Discussion:
Banach–Tarski paradox
(too old to reply)
peteolcott
2018-08-20 17:51:56 UTC
Permalink
We're not saying the sets in the partition are spherical. Their union is a ball, and the union of their images under a bunch of rotational isometries is two balls. The pieces are made using the axiom of Choice, they are not Boreland sets.
Ah so so are trying to get away with fudging the meaning of the words.
If by ball you mean a physically existing object then we are not dealing
with points we are dealing with atoms. If you take away all of the atoms
and divide them into two balls of the same size and shape they now have
half as much mass.
Talking about points on a physically exiting ball is incongruous thus
incoherent. It is either atoms of a ball or points on a sphere. Either
way Banach–Tarski is simply a silly mistake.
Copyright 2018 Pete Olcott
No, I made no reference to physical reality at all, and there was no fudging of the meanings of words. The Banach-Tarski paradox is provable in ZFC. It says that there exists a partition of the solid ball in 3-space into five subsets, four of which have infinitely many points and which are not Borel sets. What you are saying is rubbish.
You would have to point out an actual error of my reasoning for your
assertion that it is rubbish to have any basis what-so-ever.
So you're putting forward a claim to having engaged in reasoning?
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.

I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.

My proof of that is that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.

On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.

Copyright 2018 Pete Olcott
peteolcott
2018-08-20 19:39:13 UTC
Permalink
Post by peteolcott
On Saturday, August 18, 2018 at 6:54:18 PM UTC+2,
You would have to point out an actual error of my reasoning
for your assertion that it is rubbish to have any basis
what-so-ever.
So you're putting forward a claim to having engaged in
reasoning?
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states
that it is possible to decompose a ball into six pieces
which can be reassembled by rigid motions to form two
balls of the same size as the original. The number of
pieces was subsequently reduced to five by Robinson (1947),
although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere
into any subsets and recompose two identical spheres from
these same points.
My *proof* of that is that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) *it is self-evident* that no method for achieving
the desired end-result can possibly exist.
*emphasis added*
"It is self-evident" is not a proof of your claim.
AKA axiomatic. The point is totally proven entirely on the basis
of the meaning of its words.
Your attempt to use the phrase as a proof does prove that
you have not the faintest idea what a proof is, but that's all.
Mutually interlocking semantic specifications. Not the sort of
thing that you would have much knowledge of or experience with.
Post by peteolcott
On the other hand, not having these points georeferenced
would allow the possibility of using the same points twice
to go undetected.
I would suggest that you (PO) go look at the proof in order to
see that points are not being referenced twice, except that
(1) I know you won't do that, and (2) even if you ever did,
you wouldn't understand what you're looking at.
When points are uniquely identified by georeferencing it
is obvious that only a single sphere can possibly exist
with these uniquely identified points.

When-so-ever these points are not uniquely identified then
using a point more than once is indiscernible.

Do you have some other way besides georeferencing that makes
using the same point more than once unequivocally discernible?

I contend that no such system exists, provide a counter-example
proving me wrong on this point.

This whole (Banach–Tarski paradox) only arises because people
are not using my solution (derived in 15 minutes) to totally
abolish the whole issue of: [The Identity of Indiscernibles]
https://plato.stanford.edu/entries/identity-indiscernible/

When the identity of a thing also includes its precise point
in space-time, then two different things are always discernible
and have unique identities.

An imaginary sphere may not have a location in time, yet when
anchored in space has a unique identity. When it is not anchored
in space it lacks a unique identity. Thus the whole idea of
decomposing and recomposing a specific sphere that is not anchored
in space is incorrect because it is impossible to actually
identify any such sphere.

