse
> > Kibo Parry Moroney in 1997 blows his CIA cover-- to the entire world, mind you---
> > Re: Archimedes Vanadium, America's most beloved poster
> > >> In article <5nefan$i06$9...@news.thecia.net> kibo greps <ki...@shell..thecia.net> writes:
> > > >
> >
> > On Tuesday, November 1, 2022 at 9:13:14 AM UTC-5, Michael Moroney wrote:
> > >"Court Jester of Math"
> >
> > Kibo,is Ruth Charney of AMS the model in your book, for she cannot admit the slant cut of cone is Oval, never the ellipse as proven by AP in 2016? I know it falls on deaf ears of Kibo with his 938 is 12% short of 945. And Ruth, would the AMS publish Kibo's 938 is 12% short of 945, because the AMS certainly will not publish AP's conic proof? Maybe that is all the AMS publishes-- fake math.
> >
> > On Sunday, September 11, 2022 at 12:07:15 AM UTC-5, Michael Moroney wrote:
> > > I want to fuck her corpse
> > On Monday, September 19, 2022 at 3:21:37 PM UTC-5, Michael Moroney wrote:
> > > fails at math and science:
> >
> > On Sunday, September 18, 2022 at 12:43:52 AM UTC-5, Michael Moroney wrote:
> > >"bozo"
> > > fails at math and science:
> > >"psychoceramic"
> >
> > On Tuesday, September 13, 2022 at 10:40:13 AM UTC-5, Michael Moroney wrote:
> > >"physics hater"
> > > tarded:
> > > Drag Queen of Science, especially Physics
> >
> > AP no longer tolerates 30 year nonstop stalker shitheads like Kibo Parry M.
> > > No wonder Dr.Tao is a failed mathematician who cannot see that |/\| planar cut simultaneously through cylinder and cone has to yield a Oval in cone, a ellipse in cylinder. Tao fails fails fails math, and no wonder the nitwit cannot do a geometry proof of Fundamental Theorem of Calculus.
> > >
> > > On Sunday, December 4, 2022 at 6:19:00 PM UTC-6, Michael Moroney wrote:
> > > > On 12/4/2022 5:52 PM, bwr fml wrote:
> > > > > On Sunday, December 4, 2022 at 2:39:25 PM UTC-8, Archimedes Plutonium wrote:
> > > > >>> And here you can plainly see the side EC is smaller than CF, while a ellipse requires them to be equal
> > > > >>> ......./\E
> > > > >>> ...../ c.\
> > > > >>> F / .......\
> > > > >
> > > > > If you are thinking your cone is symmetric around the z axis and the point c is on the z axis
> > > > > then you claim that Ec is less than cF is true.
> > > > > And that means absolutely nothing about whether the intersection of the cone and plane is an ellipse.
> > > > >
> > > > > The fundamental crank mistake you made a decade ago is thinking that the point c on the z axis
> > > > > has ANYTHING to do with whether the intersection is an ellipse or not.
> > > > >
> > > > > As I clearly showed a couple of days ago, the algebra CLEARLY shows that your point c needs to be
> > > > > exactly half way between the point E and the point F. Then the distances Ec and cF are equal.
> > > > > BUT, far more important, all the widths of the intersection in that plane for any distance d to the left
> > > > > and the distance d to the right of that point c are then precisely equal and that is the definition
> > > > > of a plane of symmetry and the intersection is an ellipse.
> > > > >
> > > > > But you have screeched your false claim out your crank hole for long enough and you DESPERATELY
> > > > > need to imagine that you are right and everyone else in the history of the world is wrong... that you
> > > > > probably can't ever admit that you made this mistake and just keep screeching it.
> > > > >
> > > > > Go out in your yard and scream it loud and proud:
> > > > > RETARDED, RETARDED, ARCHIE IS RETARDED
> > > > > and keep crank screaming that until they call the cops on you.
> > > > bwr is correct, Archie. Let's try again with your little diagram, fixed.
> > > > >>> .......A
> > > > >>> ....../.\E
> > > > >>> ...../.C.\
> > > > >>> ..../B....\
> > > > >>> .../.......\
> > > > >>> .F/....G....\
> > > >
