Matheology § 288

[muec...@rz.fh-augsburg.de]

Matheology § 288

Here are the differences in the premises which lead to differences in the results od mathematics and matheology.

Matheology requires:

1) The Binary Tree

0.
/ \
0 1
/ \ / \
0 10 1
...

containing all rational numbers of the unit interval also contains all irrational numbers. If the rationals are written in the usual manner this is not the case.

2) The triangle construcuted in 3-symmetry is equilateral.

d
dc
dac
dbbc
...

If however, the triangle is constructed such that alsway one and the same side is expanded, then it loses 3-symmetry "in the limit".

a
bb
ccc
...

3) For the union of the sequence of sets
U({1}, {1, 2}, {1, 2, 3} , ..., {1, 2, 3, ..., n}} = {1, 2, 3, ..., n}
equality holds but not in the limit.

In mathematics all these premises lead to different results:

1) The Binary Tree containing all rational numbers of the unit interval does not contain any irrational number.

2) The triangle construcuted in 3-symmetry is and always remains equilateral.

3) For the union of the sequence of unions of preceding sets equality holds in the limit too.

Regards, WM

[Julio Di Egidio]

<muec...@rz.fh-augsburg.de> wrote in message
news:4344a7d8-ccf7-4bf0...@googlegroups.com...

> Matheology § 288
>
> Here are the differences in the premises which lead to differences in the
> results od mathematics and matheology.
>
> Matheology requires:
>
> 1) The Binary Tree
>
> 0.
> / \
> 0 1
> / \ / \
> 0 10 1
> ...
>
> containing all rational numbers of the unit interval also contains all
> irrational numbers. If the rationals are written in the usual manner this
> is not the case.

Please show what relationship you have in mind, between the complete binary
tree and the rationals "written in the usual manner", as I cannot guess what
you have in mind.

> 2) The triangle construcuted in 3-symmetry is equilateral.
>
> d
> dc
> dac
> dbbc
> ...
>
> If however, the triangle is constructed such that alsway one and the same
> side is expanded, then it loses 3-symmetry "in the limit".
>
> a
> bb
> ccc
> ...

It's rather you who keep claiming, contrary to all evidence, that,
standardly, that symmetry is lost in the limit.

> 3) For the union of the sequence of sets
> U({1}, {1, 2}, {1, 2, 3} , ..., {1, 2, 3, ..., n}} = {1, 2, 3, ..., n}
> equality holds but not in the limit.

It's rather you who keep claiming, contrary to all evidence, that,
standardly, that equality does not hold in the limit.

> In mathematics all these premises lead to different results:
>
> 1) The Binary Tree containing all rational numbers of the unit interval
> does not contain any irrational number.

I'd venture that all you can have in mind is a bijection of nodes to
rationals, and a trivial one anyway, as of course the set of nodes is
countable. In the complete binary tree we are rather interested in the
correlation of infinite paths (or infinite binary strings) to reals: but, in
your realm, where infinite paths do not exist at all, irrationals just do
not exist either.

> 2) The triangle construcuted in 3-symmetry is and always remains
> equilateral.

As well as, in your realm, always finite, as limits neither exist, nor there
is an inductive set N.

> 3) For the union of the sequence of unions of preceding sets equality
> holds in the limit too.

Again, in your realm, limits just do not exist and not even infinite unions.

Julio

[Zeit Geist]

You are incorrect.

>
> 2) The triangle construcuted in 3-symmetry is and always remains equilateral.
>

Yes, but the stament above is wrong.

>
> 3) For the union of the sequence of unions of preceding sets equality holds in the limit too.
>

Wrong, it holds.

>
> Regards, WM

ZG

[muec...@rz.fh-augsburg.de]

On Friday, 14 June 2013 10:11:40 UTC+2, Julio Di Egidio wrote:

>>The Binary Tree

> Please show what relationship you have in mind, between the complete binary tree and the rationals "written in the usual manner", as I cannot guess what you have in mind.

In the usual way, you write the numbers separately. Example: In the list
0.1
0.11
0.111
...
you can look at every line, you will not find 1/9.
If you write all numbers into the same line, like the nodes in a path of atree, you have, according to matheology, an infinite string of aleph_0 digits which can be abbreviated by 0.111... or 1/9.

> > 2) The triangle construcuted in 3-symmetry is equilateral.

> It's rather you who keep claiming, contrary to all evidence, that, standardly, that symmetry is lost in the limit.

No.

The list given above does not contain aleph_0 1's in any direction.

Regards, WM

[muec...@rz.fh-augsburg.de]

On Friday, 14 June 2013 10:16:05 UTC+2, Zeit Geist wrote:

> > 1) The Binary Tree containing all rational numbers of the unit interval does not contain any irrational number.

> You are incorrect.

> > 2) The triangle construcuted in 3-symmetry is and always remains equilateral.

> Yes, but the stament above is wrong.

No, it is the same fact. Let the triangle remain equilateral. A line does not contain aleph_0 1's. Hence the first column and diagonal do not contain aleph_0 1's either. A path in the Binary Tree does not contain aleph_0 nodes. The Binary Tree does not contain paths of irrational numbers.

Regards, WM

[Julio Di Egidio]

<muec...@rz.fh-augsburg.de> wrote in message
news:c3c1dea5-20e4-4efd...@googlegroups.com...

> On Friday, 14 June 2013 10:11:40 UTC+2, Julio Di Egidio wrote:
>
>>>The Binary Tree
>
>> Please show what relationship you have in mind, between the complete
>> binary tree and the rationals "written in the usual manner", as I cannot
>> guess what you have in mind.
>
> In the usual way, you write the numbers separately. Example: In the list
> 0.1
> 0.11
> 0.111
> ...
> you can look at every line, you will not find 1/9.
> If you write all numbers into the same line, like the nodes in a path of
> atree, you have, according to matheology, an infinite string of aleph_0
> digits which can be abbreviated by 0.111... or 1/9.

I see. What a fraud, your points and numbers.

>> > 2) The triangle construcuted in 3-symmetry is equilateral.
>
>> It's rather you who keep claiming, contrary to all evidence, that,
>> standardly, that symmetry is lost in the limit.
>
> No.
>
> The list given above does not contain aleph_0 1's in any direction.

So you keep saying.

Never mind, have fun.

Julio

[Julio Di Egidio]

<muec...@rz.fh-augsburg.de> wrote in message
news:cd5dd0f2-be27-437d...@googlegroups.com...

It's in your realm, where infinite paths do not exist at all, that

irrationals just do not exist either.

It's in your realm that objects are only finite, limits do not exist, nor

there is an inductive set N.

It's in your realm that limits just do not exist and not even infinite
unions.

In fact, it's you who are a complete fraud, indeed a mixture of a spammer
and a troll.

Julio

[muec...@rz.fh-augsburg.de]

On Friday, 14 June 2013 18:39:35 UTC+2, Julio Di Egidio wrote:
>> The Binary Tree does not contain paths of irrational numbers.

> It's in your realm, where infinite paths do not exist at all, that irrationals just do not exist either.

You have not yet understood, but you are not the only one. It takes a very bright mind or much time. (I needed much time.)
Irrational numbers exist in mathemaics. But they have no decimal fraction.

> It's in your realm that objects are only finite, limits do not exist, nor there is an inductive set N.

You have not yet understood. There are limits like SUM1/n! or 0.111... But they have no decimal expansion. In principle no fraction has a pure decimal expansion. When you write 0.25, then you have the additional finite definition that only zeros will follow.

> It's in your realm that limits just do not exist

Wrong. See above.

> and not even infinite unions.

Also that is wrong. There are infinite unions, namely finite unions that surpassing every finite one.
For every n there is m > n
Contrary to matheology which requires
There is M such that for every n : M >= n.

> In fact, it's you who are a complete fraud, indeed a mixture of a spammer and a troll.

In fact you have shown that you have not yet understood. You show the typical reaction.

Regards, WM

[Julio Di Egidio]

<muec...@rz.fh-augsburg.de> wrote in message
news:58e50244-e695-4d4a...@googlegroups.com...

It's a bit now that I look at your posts and participate into ensuing
discussion only as a matter of personal knowledge and exercise, not because
I anymore find any intrinsic merit into what you keep blathering. But,
eventually, your patent inability and/or unwillingness to present and/or
follow a coherent argument that is one makes me realise: just do not feed
the spammer and troll.

All the best,

Julio

[Zeit Geist]

On Friday, June 14, 2013 7:07:43 AM UTC-7, muec...@rz.fh-augsburg.de wrote:
> On Friday, 14 June 2013 10:11:40 UTC+2, Julio Di Egidio wrote:
>
>
>
> >>The Binary Tree
>
>
>
> > Please show what relationship you have in mind, between the complete binary tree and the rationals "written in the usual manner", as I cannot guess what you have in mind.
>
>
>
> In the usual way, you write the numbers separately. Example: In the list
>
> 0.1
>
> 0.11
>
> 0.111
>
> ...
>
> you can look at every line, you will not find 1/9.
>
> If you write all numbers into the same line, like the nodes in a path of atree, you have, according to matheology, an infinite string of aleph_0 digits which can be abbreviated by 0.111... or 1/9.
>

Writing all numbers one line, .111111..., is a Union.
The "entire" first column is a Union.
The "entire" diagonal is a Union.


You are a fool.

>
>
> > > 2) The triangle construcuted in 3-symmetry is equilateral.
>
>
>
> > It's rather you who keep claiming, contrary to all evidence, that, standardly, that symmetry is lost in the limit.
>
>
>
> No.
>
>
>
> The list given above does not contain aleph_0 1's in any direction.
>

Really, can you write the whole list then?

If its finite you should be able to.

>
> Regards, WM

ZG

[Virgil]

In article <4344a7d8-ccf7-4bf0...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:


>
> Matheology requires:
>
> 1) The Binary Tree
>
> 0.
> / \
> 0 1
> / \ / \
> 0 10 1
> ...
>
> containing all rational numbers of the unit interval also contains all
> irrational numbers. If the rationals are written in the usual manner this is
> not the case.

In order for that tree to contain all binary rationals rationals in
[0,1] at all. it must contain all but numbers 0 an 1 twice each, once
with a "tail" of infinitely many 0's and once with a "trail" of
infinitely many 1's, and, into the bargain, all paths between two such
paths. Thus containing irrationals as well.

>
> 2) The triangle construcuted in 3-symmetry is equilateral.
>
> d
> dc
> dac
> dbbc
> ...
>

It is not a triangle when it only has two sides.

So it is only an angle until you can show us a third side..

>
> 3) For the union of the sequence of sets
> U({1}, {1, 2}, {1, 2, 3} , ..., {1, 2, 3, ..., n}} = {1, 2, 3, ..., n}
> equality holds but not in the limit.

The limit, if it is to exist at all, of any strictly increasing infinite
sequence cannot be a member of the sequence

>
> In mathematics all these premises lead to different results:
>
> 1) The Binary Tree containing all rational numbers of the unit interval does
> not contain any irrational number.
>
> 2) The triangle construcuted in 3-symmetry is and always remains equilateral.
>
> 3) For the union of the sequence of unions of preceding sets equality holds
> in the limit too.