Copyright 2016, 2017, 2018 Pete Olcott
This isn't much different from your argument for the
invalidity of Godel's and Tarski's results on formal
incompleteness and undefinability of truth: You don't
know what is being said, so you make something up.
At that point, it is "self-evident" to you that the thing
you made up is wrong.
I'm not sure I've ever seen you make an argument that is
 "I (PO) don't know what you're saying, so I'll pretend this
 other thing is what you're saying, and show that the
 other thing is wrong."
https://en.wikipedia.org/wiki/Dunning%E2%80%93Kruger_effect
peteolcott
2018-08-21 15:37:48 UTC
Permalink
Post by peteolcott
When the identity of a thing also includes its precise point
in space-time, then two different things are always discernible
and have unique identities.
Your mention of space-time must mean that you're referring
to the actual world. In the actual world, some objects
(fundamental particles, for example) can be different and
also indiscernible.
It's a quantum thing. Suppose we want to know how many
states W the electrons in a piece of silicon crystal _could_
be in, if the total energy of the electrons were U.
 (W is essentially the entropy S of the electrons,
 S = k log W, a very important physical measurement.)
If we count the possible states while assuming that the
electrons are all discernible from each other, _we get_
_the wrong number_ as revealed by our measurements. If we
count while assuming electrons are _not_ discernible,
_we get the right number_ .
For example, suppose we have three orbitals that _could_
accept an electron, and two electrons -- discernible,
we assume, so e1 and e2. How many states are possible?
 [e1][e2][  ],  [e1][  ][e2],  [  ][e1][e2],
 [e2][e1][  ],  [e2][  ][e1],  [  ][e2][e1]
So, six states.
How many states if the electrons are indiscernible?
 [e][e][  ],  [e][  ][e],  [  ][e][e],
 [e][e][  ],  [e][  ][e],  [  ][e][e]
This time, *three states*, because swapping the positions
of indiscernible electrons leaves the state unaltered.
It's an odd thing to be correcting the Creator of the
Universe about, but there you have it. It would be an
understandable mistake to come from one of us non-Creators.
----
Of course, the Banach-Tarski paradox is not about the
actual physical universe, so both your comment and my
rejoinder are irrelevant to that. I just mean to correct
that one mistake in this post, and leave your others
for later.
This is related to sci.lang in that it further elaborates the mathematics
of semantics which is the broader subject of the mathematics of the
meaning of words, AKA formal semantics of linguistics.

When we imagine Banach-Tarski and do not provide some way or another
of uniquely identifying the sphere in question (such as georeferencing)
then we inadvertently conflate one sphere with another and through this
conflation confuse ourselves into thinking that one sphere can
be decomposed into pieces and then subsequently recomposed into two
different spheres.

Yes it is quite unconventional to apply georeferencing to mathematical
objects, new knowledge always tends to be quite unconventional. The
only possible way to eliminate the issue of the identity of (otherwise)
indiscernibles is to specify some set of properties such that a distinction
can always be made between two otherwise indiscernible objects.

It occurred to me less then 15 minutes after first encountering the identity
of indiscernibles through Mitch that otherwise indiscernible objects might
always be made discernible when one considers the property of their point in
time and their points in space.

The whole subject of quantum mechanics as illustrated by Schrödinger's cat
has its ultimate ground of being in the actual true nature of reality as
opposed to common misconceptions of this nature of reality.

Since these answers delve into religion they are off-topic here because none
of you has a sufficient basis to begin to understand them. Only the first-hand
direct experience of Buddhist enlightenment adequately provides these answers
in a way accessible to the human mind.

Copyright 2016, 2017, 2018 Pete Olcott
Peter Percival
2018-08-21 17:03:42 UTC
Permalink
Post by peteolcott
The whole subject of quantum mechanics as illustrated by Schrödinger's cat
has its ultimate ground of being in the actual true nature of reality as
opposed to common misconceptions of this nature of reality.
Schrödinger with his cat intended to refute the Copenhagen
interpretation of qm. It does not illustrate the whole subject.
peteolcott
2018-08-21 19:30:04 UTC
Permalink
Post by peteolcott
The whole subject of quantum mechanics as illustrated by Schrödinger's cat
has its ultimate ground of being in the actual true nature of reality as
opposed to common misconceptions of this nature of reality.
Schrödinger with his cat intended to refute the Copenhagen interpretation of qm.  It does not illustrate the whole subject.
By what possible means could a cat actually be simultaneously
alive and dead?