> > > > You chose point C to be the intersection of line segment EF and the axis
> > > > of the cone AG. As you say segment EC is smaller than CF. Not in
> > > > dispute. But that doesn't show that the intersecting curve isn't
> > > > symmetric around EF. C is not the center of EF. To see if the
> > > > intersection figure is symmetric around EF, you obviously have to start
> > > > at the center, the halfway point of EF. Here I called it B, and the
> > > > length BE = length BF. Now you need to show that the curve is or is not
> > > > symmetric around B. That the width at B+d is or is not = the width at
> > > > B-d, as bwr stated. Not quite as simple.
> > >
> > > AP comments: so look at the moron Moroney (Kibo Parry) being a spokesperson for failed Dr. Tao with his "move around axis of the cone-- say Kibo and Tao is there a witch flying around on a broomstick in that cone also???
> >
> > > > > Universitat Augsburg, Germany, rector Sabine Doering-Manteuffel
> > > > > Math dept Ronald H.W.Hoppe, B. Schmidt, Sarah Friedrich, Stefan Grosskinsky, Friedrich Pukelsheim, Mirjam Dur, Ralf Werner.
> > > > >
> > > > > Hochschule Augsburg, Wolfgang Mueckenheim
> > > > >
> > > > > Gottingen Univ math
> > > > >
> > > > > Dorothea Bahns, Laurent Bartholdi, Valentin Blomer, Jorg Brüdern, Stefan Halverscheid, Harald Andres Helfgott, Madeleine Jotz Lean, Ralf Meyer, Preda Mihailescu, Walther Dietrich Paravicini, Viktor Pidstrygach, Thomas Schick, Evelina Viada, Ingo Frank Witt, Chenchang Zhu
> > > > >
> > > > > Eternal-September.org
> > > > > Wolfgang M. Weyand
> > > > > Berliner Strasse
> > > > > Bad Homburg
> > > > >
> > > > > Goethe Universitat Physics dept
> > > > >
> > > > > Brigitta Wolff president
> > > > >
> > > > > Jurgen Habermass
> > > > > Horst Stocker
> > > > > Gerd Binnig
> > > > > Horst Ludwig Stormer
> > > > > Peter Grunberg
> > > > >
> > > > > math
> > > > > Alex Kuronya
> > > > > Martin Moller
> > > > > Jakob Stix
> > > > > Annette Werner
> > > > > Andreas Bernig
> > > > > Esther Cabezas-Rivas
> > > > > Hans Crauel
> > > > > Thomas Gerstner
> > > > > Bastian von Harrach
> > > > > Thomas Mettler
> > > > > Tobias Weth
> > > > > Amin Coja-Oghlan
> > > > > Raman Sanyal
> > > > > Thorsten Theobald
> > > > > Yury Person
> > > > >
> > > > >
> > > > > Gottingen Univ physics
> > > > > Prof. Dr. Karsten Bahr
> > > > > Prof. Dr. Peter Bloechl
> > > > > Prof. Dr. Eberhard Bodenschatz
> > > > > Prof. Laura Covi, PhD
> > > > > Prof. Dr. Andreas Dillmann
> > > > > Prof. Dr. Stefan Dreizler
> > > > > Prof. Dr. Jörg Enderlein
> > > > > Prof. Dr. Laurent Gizon
> > > > > Prof. Dr. Ariane Frey
> > > > > apl. Prof. Dr. Wolfgang Glatzel
> > > > > Prof. Dr. Fabian Heidrich-Meisner
> > > > > Prof. Dr. Hans Christian Hofsäss
> > > > > Prof. Dr. Andreas Janshoff
> > > > > Prof. Dr. Christian Jooß
> > > > > Prof. Dr. Stefan Kehrein
> > > > > Prof. Dr. Stefan Klumpp
> > > > > Prof. Dr. Sarah Köster
> > > > > Prof. Dr. Reiner Kree
> > > > > Prof. Dr. Matthias Krüger
> > > > > Prof. Dr. Stanley Lai
> > > > > Prof. Dr. Stefan Mathias
> > > > > apl. Prof. Dr. Vasile Mosneaga
> > > > > Prof. Dr. Marcus Müller
> > > > > Prof. Dr. Jens Niemeyer
> > > > > apl. Prof. Dr. Astrid Pundt
> > > > > Prof. Dr. Arnulf Quadt
> > > > > apl. Prof. Dr. Karl-Henning Rehren
> > > > > Prof. Dr. Ansgar Reiners
> > > > > Prof. Dr. Angela Rizzi
> > > > > Prof. Dr. Claus Ropers
> > > > > Prof. Dr. Tim Salditt
> > > > > Prof. Dr. Konrad Samwer
> > > > > Prof. Dr. Christoph Schmidt
> > > > > apl. Prof. Dr. Susanne Schneider
> > > > > Prof. Dr. Steffen Schumann
> > > > > Prof. Dr. Simone Techert
> > > > > apl. Prof. Dr. Michael Seibt
> > > > > Prof. Dr. Peter Sollich
> > > > > Prof. Dr. Andreas Tilgner
> > > > > Prof. Cynthia A. Volkert
> > > > > Prof. Dr. Florentin Wörgötter
> > > > > Prof. Dr. Annette Zippelius