If those were to hold, then everything and anything would also hold, so
those do not hold.

At least not outside the wild weird world of WMytheology.
>
> Regards, WM
--

[Zeit Geist]

Your lack of intellect prevents you from forming the proper intuitions
concerning the Infinite. This in turn, causes you to write statements
which you think are internal contradictions in Set Theory, even though
all you write is garbage.

>
> Regards, WM

ZG

[Virgil]

In article <cd5dd0f2-be27-437d...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Friday, 14 June 2013 10:16:05 UTC+2, Zeit Geist wrote:
>
> > > 1) The Binary Tree containing all rational numbers of the unit interval
> > > does not contain any irrational number.
>
> > You are incorrect.
>
> > > 2) The triangle construcuted in 3-symmetry is and always remains
> > > equilateral.

The triangles, at least as presented in WM's diagrams, have all been
right triangles, so that in his WMytheology, WM claims to have
equilateral right triangles, things that exist nowhere else.
--

[muec...@rz.fh-augsburg.de]

On Friday, 14 June 2013 20:08:07 UTC+2, Zeit Geist wrote:
> Writing all numbers one line, .111111..., is a Union.

And you believe that a union over sets that are unions of all preceding sets yields more than these sets. Your choice. But not a rational idea.

>> The list given above does not contain aleph_0 1's in any direction.

> Really, can you write the whole list then?

That is not relevant as an argument against anti-symmetry.
You should be able to see from the symmetrical construction that symmetry has to prevail.

> If its finite you should be able to.

"Finite" is not the complement of aleph_0. Finite is the complement of infinite, that means there is no finish, you cannot finish the list. Of course there is no whole list in the sense that no line can be added.

Regards, WM

[muec...@rz.fh-augsburg.de]

On Friday, 14 June 2013 20:14:16 UTC+2, Zeit Geist wrote:
> > Your lack of intellect prevents you from forming the proper intuitions concerning the Infinite.

There is no intuition required to prove that the triangle is equilateral in every step. So the limit is equilateral too - in mathematics.

> This in turn, causes you to write statements which you think are internal contradictions in Set Theory, even though all you write is garbage.

Sounds like withdrawal symptoms.

Regards, WM

[Virgil]

In article <58e50244-e695-4d4a...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Friday, 14 June 2013 18:39:35 UTC+2, Julio Di Egidio wrote:
> >> The Binary Tree does not contain paths of irrational numbers.
>
> > It's in your realm, where infinite paths do not exist at all, that
> > irrationals just do not exist either.
>
> You have not yet understood, but you are not the only one. It takes a very
> bright mind or much time. (I needed much time.)
> Irrational numbers exist in mathemaics. But they have no decimal fraction.

Only fractions expressible with denominators of form 2^m*5^n are decimal
fractions, which leaves "most" rational numbers out.

>
> > It's in your realm that objects are only finite, limits do not exist, nor
> > there is an inductive set N.
>
> You have not yet understood.

To be not understanding of the machinations inside the wild weird world
of WMytheology is wisdom.

>
> > It's in your realm that limits just do not exist
>
> Wrong. See above.

An infinite sequence cannot exist without having a definition of its
terms valid for all infinitely many terms. And without all those
infinitely many terms actually existing, there can be no such thing as a
limit to any such incomplete sequence.

>

>
> Also that is wrong. There are infinite unions, namely finite unions that
> surpassing every finite one.

So how does any finite union exceed itself in WMytheology?

Outside of WMytheology they don't!

> For every n there is m > n

Outside of WMytheology, there is then
a set of all those m greater than that n.

> Contrary to matheology which requires
> There is M such that for every n : M >= n.

Then matheology must then be a sub-ology of WMytheology, as that does
not hold anywhere else.

>
> > In fact, it's you who are a complete fraud, indeed a mixture of a spammer
> > and a troll.
>
> In fact you have shown that you have not yet understood. You show the typical
> reaction.

The typical, and almost universal, reaction to the wild weird world of
WMytheology is, quite properly, rejection of it.
--

[muec...@rz.fh-augsburg.de]

On Friday, 14 June 2013 20:09:57 UTC+2, Virgil wrote:

> In order for that tree to contain all binary rationals rationals in [0,1] at all. it must contain all but numbers 0 an 1 twice each, once with a "tail" of infinitely many 0's and once with a "trail" of infinitely many 1's, and, into the bargain, all paths between two such paths. Thus containing irrationals as well.

They come in on angels wings - if actual infinity exists.

> d
> dc
> dac
> dbbc
> ...

> It is not a triangle when it only has two sides. So it is only an angle until you can show us a third side.

If you look at the letters, you see that every side is treated equally when extending the triangle. It has three equal sides in every finite step.

> > 3) For the union of the sequence of sets > U({1}, {1, 2}, {1, 2, 3} , ..., {1, 2, 3, ..., n}} = {1, 2, 3, ..., n} > equality holds but not in the limit.

> The limit, if it is to exist at all,

just that is contradicted here

> of any strictly increasing infinite sequence cannot be a member of the sequence.

So it is. But to get in a second union more than in the first, without adding anything in between, shows that there is no limit.

Regards, WM

[muec...@rz.fh-augsburg.de]

On Friday, 14 June 2013 20:14:12 UTC+2, Virgil wrote:

> The triangles, at least as presented in WM's diagrams, have all been right triangles, so that in his WMytheology, WM claims to have equilateral right triangles, things that exist nowhere else.

With respect to the topology that I use, namely the number of letters per side is the measure, the triangles are equilateral.

Regards, WM

[Virgil]

In article <c3c1dea5-20e4-4efd...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Friday, 14 June 2013 10:11:40 UTC+2, Julio Di Egidio wrote:
>
> >>The Binary Tree
>
> > Please show what relationship you have in mind, between the complete binary
> > tree and the rationals "written in the usual manner", as I cannot guess
> > what you have in mind.
>
> In the usual way, you write the numbers separately. Example: In the list
> 0.1
> 0.11
> 0.111
> ...
> you can look at every line, you will not find 1/9.

You can also look at every line of that list above and not find most of
the binary rationals that should properly appear even in WM's supposed
trees.

> If you write all numbers into the same line,

then you will not have a tree at all.

>
>
> > > 2) The triangle construcuted in 3-symmetry is equilateral.

How does WM manage to have equilateral triangles each with one right
angle and a couple of half-right angles in the finite case, and only one
right angle and no other angles in the infinite case??
--

[Bergholt Stuttley Johnson]

muec...@rz.fh-augsburg.de wrote:
>
> With respect to the topology that I use

LOL
What are the open sets?
You are a moron!

--
fix$(<$>)<$>(:)<*>((<$>((:[])<$>))(=<<)<$>(*)<$>(>>=)(+)($))$1

[muec...@rz.fh-augsburg.de]

On Friday, 14 June 2013 20:39:08 UTC+2, Virgil wrote:
> In article <c3c1dea5-20e4-4efd...@googlegroups.com>, muec...@rz.fh-augsburg.de wrote: > On Friday, 14 June 2013 10:11:40 UTC+2, Julio Di Egidio wrote: > > >>The Binary Tree > > > Please show what relationship you have in mind, between the complete binary > > tree and the rationals "written in the usual manner", as I cannot guess > > what you have in mind. > > In the usual way, you write the numbers separately. Example: In the list > 0.1 > 0.11 > 0.111 > ... > you can look at every line, you will not find 1/9. You can also look at every line of that list above and not find most of the binary rationals that should properly appear even in WM's supposed trees.

The list above contains only one path of the tree, that one at the outmost right side.

2) The triangle construcuted in 3-symmetry is equilateral. How does WM manage to have equilateral triangles each with one right angle and a couple of half-right angles in the finite case, and only one right angle and no other angles in the infinite case??

Not at all. There is no infinite case.


a


a
bb


c
ac
bbc


infinite means, it goes on and on without end or limit or limit case.

Regards, WM

[Virgil]

In article <6a13b6b0-f200-46fe...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> The triangle construcuted in 3-symmetry is equilateral.

Then show it to us!

The trinangles you show us are not equilateral.

> > How does WM manage
> > to have equilateral triangles each with one right angle and a couple of
> > half-right angles in the finite case, and only one right angle and no other
> > angles in the infinite case??
>
> Not at all. There is no infinite case.

There is everywhere but in the the wild weird world of WMytheology

>
>
> a
>
>
> a
> bb
>
>
> c
> ac
> bbc
>
>
> infinite means, it goes on and on without end or limit or limit case.

If there is no limit case, then WM's claim tho have the rationals
well-ordered in standard order is a lie.

Of course, it is a lie anyway, but this shows that WM knows it is a lie
and chooses to lie.
--

[Virgil]

In article <71db6768-b655-4283...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Friday, 14 June 2013 20:14:16 UTC+2, Zeit Geist wrote:
> > > Your lack of intellect prevents you from forming the proper intuitions
> > > concerning the Infinite.
>
> There is no intuition required to prove that the triangle is equilateral in
> every step. So the limit is equilateral too - in mathematics.

To assume that every property of the finite cases is reflected in an
infinite limit case would require the union of all FISONs to be finite,
which, since such reflection is only true in the the wild weird world
of WMytheology, does not hold in any limiting process outside of the
wild weird world of WMytheology.

>
> > This in turn, causes you to write statements which you think are internal
> > contradictions in Set Theory, even though all you write is garbage.
>
> Sounds like withdrawal symptoms.

Apparently, living in the the wild weird world of WMytheology distorts
WM's hearing as well as his thinking.
--

[Virgil]

In article <ae6d3648-9b6e-4be9...@googlegroups.com>,

But in the wild weird world of WMytheology, unlike real mathematics,
that infinite sequence of diagrams has no limit/union diagram, so the
whole thing is irrelevant to any real mathematics.
--

[Zeit Geist]

On Friday, June 14, 2013 11:23:28 AM UTC-7, muec...@rz.fh-augsburg.de wrote:
> On Friday, 14 June 2013 20:08:07 UTC+2, Zeit Geist wrote:
>
> > Writing all numbers one line, .111111..., is a Union.
>
>
>
> And you believe that a union over sets that are unions of all preceding sets yields more than these sets. Your choice. But not a rational idea.
>

The amount of 1's in 1/9 is the same, not more, as the number of lines in being unioned.
It's just more than any single finite line has.
So?

>
> >> The list given above does not contain aleph_0 1's in any direction.
>
>
>
> > Really, can you write the whole list then?
>
>
>
> That is not relevant as an argument against anti-symmetry.
>
> You should be able to see from the symmetrical construction that symmetry has to prevail.
>

The symmetry does prevIail.
The three Unions in the "completed" list all have same number of elements.
It's just like in any finite sub-list.