Peter T. Daniels
2018-08-21 18:08:16 UTC
Permalink
Post by peteolcott
This is related to sci.lang in that it further elaborates the mathematics
of semantics which is the broader subject of the mathematics of the
meaning of words, AKA formal semantics of linguistics.
Nothing to do with human language. There's no such thing as "formal semantics
of linguistics."
DKleinecke
2018-08-21 18:27:32 UTC
Permalink
Post by Peter T. Daniels
Post by peteolcott
This is related to sci.lang in that it further elaborates the mathematics
of semantics which is the broader subject of the mathematics of the
meaning of words, AKA formal semantics of linguistics.
Nothing to do with human language. There's no such thing as "formal semantics
of linguistics."
There does seem to be a group of people who try to add
semantics to Chomskian formalism and call the result
"formal semantics". Montague started it and it doesn't
seem to have died out yet.
wugi
2018-08-20 19:40:48 UTC
Permalink
Post by peteolcott
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.
Even if formulated in a more challenging way than the following
examples, I see hardly a difference with the fact that any line segment
is equipotent (or what's it called) to any other line segment, yes to
any surface, to any volume, and so on: R ~ R^n.

So let's take two concentric circles with radii r and 2r. Circle r can
self-evidently be transformed onto circle 2r. Each point of circle r
finds its own place on circle 2r by a simple radial transfer (and vice
versa). Afterwards, from the circle 2r curve you can make two new
circles with radius r.

QED
--
"copywrite" guido wugi
peteolcott
2018-08-20 20:12:17 UTC
Permalink
Post by peteolcott
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.
Even if formulated in a more challenging way than the following examples, I see hardly a difference with the fact that any line segment is equipotent (or what's it called) to any other line segment, yes to any surface, to any volume, and so on: R ~ R^n.
So let's take two concentric circles with radii r and 2r. Circle r can self-evidently be transformed onto circle 2r. Each point of circle r finds its own place on circle 2r by a simple radial transfer (and vice versa). Afterwards, from the circle 2r curve
you can make two new circles with radius r.
QED
This whole (Banach–Tarski paradox) only arises because people
are not using my solution (derived in 15 minutes) to totally
abolish the whole issue of: [The Identity of Indiscernibles]
https://plato.stanford.edu/entries/identity-indiscernible/

When the identity of a thing also includes its precise point
in space-time, then two different things are always discernible
and have unique identities.

An imaginary sphere may not have a location in time, yet when
anchored in space has a unique identity. When it is not anchored
in space it lacks a unique identity. Thus the whole idea of
decomposing and recomposing a specific sphere that is not anchored
in space is incorrect because it is impossible to actually
identify any such sphere.

Copyright 2016, 2017, 2018 Pete Olcott
Peter Percival
2018-08-21 12:47:03 UTC
Permalink
Post by peteolcott
Post by wugi
Post by peteolcott
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.
Even if formulated in a more challenging way than the following
examples, I see hardly a difference with the fact that any line
segment is equipotent (or what's it called) to any other line segment,
yes to any surface, to any volume, and so on: R ~ R^n.
So let's take two concentric circles with radii r and 2r. Circle r can
self-evidently be transformed onto circle 2r. Each point of circle r
finds its own place on circle 2r by a simple radial transfer (and vice
versa). Afterwards, from the circle 2r curve you can make two new
circles with radius r.
QED
This whole (Banach–Tarski paradox) only arises because people
are not using my solution (derived in 15 minutes) to totally
abolish the whole issue of: [The Identity of Indiscernibles]
https://plato.stanford.edu/entries/identity-indiscernible/
When the identity of a thing also includes its precise point
in space-time
What have points in space-time got to do with the Banach–Tarski paradox?
It seems that you do not know what mathematicians mean by "sphere".
Post by peteolcott
, then two different things are always discernible
and have unique identities.
An imaginary sphere may not have a location in time, yet when
anchored in space has a unique identity. When it is not anchored
in space it lacks a unique identity. Thus the whole idea of
decomposing and recomposing a specific sphere that is not anchored
in space is incorrect because it is impossible to actually
identify any such sphere.
Copyright 2016, 2017, 2018 Pete Olcott
peteolcott
2018-08-21 14:11:02 UTC
Permalink
Post by peteolcott
Post by peteolcott
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.
Even if formulated in a more challenging way than the following examples, I see hardly a difference with the fact that any line segment is equipotent (or what's it called) to any other line segment, yes to any surface, to any volume, and so on: R ~ R^n.
So let's take two concentric circles with radii r and 2r. Circle r can self-evidently be transformed onto circle 2r. Each point of circle r finds its own place on circle 2r by a simple radial transfer (and vice versa). Afterwards, from the circle 2r
curve you can make two new circles with radius r.
QED
This whole (Banach–Tarski paradox) only arises because people
are not using my solution (derived in 15 minutes) to totally
abolish the whole issue of: [The Identity of Indiscernibles]
https://plato.stanford.edu/entries/identity-indiscernible/
When the identity of a thing also includes its precise point
in space-time
What have points in space-time got to do with the Banach–Tarski paradox?  It seems that you do not know what mathematicians mean by "sphere".
I augmented the conception of a sphere such that an individual
sphere can be uniquely identified and thus not inadvertently
conflated with other different spheres having the exact same size.