On 1/11/2023 6:29 PM, Ben Bacarisse wrote:
> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>
>> No shit. For some reason I don't think WM can even describe what an
>> infinite n-ary tree is.
>
> You have that right. In fact, there is a degree of deception going on
> because what WM is talking about is not the tree of graph theory. He is
> borrowing the term to mean some weird nonsense of his own. For example,
> in a WMtree removing a node does not remove every path containing it.
>

Lets take a finite tree... ;^)

0
/ \
/ \
1 2
/ \ / \
3 4 5 6

Removing node 2 gets us here:

0
/
/
1
/ \
3 4

How would a WMtree differ from this?

On 1/12/2023 1:15 AM, Chris M. Thomasson wrote:
> On 1/11/2023 6:29 PM, Ben Bacarisse wrote:
>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>
>>> No shit. For some reason I don't think WM can even describe what an
>>> infinite n-ary tree is.
>>
>> You have that right.  In fact, there is a degree of deception going on
>> because what WM is talking about is not the tree of graph theory.  He is
>> borrowing the term to mean some weird nonsense of his own.  For example,
>> in a WMtree removing a node does not remove every path containing it.
>>
>
> Lets take a finite tree... ;^)
>
>      0
>     / \
>    /   \
>   1     2
>  / \   / \
> 3   4 5   6
>
> Removing node 2 gets us here:
>
>      0
>     /
>    /
>   1
>  / \
> 3   4
>
> How would a WMtree differ from this?

0
/ \
/ \
1 2
/ \ / \
3 4 5 6

remove node 2:

0
/ \
/ \
1 5
/ \ \
3 4 6

Chris M. Thomasson explained on 1/11/2023 :
> On 1/11/2023 3:59 AM, WM wrote:
>> For every line n of the Cantor-list we see that the antidiagonal differs
>> from its entry. For every level of the Binary Tree we see that the number
>> of paths that can be distinguished by nodes is finite.
>
> Huh? I am quite fond of trees, and an infinite binary tree has no leaves. Do
> you know why?

It's winter?

"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:

> On 1/11/2023 6:29 PM, Ben Bacarisse wrote:
>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>
>>> No shit. For some reason I don't think WM can even describe what an
>>> infinite n-ary tree is.
>> You have that right. In fact, there is a degree of deception going on
>> because what WM is talking about is not the tree of graph theory. He is
>> borrowing the term to mean some weird nonsense of his own. For example,
>> in a WMtree removing a node does not remove every path containing it.
>
> Lets take a finite tree... ;^)
>
> 0
> / \
> / \
> 1 2
> / \ / \
> 3 4 5 6
>
> Removing node 2 gets us here:
>
> 0
> /
> /
> 1
> / \
> 3 4

Actually, in graph theory, removing node 2 gives us

0
/
/
1
/ \
3 4 5 6

We are left with two disconnected nodes

> How would a WMtree differ from this?