>
> > If its finite you should be able to.
>
>
>
> "Finite" is not the complement of aleph_0. Finite is the complement of infinite, that means there is no finish, you cannot finish the list. Of course there is no whole list in the sense that no line can be added.
>

There is in ZFC, and that is where you supposedly are trying to find a contradiction.

Saying that infinity does not exist in "Reality" and ZFC assumes infinity,
does make make ZFC an invalid Mathematical construct.

>
> Regards, WM

ZG

[Zeit Geist]

On Friday, June 14, 2013 11:28:15 AM UTC-7, muec...@rz.fh-augsburg.de wrote:
> On Friday, 14 June 2013 20:14:16 UTC+2, Zeit Geist wrote:
>
> > > Your lack of intellect prevents you from forming the proper intuitions concerning the Infinite.
>
>
>
> There is no intuition required to prove that the triangle is equilateral in every step. So the limit is equilateral too - in mathematics.
>

And I showed how it is an "aleph_0 equilateral triangle".

>
>
> Regards, WM

[Zeit Geist]

Topology? Can you even define Topology?
And then explain what the definition means?

>
> Regards, WM

ZG

[Virgil]

In article <b91f9bf3-219a-410a...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Friday, 14 June 2013 20:09:57 UTC+2, Virgil wrote:
>
> > In order for that tree to contain all binary rationals rationals in [0,1]
> > at all. it must contain all but numbers 0 an 1 twice each, once with a
> > "tail" of infinitely many 0's and once with a "trail" of infinitely many
> > 1's, and, into the bargain, all paths between two such paths. Thus
> > containing irrationals as well.
>
> They come in on angels wings - if actual infinity exists.

It does quite standardly in the world of mathematics outside of the wild
weird world of WM's WMytheology

>
>
> > d
> > dc
> > dac
> > dbbc
> > ...
>
> > It is not a triangle when it only has two sides. So it is only an angle
> > until you can show us a third side.
>
> If you look at the letters, you see that every side is treated equally when
> extending the triangle. It has three equal sides in every finite step.

But the whole of it does. not

>
> > > 3) For the union of the sequence of sets > U({1}, {1, 2}, {1, 2, 3} ,
> > > ..., {1, 2, 3, ..., n}} = {1, 2, 3, ..., n} > equality holds but not in
> > > the limit.
>
> > The limit, if it is to exist at all,
>
> just that is contradicted here
>
> > of any strictly increasing infinite sequence cannot be a member of the
> > sequence.
>
> So it is. But to get in a second union more than in the first, without adding
> anything in between, shows that there is no limit.

WM's "first" limit is not a proper limit at all, as it does not take
account of ALL the members of the relevant sequences, but always stops
with members of the sequence still unused.

In order to get a proper limit of an infinite sequence, one must take
into account all infinitely many terms of that sequence.

If living in the wild weird world of WMytheology prevents WM from taking
all infinitely many terms of an infinite sequence into account then WM
is incapable of dealing with such limiting processes correctly, as we
have seen, and keep seeing.
--

[Virgil]

In article <03b31204-4e19-44bc...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Friday, 14 June 2013 20:08:07 UTC+2, Zeit Geist wrote:
> > Writing all numbers one line, .111111..., is a Union.
>
> And you believe that a union over sets that are unions of all preceding sets
> yields more than these sets.

WM may believe ZG believes that, but so many of the things that WM
beleives are so obviously contrary to fact tha no one should accept any
to WM's beliefs as factual without independent evidence.

In the above instance, WM's claim rests on so many unwarranted
assumptions that its falsehood is obvious.

At least to everyone except WM himself.
--

[Zeit Geist]

How naive you are. Are still living in Ancient Greece.

Definition: A set, X, is infinite iff there exists a function, F and a
subset Y of X, such that F: X --> Y and F is onto.

Theorem: In ZFC, if X is an infinite set, then for all n e N, there
exists a Y c X such that there is function that map X onto Y.

Theorem: In ZFC, if X is an infinite set, then for all n e N, the
cardinality of X is greater than n.

This is ZFC. If you want to create a consistent theory that is
In opposition to ZFC, feel free. This, however, will NOT effect the
validity of ZFC.

>
> Regards, WM

ZG

[Virgil]

In article <5a693a23-8c97-4ff4...@googlegroups.com>,

X need only be infinite if the set Y is a PROPER subset of X.

>
> Theorem: In ZFC, if X is an infinite set, then for all n e N, there
> exists a Y c X such that there is function that map X onto Y.

In ZFC, if X is an Finite set, then for all n e N, there

exists a Y c X such that there is function that map X onto Y.

Namely whenever Y = X

>
> Theorem: In ZFC, if X is an infinite set, then for all n e N, the
> cardinality of X is greater than n.

That one is both correct and meamingful.

>
> This is ZFC. If you want to create a consistent theory that is
> In opposition to ZFC, feel free. This, however, will NOT effect the
> validity of ZFC.

If you want to create a consistent theory other
than ZFC, try ZF.

Standard mathematics is almost all compatible with ZF, with or without C.
A very large part of standard mathematics is incompatible with WM's
WMytheology.
--

[Virgil]

On Friday, June 14, 2013 11:36:39 AM UTC-7, muec...@rz.fh-augsburg.de
wrote:
> On Friday, 14 June 2013 20:14:12 UTC+2, Virgil wrote:
>
>
>
> > The triangles, at least as presented in WM's diagrams, have all
> > been right triangles, so that in his WMytheology, WM claims to have
> > equilateral right triangles, things that exist nowhere else.
>
>
>
> With respect to the topology that I use, namely the number of letters
> per side is the measure, the triangles are equilateral.

Then why display them with a right angle?
--

[Zeit Geist]

On Friday, June 14, 2013 3:04:22 PM UTC-7, Virgil wrote:
> In article <5a693a23-8c97-4ff4...@googlegroups.com>,
>
> Zeit Geist <tucso...@me.com> wrote:
>
> > Definition: A set, X, is infinite iff there exists a function, F and a
>
> > subset Y of X, such that F: X --> Y and F is onto.
>
>
>
> X need only be infinite if the set Y is a PROPER subset of X.
>

Y c X is meant to read "Y is a proper subset of X"
Just like a < b reads as simply "less than" when a and b are real numbers.

>
> > Theorem: In ZFC, if X is an infinite set, then for all n e N, there
>
> > exists a Y c X such that there is function that map X onto Y.
>
>
>
> In ZFC, if X is an Finite set, then for all n e N, there
>
> exists a Y c X such that there is function that map X onto Y.
>
> Namely whenever Y = X
>

Clarified above.

>
> > Theorem: In ZFC, if X is an infinite set, then for all n e N, the
>
> > cardinality of X is greater than n.
>
>
>
> That one is both correct and meamingful.
>

All are correct with clarification.
When the improper subset of Y = X is permitted,
I, and most I believe, write Y c= X.

I try to be more clear in the future.
When writing this i thought somebody might say this.
All good.

>
> > This is ZFC. If you want to create a consistent theory that is
>
> > In opposition to ZFC, feel free. This, however, will NOT effect the
>
> > validity of ZFC.
>
>
>
> If you want to create a consistent theory other
>
> than ZFC, try ZF.
>
>
>
> Standard mathematics is almost all compatible with ZF, with or without C.
>
> A very large part of standard mathematics is incompatible with WM's
>
> WMytheology.
>

I never said he was coming anywhere close to creating an alternative
consistent theory of any sort.

>--

ZG

[Virgil]

On Friday, June 14, 2013 11:28:15 AM UTC-7, muec...@rz.fh-augsburg.de
wrote:
> On Friday, 14 June 2013 20:14:16 UTC+2, Zeit Geist wrote:
>
> > > Your lack of intellect prevents you from forming the proper
> > > intuitions concerning the Infinite.
>
>
>
> There is no intuition required to prove that the triangle is
> equilateral in every step. So the limit is equilateral too - in
> mathematics.
>

That limit has no third side, at least none outside of the wild weird
world of WMytheology, so your TRI-angle degenerates in the limit into,
at best, a MONO-angle.

If there were any third side to WM's triangles in the limit, that third
side would have to have a length, but does not
And in WM's limit, neither of the of the other sides have lengths
either, since in WMytheology infinite lengths are outlawed, and your
alleged limit third side in WMytheology does not have either a position
or a length, so is totally mythical.

>
>
> Regards, WM
--

[Virgil]

In article <d2c0ba71-6f46-4dcf...@googlegroups.com>,

Zeit Geist <tucso...@me.com> wrote:

> On Friday, June 14, 2013 11:23:28 AM UTC-7, muec...@rz.fh-augsburg.de wrote:
> > On Friday, 14 June 2013 20:08:07 UTC+2, Zeit Geist wrote:
> >
> > > Writing all numbers one line, .111111..., is a Union.
> >
> >
> >
> > And you believe that a union over sets that are unions of all preceding
> > sets yields more than these sets. Your choice. But not a rational idea.
> >
>
> The amount of 1's in 1/9 is the same, not more, as the number of lines in
> being unioned.

What does "the number of lines in being unioned" mean in proper English?

> It's just more than any single finite line has.
> So?

In your wild weird world of WMytheology, more than finite is not
allowed, so your triangles must all have last lines which are finite.

And all such idiot restrictions inside WMytheology are irrelevant
outside of WMytheology.

>
> >
> > >> The list given above does not contain aleph_0 1's in any direction.
> >
> >
> >
> > > Really, can you write the whole list then?

One can describe it completely without having to write it all out digit
by digit, and 0.111..., at least to those who understand standard
mathematical notation, is just such a description.

> >
> >
> >
> > That is not relevant as an argument against anti-symmetry.
> >
> > You should be able to see from the symmetrical construction that symmetry
> > has to prevail.
> >
>
> The symmetry does prevIail.

NOT as you have diagrammed it. In ALL your diagramming, all finite
triangles are represented as right triangles with one set of sides
vertical and the other horizontal.

> The three Unions in the "completed" list all have same number of elements.
> It's just like in any finite sub-list.

Except that, at least outside WMytheology, the vertical and diagonal
sides of the limit diagram are endless so that any imagined third side
must be missing both ends, and your alleged triangle has only one vertex.

>

>
> Saying that infinity does not exist in "Reality" and ZFC assumes infinity,
> does make make ZFC an invalid Mathematical construct.

Mathematics is not about physical realities, but about mental realities,
which are quite different, and both are quite different from
WMytheology.
--

[Virgil]

In article <7d33581a-40f3-4b20...@googlegroups.com>,

Zeit Geist <tucso...@me.com> wrote:

> On Friday, June 14, 2013 3:04:22 PM UTC-7, Virgil wrote:
> > In article <5a693a23-8c97-4ff4...@googlegroups.com>,
> >
> > Zeit Geist <tucso...@me.com> wrote:
> >
> > > Definition: A set, X, is infinite iff there exists a function, F and a
> >
> > > subset Y of X, such that F: X --> Y and F is onto.
> >
> >
> >
> > X need only be infinite if the set Y is a PROPER subset of X.
> >
>
> Y c X is meant to read "Y is a proper subset of X"
> Just like a < b reads as simply "less than" when a and b are real numbers.