When I add this required extra degree of discernment, Banach–Tarski
cannot slip through the cracks of vagueness, thus ceases to exist.

Copyright 2018 Pete Olcott
Post by peteolcott
, then two different things are always discernible
and have unique identities.
An imaginary sphere may not have a location in time, yet when
anchored in space has a unique identity. When it is not anchored
in space it lacks a unique identity. Thus the whole idea of
decomposing and recomposing a specific sphere that is not anchored
in space is incorrect because it is impossible to actually
identify any such sphere.
Copyright 2016, 2017, 2018 Pete Olcott
Peter Percival
2018-08-21 15:04:24 UTC
Permalink
Post by peteolcott
Post by Peter Percival
Post by peteolcott
Post by wugi
Post by peteolcott
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.
Even if formulated in a more challenging way than the following
examples, I see hardly a difference with the fact that any line
segment is equipotent (or what's it called) to any other line
segment, yes to any surface, to any volume, and so on: R ~ R^n.
So let's take two concentric circles with radii r and 2r. Circle r
can self-evidently be transformed onto circle 2r. Each point of
circle r finds its own place on circle 2r by a simple radial
transfer (and vice versa). Afterwards, from the circle 2r curve you
can make two new circles with radius r.
QED
This whole (Banach–Tarski paradox) only arises because people
are not using my solution (derived in 15 minutes) to totally
abolish the whole issue of: [The Identity of Indiscernibles]
https://plato.stanford.edu/entries/identity-indiscernible/
When the identity of a thing also includes its precise point
in space-time
What have points in space-time got to do with the Banach–Tarski
paradox?  It seems that you do not know what mathematicians mean by
"sphere".
I augmented the conception of a sphere
That's very obliging of you. The Banach–Tarski paradox is about
mathematical spheres, it is not about your augmentation.
Post by peteolcott
such that an individual
sphere can be uniquely identified and thus not inadvertently
conflated with other different spheres having the exact same size.
When I add this required extra degree of discernment, Banach–Tarski
cannot slip through the cracks of vagueness, thus ceases to exist.
Copyright 2018 Pete Olcott
Post by Peter Percival
Post by peteolcott
, then two different things are always discernible
and have unique identities.
An imaginary sphere may not have a location in time, yet when
anchored in space has a unique identity. When it is not anchored
in space it lacks a unique identity. Thus the whole idea of
decomposing and recomposing a specific sphere that is not anchored
in space is incorrect because it is impossible to actually
identify any such sphere.
Copyright 2016, 2017, 2018 Pete Olcott
Mike Terry
2018-08-20 20:34:51 UTC
Permalink
Post by wugi
Post by peteolcott
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.
Even if formulated in a more challenging way than the following
examples, I see hardly a difference with the fact that any line segment
is equipotent (or what's it called) to any other line segment, yes to
any surface, to any volume, and so on: R ~ R^n.
So let's take two concentric circles with radii r and 2r. Circle r can
self-evidently be transformed onto circle 2r. Each point of circle r
finds its own place on circle 2r by a simple radial transfer (and vice
versa). Afterwards, from the circle 2r curve you can make two new
circles with radius r.
QED
True, but here you're partitioning the cirlce into infinitely many
partitions, which is missing a key point.