He won't say, because he won't define his terms. Here is an example of
the kind of nonsense he claims pertains to WMtrees:

| When a node a_n is chosen to be removed, then one path p containing
| this node a_n is removed to. All other paths remain, but they have
| lost all nodes covered by p.

All cranks learn quickly to avoid precision. If he defined WMtrees as
precisely as graph theorists define graphs (including trees) he would
not be able to play his word games.

--
Ben.

"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:

> On 1/12/2023 1:15 AM, Chris M. Thomasson wrote:
>> On 1/11/2023 6:29 PM, Ben Bacarisse wrote:
>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>
>>>> No shit. For some reason I don't think WM can even describe what an
>>>> infinite n-ary tree is.
>>>
>>> You have that right.  In fact, there is a degree of deception going on
>>> because what WM is talking about is not the tree of graph theory.  He is
>>> borrowing the term to mean some weird nonsense of his own.  For example,
>>> in a WMtree removing a node does not remove every path containing it.
>>>
>> Lets take a finite tree... ;^)
>>      0
>>     / \
>>    /   \
>>   1     2
>>  / \   / \
>> 3   4 5   6
>> Removing node 2 gets us here:
>>      0
>>     /
>>    /
>>   1
>>  / \
>> 3   4
>> How would a WMtree differ from this?
>
> 0
> / \
> / \
> 1 2
> / \ / \
> 3 4 5 6
>
>
> remove node 2:
>
> 0
> / \
> / \
> 1 5
> / \ \
> 3 4 6

No. In graph theory you don't gain any edges when you remove a node.
Neither (0,5), (5,6) where in the graph's edge set to start with.

--
Ben.

On Thursday, January 12, 2023 at 1:20:00 PM UTC+1, Ben Bacarisse wrote:

> No. In graph theory you don't gain any edges when you remove a node.
> Neither (0,5), (5,6) where in the graph's edge set to start with.

If a graph is a set G = {N, E} where N is the sets of nodes and G the set of edgets between them, the tree we start with

0
/ \
1 2
/ \ / \
3 4 5 6

would be: T = {{0, 1, 2, 3, 4, 5, 6}, {(0, 1), (0, 2), (1, 3), (1, 4), (2, 5), (2, 6)}.

Moreover there are some conditions on E = {(0, 1), (0, 2), (1, 3), (1, 4), (2, 5), (2, 6)} such that it is ensured that T is graph (or tree).

Clearly one of these conditions is that for any (x, y) e E: x, y e N. We don't wan't to have edges in our graph/tree which "connects" non existing nodes.

So what does this mean? This means that "removing a node from a graph" [which is an undefined term by now] should not just affect N, but E too! (Otherwise we might end with some set G' = {N', E} which is not a graph.) [ "removing a node from a graph" should not be identified with "removing a node from the node-set of a graph". The meaning of the latter is clear, but not the meaning of the former - without some agreement, I'd say. We might agree that "removing a node x from a graph (N, E)" results in a graph (N', E') where N' = N \ {x} and E' is E without all edges (a, b) where a = x or b = x. ]

Even worse, in our case we are dealing with a special graph which is called a (binary) tree. There we have additional conditions on E such that T = {N, E} is a tree.

One of these conditions is that for any y e N \ {0} there is exactly on x e N such that (x, y) e E. In other words, any node which is not the root node (in our case 0) has a "parent" (to which it is conected by an edge).

Here we will face the problem that "removing a certain node from a tree" [in the sense defined above for graphs in general] might not result in a tree (any more). For example, if we "remove 2 from the tree" mentioned above we would get:

0
/ 1
/ \
3 4 5 6

T' = {{0, 1, 3, 4, 5, 6}, {(0, 1), (1, 3), (1, 4)}.