Then you should explain that Y c= X means ordinary subset.

>
> >
> > > Theorem: In ZFC, if X is an infinite set, then for all n e N, there
> >
> > > exists a Y c X such that there is function that map X onto Y.
> >
> >
> >
> > In ZFC, if X is an Finite set, then for all n e N, there
> >
> > exists a Y c X such that there is function that map X onto Y.
> >
> > Namely whenever Y = X
> >
>
> Clarified above.

Not so. If by Y c X you mean PROPER subset then your statement above
using 'Y c X' is false!

>
> >
> > > Theorem: In ZFC, if X is an infinite set, then for all n e N, the
> >
> > > cardinality of X is greater than n.
> >
> >
> >
> > That one is both correct and meamingful.
> >
>
> All are correct with clarification.
> When the improper subset of Y = X is permitted,
> I, and most I believe, write Y c= X.

You missed one! But is is easy enough to do.

>
> I try to be more clear in the future.
> When writing this i thought somebody might say this.
> All good.
>
> >
> > > This is ZFC. If you want to create a consistent theory that is
> >
> > > In opposition to ZFC, feel free. This, however, will NOT effect the
> >
> > > validity of ZFC.
> >
> >
> >
> > If you want to create a consistent theory other
> >
> > than ZFC, try ZF.
> >
> >
> >
> > Standard mathematics is almost all compatible with ZF, with or without C.
> >
> > A very large part of standard mathematics is incompatible with WM's
> >
> > WMytheology.
> >
>
> I never said he was coming anywhere close to creating an alternative
> consistent theory of any sort.

Understood!
>
> >--
>
> ZG
--

[Zeit Geist]

On Friday, June 14, 2013 3:37:13 PM UTC-7, Virgil wrote:
> In article <d2c0ba71-6f46-4dcf...@googlegroups.com>,
>
> Zeit Geist <tucso...@me.com> wrote:
>
>
>
> > On Friday, June 14, 2013 11:23:28 AM UTC-7, muec...@rz.fh-augsburg.de wrote:
>
> > > On Friday, 14 June 2013 20:08:07 UTC+2, Zeit Geist wrote:
>
> > >
>
> > > > Writing all numbers one line, .111111..., is a Union.
>
> > >
>
> > >
>
> > >
>
> > > And you believe that a union over sets that are unions of all preceding
>
> > > sets yields more than these sets. Your choice. But not a rational idea.
>
> > >
>
> >
>
> > The amount of 1's in 1/9 is the same, not more, as the number of lines in
>
> > being unioned.
>
>
>
> What does "the number of lines in being unioned" mean in proper English?
>

Fine. How about "the cardinality of the set to which the union is applied"?

>
>
>
> > It's just more than any single finite line has.
>
> > So?
>
>
>
> In your wild weird world of WMytheology, more than finite is not
>
> allowed, so your triangles must all have last lines which are finite.
>
>
>
> And all such idiot restrictions inside WMytheology are irrelevant
>
> outside of WMytheology.
>
> >
>
> > >
>
> > > >> The list given above does not contain aleph_0 1's in any direction.
>
> > >
>
> > >
>
> > >
>
> > > > Really, can you write the whole list then?
>
>
>
> One can describe it completely without having to write it all out digit
>
> by digit, and 0.111..., at least to those who understand standard
>
> mathematical notation, is just such a description.
>

Yes, I know that. I was challenging WM to write it out.

>
> > >
>
> > >
>
> > > That is not relevant as an argument against anti-symmetry.
>
> > >
>
> > > You should be able to see from the symmetrical construction that symmetry
>
> > > has to prevail.
>
> > >
>
> >
>
> > The symmetry does prevIail.
>
>
>
>
>
> NOT as you have diagrammed it. In ALL your diagramming, all finite
>
> triangles are represented as right triangles with one set of sides
>
> vertical and the other horizontal.
>

No my diagram at is WM's. My arguments use Unions.

>
> > The three Unions in the "completed" list all have same number of elements.
>
> > It's just like in any finite sub-list.
>

That is exactly what I said.

>
> Except that, at least outside WMytheology, the vertical and diagonal
>
> sides of the limit diagram are endless so that any imagined third side
>
> must be missing both ends, and your alleged triangle has only one vertex.
>

I don't know of any theory that defines "triangle" in such a manner.
And WM certainly has not provide a clear definition for it in this setting.

>
>
> >
>
> > Saying that infinity does not exist in "Reality" and ZFC assumes infinity,
>
> > does make make ZFC an invalid Mathematical construct.
>
>
>
> Mathematics is not about physical realities, but about mental realities,
>
> which are quite different, and both are quite different from
>
> WMytheology.

Big OOPS!!!
That should read "does NOT make ZFC an invalid Math...".

I actually find the ironic fact that, the Mathematics that do NOT
correspond to reality to be the most fruitful in explaining physical
reality.

> --

ZG

[Zeit Geist]

On Friday, June 14, 2013 3:44:47 PM UTC-7, Virgil wrote:
> In article <7d33581a-40f3-4b20...@googlegroups.com>,
>
> Zeit Geist <tucso...@me.com> wrote:
>
>
>
> > On Friday, June 14, 2013 3:04:22 PM UTC-7, Virgil wrote:
>
> > > In article <5a693a23-8c97-4ff4...@googlegroups.com>,
>
> > >
>
> > > Zeit Geist <tucso...@me.com> wrote:
>
> > >
>
> > > > Definition: A set, X, is infinite iff there exists a function, F and a
>
> > >
>
> > > > subset Y of X, such that F: X --> Y and F is onto.
>
> > >
>
> > >
>
> > >
>
> > > X need only be infinite if the set Y is a PROPER subset of X.
>
> > >
>
> >
>
> > Y c X is meant to read "Y is a proper subset of X"
>
> > Just like a < b reads as simply "less than" when a and b are real numbers.
>
>
>
> Then you should explain that Y c= X means ordinary subset.
>
> >
>
> > >
>
> > > > Theorem: In ZFC, if X is an infinite set, then for all n e N, there
>
> > >
>
> > > > exists a Y c X such that there is function that map X onto Y.
>
> > >
>
> > >
>
> > >
>
> > > In ZFC, if X is an Finite set, then for all n e N, there
>
> > >
>
> > > exists a Y c X such that there is function that map X onto Y.
>
> > >
>
> > > Namely whenever Y = X
>
> > >
>
> >
>
> > Clarified above.
>
>
>
> Not so. If by Y c X you mean PROPER subset then your statement above
>
> using 'Y c X' is false!
>

Shit!!! Bad typing day. That was supposed to convey that an INFINITE
set can be mapped onto any finite set.

Damn, damn, damn.

>
> > >
>
> > > > Theorem: In ZFC, if X is an infinite set, then for all n e N, the
>
> > >
>
> > > > cardinality of X is greater than n.
>
> > >
>
> > >
>
> > >
>
> > > That one is both correct and meamingful.
>
> > >
>
> >
>
> > All are correct with clarification.
>
> > When the improper subset of Y = X is permitted,
>
> > I, and most I believe, write Y c= X.
>
>
>
> You missed one! But is is easy enough to do.
>
> >
>
> > I try to be more clear in the future.
>
> > When writing this i thought somebody might say this.
>
> > All good.
>
> >
>
> > >
>
> > > > This is ZFC. If you want to create a consistent theory that is
>
> > >
>
> > > > In opposition to ZFC, feel free. This, however, will NOT effect the
>
> > >
>
> > > > validity of ZFC.
>
> > >
>
> > >
>
> > >
>
> > > If you want to create a consistent theory other
>
> > >
>
> > > than ZFC, try ZF.
>
> > >
>
> > >
>
> > >
>
> > > Standard mathematics is almost all compatible with ZF, with or without C.
>
> > >
>
> > > A very large part of standard mathematics is incompatible with WM's
>
> > >
>
> > > WMytheology.
>
> > >
>
> >
>
> > I never said he was coming anywhere close to creating an alternative
>
> > consistent theory of any sort.
>
>
>
> Understood!
>

I greatly apologize for the mistakes.
I shouldn't be doing this while doing "real" work.

>
> > >--
>
> >
>
> > ZG
>
> --

ZG

[muec...@rz.fh-augsburg.de]

On Saturday, 15 June 2013 00:06:59 UTC+2, Virgil wrote:


> > With respect to the topology that I use, namely the number of letters per side is the measure, the triangles are equilateral.

> Then why display them with a right angle? --

Because the angle is without interest but easy to display.

Regards, WM

[muec...@rz.fh-augsburg.de]

On Friday, 14 June 2013 21:55:19 UTC+2, Zeit Geist wrote:
> If you want to create a consistent theory that is In opposition to ZFC, feel free. This, however, will NOT effect the validity of ZFC.

No. ZFC cannot be saved by any theory because it has been shown invalid.

Compare the silly assumption that a union over unioned sets will create more than has been there before.

Compare the silly assumption that the equilateral triangle will get perverted "in the limit".

Regards, WM ZG

[muec...@rz.fh-augsburg.de]

On Friday, 14 June 2013 21:13:51 UTC+2, Zeit Geist wrote:

> > There is no intuition required to prove that the triangle is equilateral in every step. So the limit is equilateral too - in mathematics.

> And I showed how it is an "aleph_0 equilateral triangle".

Fine. But if we forget to rotate it from step to step, then it is no longer an equilateral aleph_0 triangle, but has only two sides of measure aleph_0. So ZFC is not rotation-invariant.

Some effects of gravity seem to enter the play.

Regards, WM

[muec...@rz.fh-augsburg.de]

On Friday, 14 June 2013 21:11:16 UTC+2, Zeit Geist wrote:
> The symmetry does prevIail. The three Unions in the "completed" list all have same number of elements.

1
11
111
...

The hight and diagonal have aleph_0 elements. Is there a line with aleph_0 elements?

Regards, WM

[Zeit Geist]

On Saturday, June 15, 2013 1:26:36 AM UTC-7, muec...@rz.fh-augsburg.de wrote:
> On Friday, 14 June 2013 21:55:19 UTC+2, Zeit Geist wrote:
>
> > If you want to create a consistent theory that is In opposition to ZFC, feel free. This, however, will NOT effect the validity of ZFC.
>
>
>
> No. ZFC cannot be saved by any theory because it has been shown invalid.
>
>
>
> Compare the silly assumption that a union over unioned sets will create more than has been there before.
>

This is definitely not an assumption.
If you claim ZFC proves that, please state it rigorously and show the proof.

>
>
> Regards, WM ZG

ZG

[muec...@rz.fh-augsburg.de]

On Friday, 14 June 2013 21:11:16 UTC+2, Zeit Geist wrote:

> The amount of 1's in 1/9 is the same, not more, as the number of lines in being unioned. It's just more than any single finite line has. So?

So the union over infinitely many finite sets, each of which is the union of all its predecessors yields more than every finite set contributes. A fine credo within matheology but impossible in mathematics or any science.