Try again, but dividing the circle into only a finite number of pieces!

Regards,
Mike.
peteolcott
2018-08-21 12:31:46 UTC
Permalink
Post by wugi
Post by peteolcott
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.
Even if formulated in a more challenging way than the following
examples, I see hardly a difference with the fact that any line segment
is equipotent (or what's it called) to any other line segment, yes to
any surface, to any volume, and so on: R ~ R^n.
So let's take two concentric circles with radii r and 2r. Circle r can
self-evidently be transformed onto circle 2r. Each point of circle r
finds its own place on circle 2r by a simple radial transfer (and vice
versa). Afterwards, from the circle 2r curve you can make two new
circles with radius r.
QED
True, but here you're partitioning the cirlce into infinitely many partitions, which is missing a key point.
Try again, but dividing the circle into only a finite number of pieces!
Regards,
Mike.
Which cannot possibly be recomposed into two identical spheres if every point of these original two pieces has been georeferenced.

If every point in the original sphere had not been georeferenced then the original sphere would have never been uniquely identified thus even talking about it would be incorrect.
Franz Gnaedinger
2018-08-21 06:55:02 UTC
Permalink
Copyright 2018 Pete Olcott
Peter Olcott knows the absolute and complete and total truth, he is the author
of life and creator of life, he has hundred reasons to assume that he is God,
he creates our future minds in order that we can go on existing, and he is the
one Creator of the Universe (claims he made in sci.lang and which entitle him
to start ever more threads). The self-declared author of life imposes a lifeless
language on us. He castrates language in the name of mathematical logic, and
mathematical logic by dismissing proven theorems. Goedel was wrong, Turing was
wrong, Allgod is right. He calls the understanding of Goedel's proven theorems
(a pleonasm I can't repeat often enough) a religious conviction. Meaning heresy.
He is Allgod and holds a copyright on the Truth. We must believe in Him.
Peter Percival
2018-08-21 12:40:12 UTC
Permalink
Post by peteolcott
We're not saying the sets in the partition are spherical. Their
union is a ball, and the union of their images under a bunch of
rotational isometries is two balls. The pieces are made using the
axiom of Choice, they are not Boreland sets.
Ah so so are trying to get away with fudging the meaning of the words.
If by ball you mean a physically existing object then we are not dealing
with points we are dealing with atoms. If you take away all of the atoms
and divide them into two balls of the same size and shape they now have
half as much mass.
Talking about points on a physically exiting ball is incongruous thus
incoherent. It is either atoms of a ball or points on a sphere. Either
way Banach–Tarski is simply a silly mistake.
Copyright 2018 Pete Olcott
No, I made no reference to physical reality at all, and there was no
fudging of the meanings of words. The Banach-Tarski paradox is
provable in ZFC. It says that there exists a partition of the solid
ball in 3-space into five subsets, four of which have infinitely
many points and which are not Borel sets. What you are saying is
rubbish.
You would have to point out an actual error of my reasoning for your
assertion that it is rubbish to have any basis what-so-ever.
So you're putting forward a claim to having engaged in reasoning?
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced
What do you mean by "georeferenced"? It is not a word I have come
across before.
Post by peteolcott
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.
Copyright 2018 Pete Olcott
peteolcott
2018-08-21 14:06:16 UTC
Permalink
Post by peteolcott
We're not saying the sets in the partition are spherical. Their union is a ball, and the union of their images under a bunch of rotational isometries is two balls. The pieces are made using the axiom of Choice, they are not Boreland sets.
Ah so so are trying to get away with fudging the meaning of the words.
If by ball you mean a physically existing object then we are not dealing
with points we are dealing with atoms. If you take away all of the atoms
and divide them into two balls of the same size and shape they now have
half as much mass.
Talking about points on a physically exiting ball is incongruous thus
incoherent. It is either atoms of a ball or points on a sphere. Either
way Banach–Tarski is simply a silly mistake.
Copyright 2018 Pete Olcott
No, I made no reference to physical reality at all, and there was no fudging of the meanings of words. The Banach-Tarski paradox is provable in ZFC. It says that there exists a partition of the solid ball in 3-space into five subsets, four of which
have infinitely many points and which are not Borel sets. What you are saying is rubbish.
You would have to point out an actual error of my reasoning for your
assertion that it is rubbish to have any basis what-so-ever.
So you're putting forward a claim to having engaged in reasoning?
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced
What do you mean by "georeferenced"?  It is not a word I have come across before.
I already explained that:

The center of a one inch diameter sphere is exactly ten miles above
the center of the north pole.

