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Matheology § 014
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WM
May 21, 2012, 2:51:38 AM5/21/12
to
On Jan. 31, 2012 I answered in MatheOverflow a question concerning the
infinite Binary Tree:
The set of all finite paths (from the root-node to any other node)
in the complete infinite Binary Tree is countable. Therefore the
complete infinite Binary Tree has countably many paths that can be
identified by nodes.
It is impossible to identify an infinite path by nodes, because
1) every node belongs to a finite path, and
2) there is no identification unless it has been finished.
Therefore an infinite path can only be identified by a finite
expression like "always turn left", or "0.111...", or "the path which
represents 1/pi", or simply "1/3". However, the set of finite
expressions has countable cardinality.
Therefore the set of all paths in the complete infinite Binary Tree
has countable cardinality.
Regards, WM
For a real number r > 1 wouldn´t you consider "the path which
represents 1/r" to be a path? Aren´t there uncountably many real
numbers r > 1? – R DE LA VEGA
For a real number r > 1, that can be defined by a finite expression, 1/
r represents a path. But there are not more than countably many finite
expressions. – WM
Further questions were not admitted. My answer had been deleted.
Regards, WM
William Elliot
May 21, 2012, 4:03:17 AM5/21/12
to
On Sun, 20 May 2012, WM wrote:
> The set of all finite paths (from the root-node to any other node)
> in the complete infinite Binary Tree is countable. Therefore the
> complete infinite Binary Tree has countably many paths that can be
> identified by nodes.
> It is impossible to identify an infinite path by nodes, because
> 1) every node belongs to a finite path, and
Every node is within infinitely many finite paths.
> The set of all finite paths (from the root-node to any other node)
> in the complete infinite Binary Tree is countable. Therefore the
> complete infinite Binary Tree has countably many paths that can be
> identified by nodes.
> It is impossible to identify an infinite path by nodes, because
> 1) every node belongs to a finite path, and
> 2) there is no identification unless it has been finished.
sequence.
If you always wait to get to the end of an infinite sequence,
you'll accomplish nothing.
> Therefore an infinite path can only be identified by a finite
> expression like "always turn left", or "0.111...", or "the path which
> represents 1/pi", or simply "1/3". However, the set of finite
> expressions has countable cardinality.
as a finite statement.
> Therefore the set of all paths in the complete infinite Binary Tree
> has countable cardinality.
Almost all infinite paths aren't computable.
Apparently you're taking a constructionist view.
From that limited view, and only from that limited
view, you notions hold; your notions hold to a
minority of mathematicians.
On the other hand, insisting to get to the end a sequence
indicates you hold a finistic view of mathematics - a view
held by a tiny minority of mathematicians.
----
WM
May 21, 2012, 7:49:59 AM5/21/12
to
On 21 Mai, 10:03, William Elliot <ma...@panix.com> wrote:
> On Sun, 20 May 2012, WM wrote:
> > The set of all finite paths (from the root-node to any other node)
> > in the complete infinite Binary Tree is countable. Therefore the
> > complete infinite Binary Tree has countably many paths that can be
> > identified by nodes.
> > It is impossible to identify an infinite path by nodes, because
> > 1) every node belongs to a finite path, and
>
> Every node is within infinitely many finite paths.
>
> > 2) there is no identification unless it has been finished.
>
> Every path in the binary can be described by a infinite binary
> sequence.
Please describe only one path by an infinite binary sequence.
> On Sun, 20 May 2012, WM wrote:
> > The set of all finite paths (from the root-node to any other node)
> > in the complete infinite Binary Tree is countable. Therefore the
> > complete infinite Binary Tree has countably many paths that can be
> > identified by nodes.
> > It is impossible to identify an infinite path by nodes, because
> > 1) every node belongs to a finite path, and
>
> Every node is within infinitely many finite paths.
>
> > 2) there is no identification unless it has been finished.
>
> Every path in the binary can be described by a infinite binary
> sequence.
>
> If you always wait to get to the end of an infinite sequence,
> you'll accomplish nothing.
It is impossible to send or understand messages without end.
> If you always wait to get to the end of an infinite sequence,
> you'll accomplish nothing.
>
> > Therefore an infinite path can only be identified by a finite
> > expression like "always turn left", or "0.111...", or "the path which
> > represents 1/pi", or simply "1/3". However, the set of finite
> > expressions has countable cardinality.
>
> Some infinite paths are computable, expressible in some formal language,
> as a finite statement.
definition.
>
> > Therefore the set of all paths in the complete infinite Binary Tree
> > has countable cardinality.
>
> The set of all computable infinite paths is countable.