Regards, WM

[muec...@rz.fh-augsburg.de]

On Friday, 14 June 2013 21:19:23 UTC+2, Virgil wrote:
> > If you look at the letters, you see that every side is treated equally when extending the triangle. It has three equal sides in every finite step.

> But the whole of it does. not

No? Which sides are different?

> For the union of the sequence of sets > U({1}, {1, 2}, {1, 2, 3} , ..., {1, 2, 3, ..., n}} = {1, 2, 3, ..., n} equality holds but not in the limit.

WM's "first" limit is not a proper limit at all, as it does not take account of ALL the members of the relevant sequences, but always stops with members of the sequence still unused. In order to get a proper limit of an infinite sequence, one must take into account all infinitely many terms of that sequence.

That's just the limit I talked about. The above formula holds *for all n*. Is more possible in matheology?

Regrads, WM

[Zeit Geist]

For all n e N, the n-th line contains n 1's.

The diagonal is the sequence of the n-th element of the n-th line, for all n e N.
The first column is the sequence of the 1-th element of the n-th line, for all n e N.

For all n e N, the diagonal is not completed at the n-th line.
For all n e N, the first column is not completed at the n-th line.
For all n e N, the line with aleph_0 1's is not completed at the n-th line.

Previous post show all these objects are identical.

The Axiom of Infinity allows us to show these objects exist.

You may choose to reject AoI, but accepting it does not lead to
a contradiction.

>
> Regards, WM

ZG

[muec...@rz.fh-augsburg.de]

On Saturday, 15 June 2013 10:56:33 UTC+2, Zeit Geist wrote:
>> Is there a line with aleph_0 elements?

> For all n e N, the diagonal is not completed at the n-th line.

For all n: the set {1, 2, ..., n} is not infinite.

So what makes the set N infinite?

What is the difference between
lim(n --> oo) SUM(1 to n) 1/n! = 2.718...
and
SUM(n e N) 1/n! = 2.718...

You need the premise that the union over a union is larger than the first union.

You need the premise that
{1} U {1, 2} U {1, 2, 3} U ... = N
But the sequence
{1}, {1} U {1, 2}, {1} U {1, 2} U {1, 2, 3}, ...
does not contain N. The union over the terms of the sequence however, is N, more than any of its infinitely many terms.

And that is obvious Humbug.

Regards, WM


Regards, WM

[Virgil]

In article <ba4992c4-4e47-4903...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Friday, 14 June 2013 21:19:23 UTC+2, Virgil wrote:
> > > If you look at the letters, you see that every side is treated equally
> > > when extending the triangle. It has three equal sides in every finite
> > > step.
>
> > But the whole of it does. not
>
> No? Which sides are different?

In mathematics, line segment has endpoints and every ray has one
endpoint..
No endpoint -> no sides and no rays

In WM's limit there is a vertical 'ray' and a diagonal 'ray', both
identifiable by having a common endpoint but no third side or ray as
there are no endpoints for any such third side or ray.

> > WM's "first" limit is not a proper limit at all, as it does not take account
> > of ALL the members of the relevant sequences, but always stops with members
> > of the sequence still unused. In order to get a proper limit of an infinite
> > sequence, one must take into account all infinitely many terms of that
> > sequence.
>
> That's just the limit I talked about. The above formula holds *for all n*

But in WMytheology, there is always a last n in that "for all n".
but outside of WMytheology there is not.
--

[muec...@rz.fh-augsburg.de]

On Saturday, 15 June 2013 22:34:09 UTC+2, Virgil wrote:


> In mathematics, line segment has endpoints and every ray has one endpoint.. No endpoint -> no sides and no rays

In my triangle every side is finite. Therefore it has two endpoints.

> > That's just the limit I talked about.

> The above formula holds *for all n* But in WMytheology, there is always a last n in that "for all n".

No. But every n is finite.

Regards WM

[Virgil]

In article <430cc06d-bd49-43f3...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Friday, 14 June 2013 21:11:16 UTC+2, Zeit Geist wrote:
> > The amount of 1's in 1/9 is the same, not more, as the number of lines in
> > being unioned. It's just more than any single finite line has. So?
>
> So the union over infinitely many finite sets, each of which is the union of
> all its predecessors yields more than every finite set contributes.

It contributes more than ANY finite set contributes, but not
"every finite set" covers all infinitely many finite sets.


> A fine credo

And it works fine everywhere outside of WMytheology.
--

[Virgil]

In article <f6349720-6c66-4047...@googlegroups.com>,

If there isn't then WM's claimed "triangle" does not have three sides.
--

[Sam Sung]

Virgil wrote:

> In article <ba4992c4-4e47-4903...@googlegroups.com>,
> muec...@rz.fh-augsburg.de wrote:
>
>> On Friday, 14 June 2013 21:19:23 UTC+2, Virgil wrote:
>>> > If you look at the letters, you see that every side is treated equally
>>> > when extending the triangle. It has three equal sides in every finite
>>> > step.
>>
>>> But the whole of it does. not
>>
>> No? Which sides are different?
>
> In mathematics, line segment has endpoints and every ray has one
> endpoint..
> No endpoint -> no sides and no rays

Affine space vs Euclid space...

> In WM's limit there is a vertical 'ray' and a diagonal 'ray', both
> identifiable by having a common endpoint but no third side or ray as
> there are no endpoints for any such third side or ray.

> ...

Yes...

>>> WM's "first" limit is not a proper limit at all, as it does not take account
>>> of ALL the members of the relevant sequences, but always stops with members
>>> of the sequence still unused. In order to get a proper limit of an infinite
>>> sequence, one must take into account all infinitely many terms of that
>>> sequence.
>>
>> That's just the limit I talked about. The above formula holds *for all n*
>
>
> But in WMytheology, there is always a last n in that "for all n".
> but outside of WMytheology there is not.

Otherwise rays cannot meet ;)

[Virgil]

In article <56dfce87-c8e2-4fda...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Saturday, 15 June 2013 10:56:33 UTC+2, Zeit Geist wrote:
> >> Is there a line with aleph_0 elements?
>
> > For all n e N, the diagonal is not completed at the n-th line.
>
> For all n: the set {1, 2, ..., n} is not infinite.
>
> So what makes the set N infinite?

Among other things, that it can be injected into a proper subset of
itself satisfies one common definition of infiniteness.

>
> What is the difference between
> lim(n --> oo) SUM(1 to n) 1/n! = 2.718...
> and
> SUM(n e N) 1/n! = 2.718...

Notation.

>
> You need the premise that the union over a union is larger than the first
> union.
>
> You need the premise that
> {1} U {1, 2} U {1, 2, 3} U ... = N
> But the sequence
> {1}, {1} U {1, 2}, {1} U {1, 2} U {1, 2, 3}, ...
> does not contain N. The union over the terms of the sequence however, is N,
> more than any of its infinitely many terms.
>
> And that is obvious Humbug.

No more humbug than that the limit, if it exists as all, of any strictly
increasing sequence is never a member of the sequence.

(Is it the case that in the wild weird world WMytheology, the limit of a
strictly increasing infinite sequence CAN be a member of that sequence?)

While that property of convergence may not be required by WM to hold
inside his wild weird world of WMytheology, be assured it does hold
everywhere else.

And only someone either foolish or mathematically corrupt would question
it.
--

[Virgil]

In article <cd6a42db-0ff4-47bb...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Friday, 14 June 2013 21:13:51 UTC+2, Zeit Geist wrote:
>
> > > There is no intuition required to prove that the triangle is equilateral
> > > in every step. So the limit is equilateral too - in mathematics.

WM is incompetent to speak for mathematics, as he is incapable of
distinguishing ti from his WMytheology.

>
> > And I showed how it is an "aleph_0 equilateral triangle".

With only one vertex.
--

[Sam Sung]

Thats it... (among others;)

[Virgil]

In article <9e973096-f74e-478d...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Friday, 14 June 2013 21:55:19 UTC+2, Zeit Geist wrote:
> > If you want to create a consistent theory that is In opposition to ZFC,
> > feel free. This, however, will NOT effect the validity of ZFC.
>
> No. ZFC cannot be saved by any theory because it has been shown invalid.

Not sown so outside of WMytheology, which has itself been shown to be

invalid.
>
> Compare the silly assumption that a union over unioned sets will create more
> than has been there before.

So that , at least in WMytheology,
{1}\/{2} unioned with {3}\/{4} cannot be larger than
either {1}\/{2} or {3}\/{4}?

>
> Compare the silly assumption that the equilateral triangle will get perverted
> "in the limit".

Everything done in accord with the pseudo-rules of WMytheology
tends to get perverted,
especially if anything beyond the finite is involved.
--

[Virgil]

In article <9a9ef219-a192-4d05...@googlegroups.com>,

The angles of a triangle are witout interest?

What about the vertices and edges?

Note that in the limit, WM's tri-angles have only one limit vertex and
two limit sides, so the limit is only a mono-angle, not a tri-angle.
--

[Sam Sung]

True, its sad.

[Sam Sung]

Foul? ;)

[Bergholt Stuttley Johnson]

Virgil wrote:
>
> What about the vertices and edges?

And the law of cosines "c^2 = a^2 + b^2 - 2ab*cos(gamma)"?

--
fix$(<$>)<$>(:)<*>((<$>((:[])<$>))(=<<)<$>(*)<$>(>>=)(+)($))$1

[Virgil]

In article <d28aa3f2-212c-428f...@googlegroups.com>,

Every triangle must have 3 vertices and 3 sides.

The limit of WM's triangle has neither, so is not a triangle.

Note that just as the limit of a sequence of sets in which each is a
proper superset of all its predecessors sets can not be a finite limit
set, the limit of those finite triangles cannot be a finite triangle, or
even a triangle, for that matter.
--

[muec...@rz.fh-augsburg.de]

On Sunday, 16 June 2013 00:11:38 UTC+2, Virgil wrote:

In my triangle every side is finite. Therefore it (the side) has two endpoints. > >
The above formula holds *for all n*. But every n is finite.

> Every triangle must have 3 vertices and 3 sides. The limit of WM's triangle has neither, so is not a triangle.

There is no limit. Neither in mine nor in Cantor's triangle
1
11
111
...

Regards, WM

[Virgil]

In article <894d3a04-4ed9-4440...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Sunday, 16 June 2013 00:11:38 UTC+2, Virgil wrote:
>
> In my triangle every side is finite. Therefore it (the side) has two
> endpoints. > >
> The above formula holds *for all n*. But every n is finite.

But then you have no last or limit triangle, merely an endless list of
triangles, at least everywhere other than in your wild weird world of
WMytheology

>
> > Every triangle must have 3 vertices and 3 sides. The limit of WM's triangle
> > has neither, so is not a triangle.
>
> There is no limit. Neither in mine nor in Cantor's triangle

Then WM does not have a last triangle, so any of wWMs claims about any
such last triangle are false.
--

[muec...@rz.fh-augsburg.de]

On Sunday, 16 June 2013 09:47:02 UTC+2, Virgil wrote:
> In article <894d3a04-4ed9-4440...@googlegroups.com>, muec...@rz.fh-augsburg.de wrote: > On Sunday, 16 June 2013 00:11:38 UTC+2, Virgil wrote:

>> In my triangle every side is finite. Therefore it (the side) has two > endpoints. The above formula holds *for all n*. But every n is finite.