THIS (depending on the definition of a tree) is not a tree any more (I'd say), after all the nodes 5 and 6 (which are not the root node 0) do not have a parent any more.

Actually, it seems to me that "removing a node" is not a sensible operation on a (infinite) binary tree (i. e. a binary tree without leafs).

Hence I'm not sure if the claim:

"For example, in a WMtree removing a node does not remove every path containing it."

is a sensible claim at all. (Not even taking into account the possible meaning(s) of "does not remove every path containing it".)

On Thursday, January 12, 2023 at 1:18:04 PM UTC+1, Ben Bacarisse wrote:

> No. In graph theory you don't gain any edges when you remove a node.
> Neither (0,5), (5,6) where in the graph's edge set to start with.

If a graph is a set G = {N, E} where N is the sets of nodes and G the set of edges between them, the tree we start with

0
/ \
1 2
/ \ / \
3 4 5 6

would be: T = {{0, 1, 2, 3, 4, 5, 6}, {(0, 1), (0, 2), (1, 3), (1, 4), (2, 5), (2, 6)}.

Moreover there are some conditions on E = {(0, 1), (0, 2), (1, 3), (1, 4), (2, 5), (2, 6)} such that it is ensured that T is graph (or tree).

Clearly one of these conditions is that for any (x, y) e E: x, y e N. We don't wan't to have edges in our graph/tree which "connects" non existing nodes.

So what does this mean? This means that "removing a node from a graph" [which is an undefined term by now] should not just affect N, but E too! (Otherwise we might end with some set G' = {N', E} which is not a graph.) [ "removing a node from a graph" should not be identified with "removing a node from the node-set of a graph". The meaning of the latter is clear, but not the meaning of the former - without some agreement, I'd say. We might agree that "removing a node x from a graph (N, E)" results in a graph (N', E') where N' = N \ {x} and E' is E without all edges (a, b) where a = x or b = x. ]

Even worse, in our case we are dealing with a special graph which is called a (binary) tree. There we have additional conditions on E such that T = {N, E} is a tree.

One of these conditions is that for any y e N \ {0} there is exactly on x e N such that (x, y) e E. In other words, any node which is not the root node (in our case 0) has a "parent" (to which it is conected by an edge).

Here we will face the problem that "removing a certain node from a tree" [in the sense defined above for graphs in general] might not result in a tree (any more). For example, if we "remove 2 from the tree" mentioned above we would get:

0
/ 1
/ \
3 4 5 6

T' = {{0, 1, 3, 4, 5, 6}, {(0, 1), (1, 3), (1, 4)}.

THIS (depending on the definition of a tree) is not a tree any more (I'd say), after all the nodes 5 and 6 (which are not the root node 0) do not have a parent any more.

Actually, it seems to me that "removing a node" is not a sensible operation on a (infinite) binary tree (i. e. a binary tree without leafs).

Hence I'm not sure if the claim:

"For example, in a WMtree removing a node does not remove every path containing it."

is a sensible claim at all. (Not even taking into account the possible meaning(s) of "does not remove every path containing it".)

Ben Bacarisse schrieb am Donnerstag, 12. Januar 2023 um 13:18:04 UTC+1:
> If he defined WMtrees as
> precisely as graph theorists define graphs (including trees) he would
> not be able to play his word games.

This is the Binay Tree:

0.
/ \
0 1
/\ /\
0 1 0 1
---

I remove the root node 0. and the path 0.111... . Removed nodes are indicated by x. Edges remain as guide for the eye.
It remains

x
/ \
0 x
/\ /\
0 1 0 x
---
That means the path 0.10... has lost its first two nodes but the rest xx0... remains.

Regards, WM

WM brought next idea :
> Ben Bacarisse schrieb am Donnerstag, 12. Januar 2023 um 13:18:04 UTC+1:
>> If he defined WMtrees as
>> precisely as graph theorists define graphs (including trees) he would
>> not be able to play his word games.
>
> This is the Binay Tree:
>
> 0.
> / \
> 0 1
> /\ /\
> 0 1 0 1
> ---

Okay, but what good is a binay tree? Can you make tea from the leaves?