> Almost all infinite paths aren't computable.
Better say, they do not exist. They are of no value for mathematics,
> > Therefore the set of all paths in the complete infinite Binary Tree
> > has countable cardinality.
>
> The set of all computable infinite paths is countable.
> Almost all infinite paths aren't computable.
because nobody could use them. They cannot result from a Cantor list
and they cannot be useful for any mathematical problem.
To insist on unknowable and unusable entities is matheology.
>
> Apparently you're taking a constructionist view.
> From that limited view, and only from that limited
> view, you notions hold; your notions hold to a
> minority of mathematicians.
No. I avoid believing in items that cannot be part of mathematics
> Apparently you're taking a constructionist view.
> From that limited view, and only from that limited
> view, you notions hold; your notions hold to a
> minority of mathematicians.
(that is a language that serves for communication and computation).
>
> On the other hand, insisting to get to the end a sequence
> indicates you hold a finistic view of mathematics - a view
> held by a tiny minority of mathematicians.
Nobody can get to the end of something that has no end.
> On the other hand, insisting to get to the end a sequence
> indicates you hold a finistic view of mathematics - a view
> held by a tiny minority of mathematicians.
Therefore everything that can appear in mathematics need to have a
finite definition.
You (and anybody else either) are not capable of naming, defining,
using one of your asserted uncomputable real numbers. Further they are
not useful to mainain Cantor's view, because every diagonal number
constructed from a Cantor-list is computable. A number does not exist
by itself. A number is a definition. Why do you believe in undefinable
definitions?
Regards, WM
WM
May 21, 2012, 11:49:52 AM5/21/12
to
> Apparently you're taking a constructionist view.
> From that limited view,
What is the advantage of uncomputable "numbers" that blows up
> From that limited view,
mylimited view? Where can you use them (except as the set of
uncomputable numbers). Can you well-order them? Can you prove that
there is not a second set of unthinkable numbers that has
supercardinality and does nothing but contradict its own existence?
Regards, WM
MoeBlee
May 21, 2012, 12:31:01 PM5/21/12
to
On May 21, 10:49 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> What is the advantage of uncomputable "numbers"
The ease, as far as formal axiomatics, of classical analysis (e.g.
> What is the advantage of uncomputable "numbers"
axiomatized by Z\R+DC - cf. Moschovakis 'Notes On Set Theory') has
been explained to you before. If you wish to eschew formal axiomatics,
then that's your choice to make.
MoeBlee
WM
May 21, 2012, 12:44:37 PM5/21/12
to
On 21 Mai, 18:31, MoeBlee <modem...@gmail.com> wrote:
> On May 21, 10:49 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > What is the advantage of uncomputable "numbers"
>
> The ease, as far as formal axiomatics, of classical analysis (e.g.
> axiomatized by Z\R+DC - cf. Moschovakis 'Notes On Set Theory') has
> been explained to you before.
As those numbers under no circumstances can appear in any mathematical
> On May 21, 10:49 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > What is the advantage of uncomputable "numbers"
>
> The ease, as far as formal axiomatics, of classical analysis (e.g.
> axiomatized by Z\R+DC - cf. Moschovakis 'Notes On Set Theory') has
> been explained to you before.
framework as individuals, they cannot serve any meaningful purpose.
Neither as limits nor as results of any computation. Just matheology.
Numbers like angels may help to trust in whatever.
Regards, WM
MoeBlee
May 21, 2012, 7:32:49 PM5/21/12
to
On May 21, 11:44 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> As those numbers under no circumstances can appear in any mathematical
> framework as individuals, they cannot serve any meaningful purpose.
It's not the use of the particular numbers themselves that is
> As those numbers under no circumstances can appear in any mathematical
> framework as individuals, they cannot serve any meaningful purpose.
important in the regard I mention. Rather, it's that we have a
comparatively facile axiomatization that provides for a calculus for
the sciences. One may say that among the "byproducts" of this
axiomatization is the theorem that there are uncountably many real
numbers bu that, in the context to which I refer, that theorem is not
itself the objective or claimed to be of use in the sciences; rather
it's that to exclude such theorems (as the uncountability of the
reals) from accruing from the axioms requires, as far as I know, one
or another more intricate axiomatizations.
Such considerations have been explained to you numerous times over
years, but in your dogmatic quest, you continue to deny even
countenancing them.
MoeBlee
George Greene
May 21, 2012, 7:52:19 PM5/21/12
to
On May 21, 2:51 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> On Jan. 31, 2012 I answered in MatheOverflow a question concerning the
> infinite Binary Tree:
> The set of all finite paths (from the root-node to any other node)
> in the complete infinite Binary Tree is countable.