> But then you have no last or limit triangle, merely an endless list of triangles,

Yes. Endless is another name for infinite.

Regards, WM

[fom]

Not necessarily.

What is common to completed infinities, as a
logical species, is their inaccessibility
from below.

If one focuses on limit ordinals, these
objects are inaccessible relative to the
successor construction.

If one focuses on the accepted construction
axioms, these objects are inaccessible
relative to the "inaccessible cardinals"
which form the notion of universe discussed
by Zermelo at the beginning of large cardinal
theory.

When considering the nature of singular judgements
in relation to universal judgements, Kant observed
that a singular judgement is to universal judgement
as an individual is to an infinity.

When completed infinities are referred to in
logical language, all that occurs is that universals
have to be interpreted in relation to "absolute
infinity".

Endlessness need not be interpreted as
you have stated. But, you respect neither
mathematics based upon axioms nor logic based
upon principles.

[muec...@rz.fh-augsburg.de]

On Sunday, 16 June 2013 11:25:27 UTC+2, fom wrote:
> On 6/16/2013 3:49 AM, muec...@rz.fh-augsburg.de wrote: > On Sunday, 16 June 2013 09:47:02 UTC+2, Virgil wrote: >> In article <894d3a04-4ed9-4440...@googlegroups.com>, muec...@rz.fh-augsburg.de wrote: > On Sunday, 16 June 2013 00:11:38 UTC+2, Virgil wrote:
>>> But then you have no last or limit triangle, merely an endless list of triangles,


> > Yes. Endless is another name for infinite.

> Not necessarily.

Necessarily! endless = without end = not ended.

> What is common to completed infinities

is that they belong to the logical species of antinomies.

Regards, WM

[muec...@rz.fh-augsburg.de]

On Sunday, 16 June 2013 11:25:27 UTC+2, fom wrote:

> But, you respect neither mathematics based upon axioms nor logic based upon

contradictions. Like the undefinability of elements and extensionality:

"Eventually, most mathematicians came to accept that definability should not be required, partly because the axiom of choice leads to nice results, but mostly because of the difficulties that arise when one tries to make notion of definability precise." (Andreas Blass)

That is a real surprise to me. Which mathematicians accepted that and when? Was there a public meeting with voting like in meta or like in the astronomy scene when Pluto has been degraded?

Wouldn't a set with undefined elements contradict the Axiom of Extensionality:

If every element of X is an element of Y and every element of Y is an element of X, then X = Y.

How could that be decided for undefined elements?

But my actual question is this: I have heard (but don't remember where) that there is another solution: The set of finite definitions is countable. That cannot be explained away, can it? But not every finite definition has a meaning. In fact, if we refrain from using common sense, we cannot even define definability, let alone the set of meaningful definitions. Therefore this set is not countable but subcountable - and if we identify subcountability with uncountability, we have won and can continue to enjoy the nice results of the axiom of choice.

Obviously to vague formulated as that matheologians could understand it - with their precisely defined definitions. Therefore deleted in MO after an hour.

Regards, WM

[fom]

On 6/16/2013 5:40 AM, muec...@rz.fh-augsburg.de wrote:
> On Sunday, 16 June 2013 11:25:27 UTC+2, fom wrote:
>> On 6/16/2013 3:49 AM, muec...@rz.fh-augsburg.de wrote: > On Sunday, 16 June 2013 09:47:02 UTC+2, Virgil wrote: >> In article <894d3a04-4ed9-4440...@googlegroups.com>, muec...@rz.fh-augsburg.de wrote: > On Sunday, 16 June 2013 00:11:38 UTC+2, Virgil wrote:
>>>> But then you have no last or limit triangle, merely an endless list of triangles,
>
>
>>> Yes. Endless is another name for infinite.
>
>> Not necessarily.
>
> Necessarily! endless = without end = not ended.
>

"endless = without end = not ended"

But no mention of infinity.

Not necessarily. These things are a
matter of definition -- something unheard
of in your subjective solipsism.

[fom]

On 6/16/2013 5:47 AM, muec...@rz.fh-augsburg.de wrote:
> On Sunday, 16 June 2013 11:25:27 UTC+2, fom wrote:
>> But, you respect neither mathematics based upon axioms nor logic based upon
>
> contradictions. Like the undefinability of elements and extensionality:
>
> "Eventually, most mathematicians came to accept that definability should
> not be required, partly because the axiom of choice leads to nice
results,
> but mostly because of the difficulties that arise when one tries
> to make notion of definability precise." (Andreas Blass)
>
> That is a real surprise to me. Which mathematicians accepted that and when?

I must concede to you on this one. Tarski wrote a paper
on definability in which he made the observation that
mathematicians are not too keen on the subject.

> Was there a public meeting with voting like in meta or like
> in the astronomy scene when Pluto has been degraded?
>

I spent a great deal of time trying to discern the origins
of "undefined language primitives" in the literature. Of
course, I have only limited means and a handful of translations.

The evidence I have been able to gather directs attention,
primarily, to Bolzano and his search for a definition of
simple substance.

The Aristotelian class hierarchy has two directions. Aristotle
asserts that genera are prior to species. Hence, his view of
class organization is a downward-directed view:

genus -> species -> individual

But, when Aristotle speaks of substance, he asserts that primary
substance is associated with individuals, secondary substance is
associated with species, and so forth. Hence, the notion of
substance is an upward-directed view:

individual -> species -> genus

Based on Leibniz' remarks, it seems that Aquinas asserted that
there are enough "properties" so that God can know every
individual. The generalization of this is Leibniz' principle
of identity of indiscernibles. But, in discussing his views
on logic, Leibniz contrasts himself with the Scholastic
tradition. Leibniz associates his views with the
downward-directed view of Aristotelian origin:

genus -> species -> individual

Thus, one must surmise that the Scholastic view is an
upward-directed view:

individual -> species -> genus

Let me call the upward-directed view "extensional" and
the downward-directed view "intensional". These are standard
terms to the best of my knowledge.

It is important to remember that Bolzano is historically
prior to the modern compositional logical systems. The
kind of definitions that Bolzano may have wished to consider
would be of the form,

"A rational man is a man"

This definition segregates the genus 'man' into the 'rational men'
and the 'irrational men'. Such is the general problem for this
syllogistic logic. In order to satisfy the general Aristotelian
requirement that "truth is the result of division and combination",
one is confronted by the fact that individuals cannot be divided
and combined.

To make matters worse, the notion of priority with regard to
language terms in definition appears to have already been
established. Thus,

"A rational man is a man"

had been admissible to Leibniz. But, for Bolzano it could not
have been because of the circular use of the term 'man'. Rather,
something along the lines of

"A bachelor is an unmarried man"

would have been more like what he considered a definition.

This kind of logic had been inappropriate for the definition of
individuals in relation to the extensional, Scholastic view that
Bolzano had been trying to implement.

Bolzano then goes on to argue for undefined language terms.

There is a second aspect to this that is discussed in the
work of De Morgan.

Specifically, the introduction of novel number systems such
as the complex numbers and the quaternions forced mathematicians
to accept the fact that arithmetical operations could be
applied more generally -- or, at least, more abstractly -- than
had been previously considered. Thus, 'number systems' begin
to be understood with respect to stipulations rather than some
intrinsic metaphysical explanation of number.

De Morgan recognized what we would now call semantic indeterminacy.
So, even familiar operations between numbers presuppose the
interpretation of abstract symbols. It is a simple step to
correlate this with Bolzano's arguments.

> Wouldn't a set with undefined elements contradict the Axiom of Extensionality:
>
> If every element of X is an element of Y and every element of Y is an element of X, then X = Y.
>
> How could that be decided for undefined elements?
>

For that I have no answer. This is, in fact, where I have non-standard
views. In my version of foundations, definability is fundamental. Of
course, I am using this notion differently from you. But, for
example, my theory begins with

AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))

AxAy(xey <-> (Az(ycz -> xez) /\ Ez(xez /\ -ycz)))

where the transitive, irreflexive order relation of 'proper part' is
prior to the 'membership' relation because the latter depends upon
the former for its definition.

In a modern interpretation, my symbols are still undefined and the
sentences above are axioms. In this view of things, any set of
axioms constitute "definitions-in-use" as opposed to the traditional
expectation whereby a defined symbol (definiendum) is related to
its definition (definiens) by a substitutivity criterion. When the
non-circularity criterion is applied (as with Bolzano) this traditional
expectation permits elimination of defined language symbols until the
only expressions remaining have undefined language symbols as
constituents.

When I say my view is different from yours, I do not care if
definitions are merely recognized in principle. So, I am
not restricting the idea to some countable set of terms. What
I consider important is to understand that the semiotics of
naming imposes a well-ordering criterion on the admissibility
of models -- to be a unique identifier, each name is restricted
from being the same as a prior name.

Logic uses names rather than numbers. Because of this, one
cannot distinguish between ordinal numbers and a unique system
of names. So, how can one speak of an inner model that cannot
be put into correspondence with the ordinal numbers?

> But my actual question is this: I have heard (but don't remember where) that there is another
> solution: The set of finite definitions is countable. That cannot be
explained away, can
> it?

Actually, it can.

That is, you are correct with regard to the limitations of
what can be expressed by "locally finite languages". But,
that is not what I am talking about.

What you are thinking of is the participation of the Lowenheim-Skolem
theorems with respect to the continuum hypothesis given by Goedel's
constructible universe (V=L).

If there is a model, then there is a countable model.

Cohen, acknowledging Shepherdson for the construction, formulates
a notion of "strongly constructible set" which, according to at
least one author, corresponds with a notion of provability concerning
the existence within the model.

I still have to look at these works more closely. My suspicion is
that this notion of provability corresponds with the notion of
provability associated with definability as discussed in Tarski's
paper mentioned above.

When I speak of what can be "explained away", I refer to the fact
that set theory ought to be logically prior to model theory. So,
I have deep reservations concerning the "model theory of set theory"
as it has been applied to prove the independence of the continuum
hypothesis.

This is why I make the distinction between set theory as a
foundational theory and set theory as "just another theory".

Since you previously cited pages from van Heijenoort, I will assume
you have it. You should look at Skolem's papers. One of them
will speak about the formability of a countable model.

> But not every finite definition has a meaning. In fact, if we
> refrain from using common sense, we cannot even define definability,
let alone the
> set of meaningful definitions.

Husserl: "What is the meaning of meaning?"

The ideas of model theory arise from the use of examples to
substantiate definitions and the use of counter-examples
to discount the universality of statements. Model theory
addresses the same questions in terms of "systems".

Unfortunately, the drive for foundations is fuel for the
skeptics of every breed.