The binary of binary tree refers to the number of choices at each node.

On 1/12/2023 2:26 AM, FromTheRafters wrote:
> Chris M. Thomasson explained on 1/11/2023 :
>> On 1/11/2023 3:59 AM, WM wrote:
>>> For every line n of the Cantor-list we see that the antidiagonal
>>> differs from its entry. For every level of the Binary Tree we see
>>> that the number of paths that can be distinguished by nodes is finite.
>>
>> Huh? I am quite fond of trees, and an infinite binary tree has no
>> leaves. Do you know why?
>
> It's winter?

;^D

If the tree has leaves, then we are looking at a finite portion of the
infinite binary tree. Fair enough?

On 1/12/2023 10:19 AM, WM wrote:
> Ben Bacarisse schrieb am Donnerstag, 12. Januar 2023 um 13:18:04 UTC+1:
>> If he defined WMtrees as
>> precisely as graph theorists define graphs (including trees) he would
>> not be able to play his word games.
>
> This is the Binay Tree:
>
> 0.
> / \
> 0 1
> /\ /\
> 0 1 0 1

The n in the n-ary of the tree is 2. 2 is binary:

0
/ \
/ \
1 2
/ \ / \
3 4 5 6

It means that a node has two children. So, WM, what would a 6-ary tree
look like?

[...]

On 1/12/2023 12:19 PM, WM wrote:
> Ben Bacarisse schrieb am Donnerstag, 12. Januar 2023 um 13:18:04 UTC+1:
>> If he defined WMtrees as
>> precisely as graph theorists define graphs (including trees) he would
>> not be able to play his word games.
>
> This is the Binay Tree:
>
> 0.
> / \
> 0 1
> /\ /\
> 0 1 0 1
> ---
>
> I remove the root node 0. and the path 0.111... . Removed nodes are indicated by x. Edges remain as guide for the eye.
> It remains
>
> x
> / \
> 0 x
> /\ /\
> 0 1 0 x

no, you changed the tree, it does not "remain"
each edge requires two nodes to exist.

> ---
> That means the path 0.10... has lost its first two nodes but the rest xx0... remains.

"remains" is the wrong word, you *changed the path* by removing nodes, you have changed the number.

>
> Regards, WM
>

On 1/12/2023 4:17 AM, Ben Bacarisse wrote:
> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>
>> On 1/11/2023 6:29 PM, Ben Bacarisse wrote:
>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>
>>>> No shit. For some reason I don't think WM can even describe what an
>>>> infinite n-ary tree is.
>>> You have that right. In fact, there is a degree of deception going on
>>> because what WM is talking about is not the tree of graph theory. He is
>>> borrowing the term to mean some weird nonsense of his own. For example,
>>> in a WMtree removing a node does not remove every path containing it.
>>
>> Lets take a finite tree... ;^)
>>
>> 0
>> / \
>> / \
>> 1 2
>> / \ / \
>> 3 4 5 6
>>
>> Removing node 2 gets us here:
>>
>> 0
>> /
>> /
>> 1
>> / \
>> 3 4
>
> Actually, in graph theory, removing node 2 gives us
>
> 0
> /
> /
> 1
> / \
> 3 4 5 6
>
> We are left with two disconnected nodes
>
>> How would a WMtree differ from this?
>
> He won't say, because he won't define his terms. Here is an example of
> the kind of nonsense he claims pertains to WMtrees:
>
> | When a node a_n is chosen to be removed, then one path p containing
> | this node a_n is removed to. All other paths remain, but they have
> | lost all nodes covered by p.
>
> All cranks learn quickly to avoid precision. If he defined WMtrees as
> precisely as graph theorists define graphs (including trees) he would
> not be able to play his word games.
>

Humm... I wonder. Is this valid, stupid, or kind of stupid?

0
/ \
/ \
1 2
/ \ / \
3 4 5 6

Let's remove node 2... We end up with this thing:

0
/ \
/ \
1 5
/ \ / \
3 4 6 7

Humm...