Not only that, the set of all finite paths covers ALL NODES in the
> On Jan. 31, 2012 I answered in MatheOverflow a question concerning the
> infinite Binary Tree:
> The set of all finite paths (from the root-node to any other node)
> in the complete infinite Binary Tree is countable.
infinite binary tree.
EVERY NODE IN THE TREE is at the end OF SOME FINITE path.
By your logic, this would mean that there could be no infinite paths
AT ALL,
not merely that there could only be countably many of them. There are
no nodes left
"to participate in" the infinite paths -- they've all ALREADY BEEN
USED UP BY the finite paths.
> Therefore the complete infinite Binary Tree has countably many paths that can be
> identified by nodes.
countably infinitely many nodes,
since THERE IS A TRIVIAL BIJECTION BETWEEN finite paths and nodes
(every path is mapped to the node where it ends, and every node is
mapped to the path that ends at it).
> It is impossible to identify an infinite path by nodes, because
> 1) every node belongs to a finite path, and
> 2) there is no identification unless it has been finished.
OBVIOUSLY it not only can be "identified" by the infinite string of
nodes, it arguably IS that infinite string.
> Therefore an infinite path can only be identified by a finite
> expression like "always turn left", or "0.111...", or "the path which
> represents 1/pi", or simply "1/3". However, the set of finite
> expressions has countable cardinality.
countable (because of the finitude of the descriptions),
BUT IT IS ALSO DIAGONALIZABLE -- FINITARILY. It is trivial to prove
that this class of infinite paths CANNOT POSSIBLY BE *all* the
infinite paths there are.
George Greene
May 21, 2012, 7:53:10 PM5/21/12
to
On May 21, 11:49 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> What is the advantage of uncomputable "numbers"
They DON'T NEED to have ANY advantages in order TO EXIST.
> What is the advantage of uncomputable "numbers"
WHICH THEY PROVABLY do.
WM
May 22, 2012, 5:24:55 AM5/22/12
to
conclude from:
A finite definition can convey an infinite string
to
An infinite string can convey a finite definition.
Learn logic. A ==> B cannot be reversed to B ==> A.
Regards, WM
WM
May 22, 2012, 5:21:34 AM5/22/12
to
On 22 Mai, 01:32, MoeBlee <modem...@gmail.com> wrote:
> On May 21, 11:44 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > As those numbers under no circumstances can appear in any mathematical
> > framework as individuals, they cannot serve any meaningful purpose.
>
> It's not the use of the particular numbers themselves that is
> important in the regard I mention.
Really? In fact? A number that cannot be used is not important in
> On May 21, 11:44 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > As those numbers under no circumstances can appear in any mathematical
> > framework as individuals, they cannot serve any meaningful purpose.
>
> It's not the use of the particular numbers themselves that is
> important in the regard I mention.
matheology? Attention! You are in danger to leave Cantor's paradies
and to become a realist.
> Rather, it's that we have a
> comparatively facile axiomatization that provides for a calculus for
> the sciences.
sciences! Astrology is more important for sciences than Cantor's
humbug.
> One may say that among the "byproducts" of this
> axiomatization is the theorem that there are uncountably many real
they are not real and not numbers. Yes. Are you really incapable of
seeing what a nonsense you proclaim?
> in the context to which I refer, that theorem is not
> itself the objective or claimed to be of use in the sciences; rather
> it's that to exclude such theorems (as the uncountability of the
> reals) from accruing from the axioms requires, as far as I know, one
> or another more intricate axiomatizations.
like mathematics, because they cannot convey information. Only finite
definitions of finite or infinite strings belong to mathematics.
Therefore the Cantor diagonal is not a mathematical object unless it
can be defined by a finite string. But then it belongs to a countably
set.
Regards, WM
WM
May 22, 2012, 5:13:56 AM5/22/12
to
On 22 Mai, 01:52, George Greene <gree...@email.unc.edu> wrote:
>
> > It is impossible to identify an infinite path by nodes, because
> > 1) every node belongs to a finite path, and
> > 2) there is no identification unless it has been finished.
>
> This is just bullshit. If the infinite path CAN EXIST AT ALL then
> OBVIOUSLY it not only can be "identified" by the infinite string of
> nodes, it arguably IS that infinite string.
If the infinite string has no chance to become a part of mathematics,
>
> > It is impossible to identify an infinite path by nodes, because
> > 1) every node belongs to a finite path, and
> > 2) there is no identification unless it has been finished.