When I finally turned to examine Aristotle and Leibniz, I understood
that the notion of definition is posterior to the deductive calculus.
Whatever linguistic analysis identifies the nature of what
transformations constitute the steps in a proof also identifies
what is admissible as a definition. In contrast to modern views,
Aristotle admits a number of notions of definition. The one
of particular relevance to my statements here are the ones he refers
to as "immediate principles".

Both of my sentences above correspond with a deductive calculus
as required by Aristotle and exemplified in Leibniz.

> Thereforethis set is not countable

> but subcountable - and if we identify subcountability with
> uncountability, we have won and can continue to enjoy the nice
results of
> the axiom of choice.
>
> Obviously to vague formulated as that matheologians could understand it - with
> their precisely defined definitions. Therefore deleted in MO
> after an hour.
>

Since I try to understand these matters with as little
deviation from classical logic and classical mathematics
as possible, I doubt that I would draw your conclusions.

But, do you have a link to where you discuss/define "subcountable"
with the intention of the interpretation you give above?

And, sorry about the long reply. I think my life would have
been easier if a committee led by Andreas Blass spoke for
all mathematicians concerning definitions and definability.

I am sure their press conference would have been on Pluto.

[muec...@rz.fh-augsburg.de]

On Sunday, 16 June 2013 17:38:56 UTC+2, fom wrote:
> On 6/16/2013 5:47 AM, muec...@rz.fh-augsburg.de wrote:

Sorry, it is really impossible for me with my Brouwser to answer your long text. Therefore only few points.

>> "Eventually, most mathematicians came to accept that definability should not be required, partly because the axiom of choice leads to nice results, but mostly because of the difficulties that arise when one tries to make notion of definability precise." (Andreas Blass)
>
>> That is a real surprise to me. Which mathematicians accepted that and when?


> I must concede to you on this one. Tarski wrote a paper
on definability in which he made the observation that
mathematicians are not too keen on the subject.

Definability is not a logical but a practical notion. Every child knows what is meant.

In 2009 a reviewer of FOM (I think it was Blass) wrote: "He's right that some statements about definability don't need details of the language, but I think he's wrong to infer (in his first sentence) "an absolute meaning of
undefinability." (Does he perhaps think there's a particular real number that is undefinable in any (countable) language?)"

That shows that Blass (if he was it) at that time did not yet know that there are undefinable real numbers (or that there has been a resolution of a majority to accept that).

>> Wouldn't a set with undefined elements contradict the Axiom of Extensionality: If every element of X is an element of Y and every element of Y is an element of X, then X = Y. How could that be decided for undefined elements?
>

> For that I have no answer. This is, in fact, where I have non-standard views.

Why? Standard views also cannot explain that. They prefer to delete the questions.

> In my version of foundations, definability is fundamental.

Then you should consider the fact that the definitions are a subset of a countable set.

> But, do you have a link to where you discuss/define "subcountable" with the intention of the interpretation you give above?

Sorry, I don't. I think it was somewhere here that I read it. But have forgotten context and possibly the intended meaning. (In my question I used that only in a rhetoric way - since every subset of a countable set is countable or empty and by no means uncountable. It was in fact not a real question.)


Regards, WM

[Virgil]

In article <91c8fab9-d815-4ffb...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Sunday, 16 June 2013 09:47:02 UTC+2, Virgil wrote:
> > In article <894d3a04-4ed9-4440...@googlegroups.com>,
> > muec...@rz.fh-augsburg.de wrote: > On Sunday, 16 June 2013 00:11:38 UTC+2,
> > Virgil wrote:
>
> >> In my triangle every side is finite. Therefore it (the side) has two >
> >> endpoints. The above formula holds *for all n*. But every n is finite.
>
> > But then you have no last or limit triangle, merely an endless list of
> > triangles,
>
> Yes. Endless is another name for infinite.

But if there ever is an end triangle, you do not ever have all the
triangles possible.

Even in WMytheology there must always be at least one more triangle, one
that is missing from your set.
--

[Virgil]

In article <e78a0d33-714d-459a...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Sunday, 16 June 2013 11:25:27 UTC+2, fom wrote:
> > But, you respect neither mathematics based upon axioms nor logic based

> > upon axioms.

>
> contradictions. Like the undefinability of elements and extensionality:
>
> "Eventually, most mathematicians came to accept that definability should not
> be required, partly because the axiom of choice leads to nice results, but
> mostly because of the difficulties that arise when one tries to make notion
> of definability precise." (Andreas Blass)
>
> That is a real surprise to me. Which mathematicians accepted that and when?
> Was there a public meeting with voting like in meta or like in the astronomy
> scene when Pluto has been degraded?

Since WM apparently claims to be able to define everything, what words
will he use in his very first definition, before he has defined any
words?

Those who are not imprisoned in WM's WMytheology realize that one cannot
define everything.

But WM is so used to his imprisonment there that he no longer sees its
walls.
--

[Virgil]

In article <6d4e80bf-a9f2-49d9...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Sunday, 16 June 2013 17:38:56 UTC+2, fom wrote:
> > On 6/16/2013 5:47 AM, muec...@rz.fh-augsburg.de wrote:
>
> Sorry, it is really impossible for me with my Brouwser to answer your long
> text. Therefore only few points.
>
> >> "Eventually, most mathematicians came to accept that definability should
> >> not be required, partly because the axiom of choice leads to nice results,
> >> but mostly because of the difficulties that arise when one tries to make
> >> notion of definability precise." (Andreas Blass)
> >
> >> That is a real surprise to me. Which mathematicians accepted that and
> >> when?
>
>
> > I must concede to you on this one. Tarski wrote a paper
> on definability in which he made the observation that
> mathematicians are not too keen on the subject.
>
> Definability is not a logical but a practical notion. Every child knows what
> is meant.

In order for any two people to agree on definitions, they must first
agree that they agree an a large numbers of undefined things, like the
meanings of the words in which they will make their first definitions.
majority to accept that).

>
> > In my version of foundations, definability is fundamental.
>
> Then you should consider the fact that the definitions are a subset of a
> countable set.

And that all agreed upon definitions presume prior, though often
unstated, agreement on the meanings of the words and grammar of some
common language.

And the meanings of the words and grammar of the language of WMytheology
is not agreed to by anyone except WM himself.

Thus no one is required to accept any of WM's definitions, nor his
interpretation or, more commonly, misinterpretations of anyone else's
meanings.

But it is sometimes amusing to watch WM's flounderings.
--

[Virgil]

In article <fc9d4d48-4f0d-46eb...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Sunday, 16 June 2013 11:25:27 UTC+2, fom wrote:
> > On 6/16/2013 3:49 AM, muec...@rz.fh-augsburg.de wrote: > On Sunday, 16
> > June 2013 09:47:02 UTC+2, Virgil wrote: >> In article
> > <894d3a04-4ed9-4440...@googlegroups.com>,
> > muec...@rz.fh-augsburg.de wrote: > On Sunday, 16 June 2013 00:11:38 UTC+2,
> > Virgil wrote:
> >>> But then you have no last or limit triangle, merely an endless list of
> >>> triangles,
>
>
> > > Yes. Endless is another name for infinite.
>
> > Not necessarily.
>
> Necessarily! endless = without end = not ended.

Infinite might be interpreted to imply endless
but but endless does not imply infinite!

Circles are endless but not infinite.
All sorts of cyclic processes are endless but not infinite.

So WM is WRONG!
AGAIN!!
AS USUAL!!!

>
> > What is common to completed infinities
>

> is that they belong to logical species

Which are extinct only within WMytheology.
--

[apoorv]

On Jun 16, 1:44 am, Virgil <vir...@ligriv.com> wrote:
> In article <f6349720-6c66-4047...@googlegroups.com>,

>
>  mueck...@rz.fh-augsburg.de wrote:
> > On Friday, 14 June 2013 21:11:16 UTC+2, Zeit Geist wrote:
> > > The symmetry does prevIail. The three Unions in the "completed" list all
> > > have same number of elements.
>
> > 1
> > 11
> > 111
> > ...
>
> > The hight and diagonal have aleph_0 elements. Is there a line with aleph_0
> > elements?
>
> If there isn't then WM's claimed "triangle" does not have three sides.
> --

This is an interesting thought.
Suppose we draw up triangles
...............................l..
...........................1....1
........................1....1....1
.....................1....1....1....1
....................................................
...........................................................
1....1....1....1....1....1....1....1....1....1....1....1
Each of the diagonal is of the order type {1,2,3....w}
The question is what is the order type of the bottom horizontal line?
-apoorv

[muec...@rz.fh-augsburg.de]

On Sunday, 16 June 2013 21:05:35 UTC+2, Virgil wrote:

> > Yes. Endless is another name for infinite.

> there must always be at least one more triangle, one that is missing from your set.

Of course. Infinite sets are never complete. It seems you are, contrary to my former impression, able to learn.

Regards, WM

[Virgil]

In article
<eae2095c-abd4-4cd9...@qn4g2000pbc.googlegroups.com>,

If your triangle has diagonal edges of a given finite order type, it has
a base edge of the same type.
A problem arises only when the edges are allowed to have the order type
of |N.
--

[muec...@rz.fh-augsburg.de]

On Sunday, 16 June 2013 21:16:06 UTC+2, Virgil wrote:
> Since WM apparently claims to be able to define everything, what words will he use in his very first definition, before he has defined any words?

The definitions are based upon the knowledge already acquired in school.

The first things you learn when soemone shows them to you and speaks their names. After a while you know a good deal of a dictionary. That requires a compartively small amount of bits. The rest of bits can be used for further definitions based upon that language. You can also learn more than one language, but not more than a finite number.

Regards, WM

[muec...@rz.fh-augsburg.de]

On Sunday, 16 June 2013 22:28:22 UTC+2, Virgil wrote:
> In article <eae2095c-abd4-4cd9...@qn4g2000pbc.googlegroups.com>, apoorv <skj...@gmail.com> wrote: > On Jun 16, 1:44áam, Virgil <vir...@ligriv.com> wrote: > > In article <f6349720-6c66-4047-9557-05fabfc326a6

-- > > This is an interesting thought. > Suppose we draw up triangles > ...............................l.. > ...........................1....1 > ........................1....1....1 > .....................1....1....1....1 > .................................................... > ........................................................... > 1....1....1....1....1....1....1....1....1....1....1....1 > Each of the diagonal is of the order type {1,2,3....w} > The question is what is the order type of the bottom horizontal line? > -apoorv

> If your triangle has diagonal edges of a given finite order type, it has a base edge of the same type. A problem arises only when the edges are allowed to have the order type of |N.

That's why there is no order type of |N and no problem.

Regards, WM

[apoorv]

On Jun 17, 1:28 am, Virgil <vir...@ligriv.com> wrote:
> In article

> <eae2095c-abd4-4cd9-8935-d345a2769...@qn4g2000pbc.googlegroups.com>,

Do you mean such a triangle does not exist?
Does the extended plane, like the extended number line exist?
How many points at infinity it has?
-Apoorv

[fom]

That is too bad.