On 1/12/2023 4:17 AM, Ben Bacarisse wrote:
> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>
>> On 1/11/2023 6:29 PM, Ben Bacarisse wrote:
>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>
>>>> No shit. For some reason I don't think WM can even describe what an
>>>> infinite n-ary tree is.
>>> You have that right. In fact, there is a degree of deception going on
>>> because what WM is talking about is not the tree of graph theory. He is
>>> borrowing the term to mean some weird nonsense of his own. For example,
>>> in a WMtree removing a node does not remove every path containing it.
>>
>> Lets take a finite tree... ;^)
>>
>> 0
>> / \
>> / \
>> 1 2
>> / \ / \
>> 3 4 5 6
>>
>> Removing node 2 gets us here:
>>
>> 0
>> /
>> /
>> 1
>> / \
>> 3 4
>
> Actually, in graph theory, removing node 2 gives us
>
> 0
> /
> /
> 1
> / \
> 3 4 5 6
>
> We are left with two disconnected nodes

[...]

disconnected wrt an n-ary tree makes blood want to shoot out of my eyes!

"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:

> Humm... I wonder. Is this valid, stupid, or kind of stupid?
>
> 0
> / \
> / \
> 1 2
> / \ / \
> 3 4 5 6
>
> Let's remove node 2... We end up with this thing:
>
> 0
> / \
> / \
> 1 5
> / \ / \
> 3 4 6 7
>
> Humm...

I am sure you could devise some definitions that have that consequence
though it's not clear that the result would be interesting (to anyone
but you, I mean). But if you ever do, you should be sure to use names
that make is clear you are not talking about conventional graph theory.

--
Ben.

Chris M. Thomasson explained on 1/12/2023 :
> On 1/12/2023 2:26 AM, FromTheRafters wrote:
>> Chris M. Thomasson explained on 1/11/2023 :
>>> On 1/11/2023 3:59 AM, WM wrote:
>>>> For every line n of the Cantor-list we see that the antidiagonal differs
>>>> from its entry. For every level of the Binary Tree we see that the number
>>>> of paths that can be distinguished by nodes is finite.
>>>
>>> Huh? I am quite fond of trees, and an infinite binary tree has no leaves.
>>> Do you know why?
>>
>> It's winter?
>
> ;^D
>
> If the tree has leaves, then we are looking at a finite portion of the
> infinite binary tree. Fair enough?

I would call them 'leaf nodes' but yes.

Fritz Feldhase <franz.fritschee.ff@gmail.com> writes:

> On Thursday, January 12, 2023 at 1:18:04 PM UTC+1, Ben Bacarisse wrote:
>
>> No. In graph theory you don't gain any edges when you remove a node.
>> Neither (0,5), (5,6) where in the graph's edge set to start with.
>
> If a graph is a set G = {N, E} where N is the sets of nodes and G the set of edges between them, the tree we start with
>
> 0
> / \
> 1 2
> / \ / \
> 3 4 5 6
>
> would be: T = {{0, 1, 2, 3, 4, 5, 6}, {(0, 1), (0, 2), (1, 3), (1, 4),
> (2, 5), (2, 6)}.

Or a graph can be defined to be just be an edge set. Allowing "self"
edges (e.g. (0,0)) alters the meanings a little, but "removing a node k"
would usually mean removing any pair (k, _) or (_, k).

The complete infinite binary tree is then just the graph with edges

{ (i, 2i+1), (i, 2i+2) | i in N }

If the edges are directed, then a binary tree would usually also include
the edges (b, a) for every edge (a, b) defined above.

The paths in a graph are the sets of connected edges.

Your node and edge set definition is the more usual one but I don't
really mind what the details, are provided they are specified. WM
operates by never specifying anything so that the metaphorical meanings
of the words are working overtime to suggest the mysterious results he
claims for real mathematics.