>
> This is just bullshit. If the infinite path CAN EXIST AT ALL then
> OBVIOUSLY it not only can be "identified" by the infinite string of
> nodes, it arguably IS that infinite string.
i.e., if it cannot be identified, addressed, used at all, then, may it
exist somewhere or not: It does not belong to mathematics. Mathematics
is a language to be written and to be understood. Infinite strings do
not belong to this language (unless they have a finite definition).
>
> > Therefore an infinite path can only be identified by a finite
> > expression like "always turn left", or "0.111...", or "the path which
> > represents 1/pi", or simply "1/3". However, the set of finite
> > expressions has countable cardinality.
>
> Well, sure, the set of "infinite paths with finite descriptions" is
> countable (because of the finitude of the descriptions),
> BUT IT IS ALSO DIAGONALIZABLE -- FINITARILY. It is trivial to prove
> that this class of infinite paths CANNOT POSSIBLY BE *all* the
> infinite paths there are.
Diagonalization produces an infinite string. But infinite strings have
> > Therefore an infinite path can only be identified by a finite
> > expression like "always turn left", or "0.111...", or "the path which
> > represents 1/pi", or simply "1/3". However, the set of finite
> > expressions has countable cardinality.
>
> Well, sure, the set of "infinite paths with finite descriptions" is
> countable (because of the finitude of the descriptions),
> BUT IT IS ALSO DIAGONALIZABLE -- FINITARILY. It is trivial to prove
> that this class of infinite paths CANNOT POSSIBLY BE *all* the
> infinite paths there are.
no meaning in mathematics and, therefore, do not belong to
mathematics, unless they can be defined by finite definitions.
Therefore diagonalization does not produce mathematical objects,
unless the Cantor-list has been completely defined by a finite
definition. But then also the diagonal is defined by a finite
definition and does belong to a finite set. It is horrible to see what
you guys worship this completely nonsensical idea of diagonalization.
Either the result has a finite definition - or it does not belong to
mathematics.
Regards, WM
MoeBlee
May 22, 2012, 4:48:34 PM5/22/12
to
On May 22, 4:21 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 22 Mai, 01:32, MoeBlee <modem...@gmail.com> wrote:
>
> > On May 21, 11:44 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > As those numbers under no circumstances can appear in any mathematical
> > > framework as individuals, they cannot serve any meaningful purpose.
>
> > It's not the use of the particular numbers themselves that is
> > important in the regard I mention.
>
> Really? In fact? A number that cannot be used is not important in
> matheology?
I said "IN THE REGARD I MENTIONED". And please read the REST OF MY
> On 22 Mai, 01:32, MoeBlee <modem...@gmail.com> wrote:
>
> > On May 21, 11:44 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > As those numbers under no circumstances can appear in any mathematical
> > > framework as individuals, they cannot serve any meaningful purpose.
>
> > It's not the use of the particular numbers themselves that is
> > important in the regard I mention.
>
> Really? In fact? A number that cannot be used is not important in
> matheology?
POST. It's clear you're just picking at each line in my post before
even reading it to see what I'm driving at.
> > Rather, it's that we have a
> > comparatively facile axiomatization that provides for a calculus for
> > the sciences.
>
> It is ridiculous to claim that matheology had anything to do with
> sciences! Astrology is more important for sciences than Cantor's
> humbug.
before even considering the point I'm making.
> > One may say that among the "byproducts" of this
> > axiomatization is the theorem that there are uncountably many real
> > numbers but that,
>
> they are not real and not numbers. Yes. Are you really incapable of
> seeing what a nonsense you proclaim?
is even such realist or physical existence of sense even in claiming
the existence of such objects, set theory happen to give certain
theorems that we read off as "There exist uncountably many reals". It
is to such theorems that I refer. My point is that even if one finds
such theorems undesirable, nonsense, or anethema, they are by product
of an axiomatization that does provide for othe theorems that are used
in the calculus for the sciences.
> > in the context to which I refer, that theorem is not
> > itself the objective or claimed to be of use in the sciences; rather
> > it's that to exclude such theorems (as the uncountability of the
> > reals) from accruing from the axioms requires, as far as I know, one
> > or another more intricate axiomatizations.
>
> Things are very simple. Infinite strings don't belong to a language
> like mathematics, because they cannot convey information. Only finite
> definitions of finite or infinite strings belong to mathematics.
> Therefore the Cantor diagonal is not a mathematical object unless it
> can be defined by a finite string. But then it belongs to a countably
> set.
I'm talking about, or to see the point I'm making, but rather you saw
my remarks as just another opportunity for your polemical
denouncements.
MoeBlee
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