Without this degree of specificity the sphere is never uniquely
identified thus indiscernible from other spheres.

When we decompose this sphere into any parts and recompose them it is
impossible to create two spheres, thus Banach–Tarski has been fully refuted.

Banach–Tarski only exists because no one ever bothered to uniquely
identify the specific sphere in question, thus got confused and made a
copy of the sphere without realizing that it was only a copy and not
the original sphere at all.

The identity of otherwise indiscernibles is always uniquely established
if anchored in points in space and/or a point in time as appropriate.
In Banach–Tarski we anchor the sphere in points in space.

Copyright 2016, 2017, 2018 Pete Olcott
Post by peteolcott
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.
Copyright 2018 Pete Olcott
Peter Percival
2018-08-21 14:54:34 UTC
Permalink
Post by peteolcott
Post by peteolcott
We're not saying the sets in the partition are spherical. Their
union is a ball, and the union of their images under a bunch of
rotational isometries is two balls. The pieces are made using
the axiom of Choice, they are not Boreland sets.
Ah so so are trying to get away with fudging the meaning of the words.
If by ball you mean a physically existing object then we are not dealing
with points we are dealing with atoms. If you take away all of the atoms
and divide them into two balls of the same size and shape they now have
half as much mass.
Talking about points on a physically exiting ball is incongruous thus
incoherent. It is either atoms of a ball or points on a sphere. Either
way Banach–Tarski is simply a silly mistake.
Copyright 2018 Pete Olcott
No, I made no reference to physical reality at all, and there was
no fudging of the meanings of words. The Banach-Tarski paradox is
provable in ZFC. It says that there exists a partition of the
solid ball in 3-space into five subsets, four of which have
infinitely many points and which are not Borel sets. What you are
saying is rubbish.
You would have to point out an actual error of my reasoning for your
assertion that it is rubbish to have any basis what-so-ever.
So you're putting forward a claim to having engaged in reasoning?
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced
What do you mean by "georeferenced"?  It is not a word I have come across before.
The center of a one inch diameter sphere is exactly ten miles above
the center of the north pole.
These physical things have nothing to do with the Banach–Tarski paradox.
Do you even know what a sphere is to a mathematician? It's not a
rhetorical question, I'd really like to know if you know.
Post by peteolcott
Without this degree of specificity the sphere is never uniquely
identified thus indiscernible from other spheres.
When we decompose this sphere into any parts and recompose them it is
impossible to create two spheres, thus Banach–Tarski has been fully refuted.
Read Banach and Tarski's paper. Where is the first error?
Post by peteolcott
Banach–Tarski only exists because no one ever bothered to uniquely
identify the specific sphere in question, thus got confused and made a
Because there is no "the" sphere. The theorem is true of all
mathematical spheres.
Post by peteolcott
copy of the sphere without realizing that it was only a copy and not
the original sphere at all.
The identity of otherwise indiscernibles is always uniquely established
if anchored in points in space and/or a point in time as appropriate.
In Banach–Tarski we anchor the sphere in points in space.
Time and space are irrelevant.

You're a bit of an idiot, aren't you? And you have no sense of shame.
Peter Percival
2018-08-21 12:43:09 UTC
Permalink
Post by peteolcott
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is
What?

Also, are you posting to appropriate newsgroups?
Post by peteolcott
that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.
Copyright 2018 Pete Olcott
Peter Percival
2018-08-21 15:33:48 UTC
Permalink
Post by Peter Percival
Post by peteolcott
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is
What?
Also, are you posting to appropriate newsgroups?
Meant inappropriate.
Post by Peter Percival
Post by peteolcott
that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.
Copyright 2018 Pete Olcott
Loading...