> Moreover there are some conditions on E = {(0, 1), (0, 2), (1, 3), (1,
> 4), (2, 5), (2, 6)} such that it is ensured that T is graph (or tree).
>
> Clearly one of these conditions is that for any (x, y) e E: x, y e
> N. We don't wan't to have edges in our graph/tree which "connects" non
> existing nodes.
>
> So what does this mean? This means that "removing a node from a graph"
> [which is an undefined term by now] should not just affect N, but E
> too! (Otherwise we might end with some set G' = {N', E} which is not a
> graph.) [ "removing a node from a graph" should not be identified with
> "removing a node from the node-set of a graph". The meaning of the
> latter is clear, but not the meaning of the former - without some
> agreement, I'd say. We might agree that "removing a node x from a
> graph (N, E)" results in a graph (N', E') where N' = N \ {x} and E' is
> E without all edges (a, b) where a = x or b = x. ]
>
> Even worse, in our case we are dealing with a special graph which is
> called a (binary) tree. There we have additional conditions on E such
> that T = {N, E} is a tree.
>
> One of these conditions is that for any y e N \ {0} there is exactly
> on x e N such that (x, y) e E. In other words, any node which is not
> the root node (in our case 0) has a "parent" (to which it is conected
> by an edge).

I agree with all this, though I think a simpler presentation would do.
Not for WM, of course -- he needs as much waffle as possible.

> Here we will face the problem that "removing a certain node from a
> tree" [in the sense defined above for graphs in general] might not
> result in a tree (any more).

Sure.

> For example, if we "remove 2 from the
> tree" mentioned above we would get:
>
> 0
> /
> 1
> / \
> 3 4 5 6
> is a sensible claim at all.>
> T' = {{0, 1, 3, 4, 5, 6}, {(0, 1), (1, 3), (1, 4)}.
>
> THIS (depending on the definition of a tree) is not a tree any more
> (I'd say), after all the nodes 5 and 6 (which are not the root node 0)
> do not have a parent any more.
>
> Actually, it seems to me that "removing a node" is not a sensible
> operation on a (infinite) binary tree (i. e. a binary tree without
> leafs).

I don't see why. If we use the simple edge definition then the infinite
binary tree with node k removed is the graph:

{ (i, 2i+1), (i, 2i+2) | i in N } \ { (k, i), (i, k) | i in N }

(There are lots of other ways to write this edge set, of course.)

> Hence I'm not sure if the claim:
>
> "For example, in a WMtree removing a node does not remove every path
> containing it." is a sensible claim at all.

Well, since it /seems/ to be what WM is saying (it is my paraphrase
though), not being sensible is the starting assumption. I think he
gives it meaning (albeit a silly meaning) by imagining separate node and
path sets (there are no edges at all).

> (Not even taking into account the possible
> meaning(s) of "does not remove every path containing it".)

--
Ben.

WM <monteuffl@t-online.de> writes:
(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
Unendlichen" at Hochschule Augsburg.)

> Ben Bacarisse schrieb am Donnerstag, 12. Januar 2023 um 13:18:04 UTC+1:
>> If he defined WMtrees as
>> precisely as graph theorists define graphs (including trees) he would
>> not be able to play his word games.
>
> This is the Binay Tree:
>
> 0.
> / \
> 0 1
> /\ /\
> 0 1 0 1
> ---
>
> I remove the root node 0. and the path 0.111... . Removed nodes are
> indicated by x. Edges remain as guide for the eye. It remains
>
> x
> / \
> 0 x
> /\ /\
> 0 1 0 x
> ---
> That means the path 0.10... has lost its first two nodes but the rest
> xx0... remains.

This is WMaths in a nutshell. They are removed but they remain in that
they can be still form part of paths. I can see now why you usually
draw the tree without the numbers of the nodes, and why you write paths
not as sequences of node numbers but as "digits". You need "0.000..."
(or "x000..." whatever that means) to remain even though the node
numbered zero is not longer part of the graph. You would not want to
write { a_0, a_1, a_3, a_5, ... } for that path once a_0 have been
"removed". Something hand waving like "x000..." is so much more
flexible.

I bet you don't tell your students that this is not the infinite binary
tree as other mathematicians define it.

--
Ben.