Sorry about the long reply.

[Virgil]

In article <4cc251da-05b0-4b86...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Sunday, 16 June 2013 21:16:06 UTC+2, Virgil wrote:
> > Since WM apparently claims to be able to define everything, what words will
> > he use in his very first definition, before he has defined any words?
>
> The definitions are based upon the knowledge already acquired in school.

In other words, all formal definitions are ultimately based on things
which are not formally defined at all.
--

[Virgil]

In article <81f99435-a15e-4fd0...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Sunday, 16 June 2013 22:28:22 UTC+2, Virgil wrote:
> > In article
> > <eae2095c-abd4-4cd9...@qn4g2000pbc.googlegroups.com>, apoorv

> > <skj...@gmail.com> wrote: > On Jun 16, 1:44įam, Virgil <vir...@ligriv.com>

> > wrote: > > In article <f6349720-6c66-4047-9557-05fabfc326a6
>
> -- > > This is an interesting thought. > Suppose we draw up triangles >
> ...............................l.. > ...........................1....1 >
> ........................1....1....1 > .....................1....1....1....1 >
> .................................................... >
> ........................................................... >
> 1....1....1....1....1....1....1....1....1....1....1....1 > Each of the
> diagonal is of the order type {1,2,3....w} > The question is what is the
> order type of the bottom horizontal line? > -apoorv
>
> > If your triangle has diagonal edges of a given finite order type, it has a
> > base edge of the same type. A problem arises only when the edges are
> > allowed to have the order type of |N.
>
> That's why there is no order type of |N and no problem.

Nonsense! The naturals, |N, the integers, |Z, the rationals, |Q, and the
reals, |R, all have their own order types, at least outside of
WMytheology. One has to go all the way up to the complexes before one
gets to a set of numbers with no built in order type.

At least when not incarcerated in WM's wild, weird world of WMytheology.

But only WM is incarcerated,and by his own actions, in that ugly place.
--

[Virgil]

In article
<cab6cf9b-f991-4f77...@y3g2000pbl.googlegroups.com>,

When two sides having a common vertex no longer have two endpoints each,
but only one common endpoint at that vertex, the resulting diagram is
called simply an angle, not a triangle, at least everywhere outside the
wild weird world of WMytheology.

> Does the extended plane, like the extended number line exist?
> How many points at infinity it has?
> -Apoorv

--

[Virgil]

In article <14c5b90b-a8c0-4619...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Sunday, 16 June 2013 21:05:35 UTC+2, Virgil wrote:
>
> > > Yes. Endless is another name for infinite.
>
> > there must always be at least one more triangle, one that is missing from
> > your set.
>
> Of course. Infinite sets are never complete.

While infinite sets may not be complete, or even existent, in the wild
weird world of WMytheology, they are common enough everywhere else in
mathematics.

Within the corruption of WM's wild, weird world of WMytheology:
there is always a largest natural in every
necessarily finite set of naturals;
there is always a largest integer in every
necessarily finite set of integers;
there is always a largest rational in every
necessarily finite set of nrational;
there is always a largest real in every
necessarily finite set of reals

Free of the corruption of WM's wild, weird world of WMytheology,
but within the aegis of more standard mathematics:
there is NOT EVER a largest natural in an actually
NOT FINITE set of naturals;
there is NOT always a largest integer in an actually
NOT FINITE set of integers;
there is NOT always a largest rational in an actually
NOT FINITE set of nrational;
there is NOT always a largest real in a an actually
NOT FINITE set of reals;
--

[Virgil]

> On 6/16/2013 12:25 PM, muec...@rz.fh-augsburg.de wrote:

> >
> >> But, do you have a link to where you discuss/define "subcountable" with
> >> the intention of the interpretation you give above?
> >
> > Sorry, I don't. I think it was somewhere here that I read it. But have
> > forgotten

Curious that when WM is called on the source some of his idiotic claims,
he manages to have forgotten its source.
--

[fom]

constructive mathematics, apparently...

http://en.wikipedia.org/wiki/Subcountability

[FredJeffries]

On Jun 16, 1:17 pm, apoorv <skj...@gmail.com> wrote:
>
> This is an interesting thought.
> Suppose we draw up triangles
> ...............................l..
> ...........................1....1
> ........................1....1....1
> .....................1....1....1....1
> ....................................................
> ...........................................................
> 1....1....1....1....1....1....1....1....1....1....1....1
> Each of the diagonal is of the order type {1,2,3....w}
> The question is what is the order type of the bottom horizontal line?

omega + omega*

[apoorv]

On Jun 17, 3:48 am, Virgil <vir...@ligriv.com> wrote:
> In article

> <cab6cf9b-f991-4f77-88ce-c3af4e353...@y3g2000pbl.googlegroups.com>,

The two sides are of the order type {1,2,3...w} I.e like
xxxxx.....x.So
Each does have a " end point".

> > Does the extended plane, like the extended number line exist?
> > How many points at infinity it has?

My earlier questions remain.
-Apoorv

[apoorv]

Then one of "1"s has no predecessor ; Which one would it be- given
that the
Two sides are an unbroken string of "1" s.
-Apoorv

[apoorv]

You mean the order type is ....4321234..... .
Thanks.
-apoorv

[Virgil]

In article
<cbbb7de6-59bd-4351...@qn4g2000pbc.googlegroups.com>,

In my world, the limit of the increasing sequence of elements of order
type {1,2,3,...,n} is of order type {1 2 3,...}, with no last member.
--

[Virgil]

In article
<f9c32179-cf5e-4e10...@q10g2000pbc.googlegroups.com>,

What leads you to suppose that there ever is a last line to be a bottom
line?

In my world, for each line there is a next line, without end.
--

[muec...@rz.fh-augsburg.de]

On Sunday, 16 June 2013 21:38:26 UTC+2, Virgil wrote:

> > Necessarily! endless = without end = not ended.

> Infinite might be interpreted to imply endless but but endless does not imply infinite! Circles are endless but not infinite.

The way on a circle is infinite.

> All sorts of cyclic processes are endless but not infinite.

These processes are infinite.

Regards, WM

[muec...@rz.fh-augsburg.de]

On Monday, 17 June 2013 00:36:03 UTC+2, Virgil wrote:
> In article <4cc251da-05b0-4b86...@googlegroups.com>, muec...@rz.fh-augsburg.de wrote: > On Sunday, 16 June 2013 21:16:06 UTC+2, Virgil wrote: > > Since WM apparently claims to be able to define everything, what words will > > he use in his very first definition, before he has defined any words? > > The definitions are based upon the knowledge already acquired in school. In other words, all formal definitions are ultimately based on things which are not formally defined at all. --

Of course. "Formal" is a very subjective word that could be attached to any kind of arguing, by sketches as well as by dancing or singing or speaking.

Today it is mainly used for the formalism of FOPL+ZFC, but it is based on that thinking that has been acquired during life and that is informal. And the important things like set or definition or truth has no formal definition. Therefore this matheological branch is as useless as the rest.

You can see best that it is rubbish by "proving" something and "proving" exactly the opposite (using other axioms). A real proof is nozt the "formal" of some nonsense from some assumed nonsense, but includes above all to use the correct premises sometimes called axioms. But this word has been perverted and besmirched by matheology such that one feels some disgusting taste when using the word axiom.

Regards, WM

Regards, WM

[FredJeffries]

No, I mean the order type is 1234......4321, the order type of the
integers "turned inside out".

The first (leftmost) 1 comes from the left outside diagonal-- the one
starting at the apex and proceeding down the left side. The first 2
comes from starting from the second 1 on the right side and proceeding
down towards the lower left.

Likewise the last 1 comes from starting at the apex and proceeding
down the right side. The last 2 from starting at the second 1 on the
left side and proceeding towards the lower right.

Perhaps if we relabel:

.................(1,1).................
............(1,2)......(2,1)...........
.......(1,3).....(2,2)......(3,1)......
.
.
.
1 2 3.............3.....2......1

[FredJeffries]

On Jun 16, 9:49 pm, Virgil <vir...@ligriv.com> wrote:
> In article

> <f9c32179-cf5e-4e10-9f69-9c2f5687b...@q10g2000pbc.googlegroups.com>,

Because he said that the order type of each diagonal is {1,2,3....w},
which I take to mean omega+1

[apoorv]

On Jun 17, 9:47 am, Virgil <vir...@ligriv.com> wrote:
> In article

> <cbbb7de6-59bd-4351-80c4-a55fba815...@qn4g2000pbc.googlegroups.com>,

The sequence 1,1/2,1/4,1/8.....0 is of the order type {1,2,3...w}.
Apoorv

[Virgil]

In article <b9c62fb3-e752-4c79...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Monday, 17 June 2013 00:36:03 UTC+2, Virgil wrote:
> > In article <4cc251da-05b0-4b86...@googlegroups.com>,
> > muec...@rz.fh-augsburg.de wrote: > On Sunday, 16 June 2013 21:16:06 UTC+2,
> > Virgil wrote: > > Since WM apparently claims to be able to define
> > everything, what words will > > he use in his very first definition, before
> > he has defined any words? > > The definitions are based upon the knowledge
> > already acquired in school. In other words, all formal definitions are
> > ultimately based on things which are not formally defined at all. --
>
> Of course. "Formal" is a very subjective word that could be attached to any
> kind of arguing, by sketches as well as by dancing or singing or speaking.
>
> Today it is mainly used for the formalism of FOPL+ZFC, but it is based on
> that thinking that has been acquired during life and that is informal. And
> the important things like set or definition or truth has no formal
> definition.

Everything we attempt to define ultimately has no formal definition

>
> You can see best that it is rubbish by "proving" something and "proving"
> exactly the opposite (using other axioms).

Why is there any problem with different assumptions leading to
different, even possible exactly opposite conclusions?

Anyone who is incapable of realizing that one's conclusions derive from
one's assumptions, and different assumption tend to lead to different
conclusions, knows nothing about mathematics or logic and had no common
sense.

Anyone who, like WM, prides himself on such a willful amalgam of
ignorance and stupidity, should be forever prohibited from teaching
anyone anything. That he has not been is evidence of corruption at
Hochschule Augsburg, University of Applied Sciences
--

[Virgil]

In article <57ca1022-ed1d-4e9d...@googlegroups.com>,

muec...@rz.fh-augsburg.de wrote:

> On Sunday, 16 June 2013 21:38:26 UTC+2, Virgil wrote:
>
> > > Necessarily! endless = without end = not ended.
>
> > Infinite might be interpreted to imply endless but but endless does not
> > imply infinite! Circles are endless but not infinite.
>
> The way on a circle is infinite.

Not unless there is an actual infinity, which WM denies.

>
> > All sorts of cyclic processes are endless but not infinite.
>
> These processes are infinite.

Not at all.

Such a process, like a circle, is itself finite, and can possibly be
repeated endlessly, but not infinitely unless there is an infinity.
--