Re: A consideration concerning the diagonal argument of G. Cantor
WM
> WM says...
>
> >Two infinite strings can be identical although their digits are
> >different.
>
> >Cantor's conclusion was: Every entry differs in at least one digit
> >from the AD. Therefore they cannot be equal. This conclusion is wrong.
> >In his 1892 paper (and later) Cantor did not consider cases like
> >1.000... = 0.111....
>
> Surely you know that this point is irrelevant to Cantor's conclusion.
> Why do you even bring it up?
I know that this point is very relevant to Cantor's conclusion. Every
modern textbook excludes replacement of 9 by 0 or so.
>
> Cantor's diagonal proof shows that there is no list containing every
> binary sequence. This *implies* that there is no list containing every
> real number.
Cantor's original diagonal proof does not hold for binary sequences of
digits.
>
> Let f(n) be a function from N to R.
> Let f(n)[m] be digit number m in the decimal representation of f(n).
> Let d be the real number whose decimal representation is given by:
>
> if f(n)[n] = 5, then d[n] = 4.
> Otherwise, d[n] = 5.
In binary representation we have 0.111... = 1.000...
>
> d has a decimal representation that is different from every
> real in the image of the function f. And d is guaranteed not
> to be a real that has multiple representations (since those
> reals always end with all 0s or 9s).
>
> You *know* that multiple representations of reals does not
> invalidate Cantor's conclusion. Why did you bring it up?
You are purporting a falsity which I dismiss herewith. In order to
obtain a Cantor-like proof for reals in binary representation, you
need to apply another replacement scheme, for instance that one first
proposed by Koenig.
But as the whole concept of actually existing infinite sets is self-
contradicting, this will not help either.
Regards, WM
Daryl McCullough
>
>On 3 Okt., 13:00, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
>> WM says...
>>
>> >Two infinite strings can be identical although their digits are
>> >different.
>>
>> >Cantor's conclusion was: Every entry differs in at least one digit
>> >from the AD. Therefore they cannot be equal. This conclusion is wrong.
>> >In his 1892 paper (and later) Cantor did not consider cases like
>> >1.000... = 0.111....
>>
>> Surely you know that this point is irrelevant to Cantor's conclusion.
>> Why do you even bring it up?
>
>I know that this point is very relevant to Cantor's conclusion. Every
>modern textbook excludes replacement of 9 by 0 or so.
Give an example of any treatment of Cantor's theorem in a textbook
that fails to take into account that multiple decimal representations
can represent the same real. There are no such treatements.
--
Daryl McCullough
Ithaca, NY
Virgil
<d533c941-4bef-4548...@i76g2000hsf.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 3 Okt., 13:00, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
> > WM says...
> >
> > >Two infinite strings can be identical although their digits are
> > >different.
> >
> > >Cantor's conclusion was: Every entry differs in at least one digit
> > >from the AD. Therefore they cannot be equal. This conclusion is wrong.
> > >In his 1892 paper (and later) Cantor did not consider cases like
> > >1.000... = 0.111....
> >
> > Surely you know that this point is irrelevant to Cantor's conclusion.
> > Why do you even bring it up?
>
> I know that this point is very relevant to Cantor's conclusion. Every
> modern textbook excludes replacement of 9 by 0 or so.
> >
> > Cantor's diagonal proof shows that there is no list containing every
> > binary sequence. This *implies* that there is no list containing every
> > real number.
>
> Cantor's original diagonal proof does not hold for binary sequences of
> digits.
It does in mathematics and in logic, regardless of how it goes in
physics.
> >
> > Let f(n) be a function from N to R.
> > Let f(n)[m] be digit number m in the decimal representation of f(n).
> > Let d be the real number whose decimal representation is given by:
> >
> > if f(n)[n] = 5, then d[n] = 4.
> > Otherwise, d[n] = 5.
>
> In binary representation we have 0.111... = 1.000...
Then use digit pairs and build the anti-diagonal effectively in base 4.
This has been pointed out before.
> >
> > d has a decimal representation that is different from every
> > real in the image of the function f. And d is guaranteed not
> > to be a real that has multiple representations (since those
> > reals always end with all 0s or 9s).
> >
> > You *know* that multiple representations of reals does not
> > invalidate Cantor's conclusion. Why did you bring it up?
>
> You are purporting a falsity which I dismiss herewith.
You are purporting a falsity which WE dismiss herewith.
> In order to
> obtain a Cantor-like proof for reals in binary representation, you
> need to apply another replacement scheme, for instance that one first
> proposed by Koenig.
The uncountability of the whole together with the countability of the
dual representations is sufficient.
Herbert Newman
>>>
>>> Cantor's diagonal proof shows that there is no list containing every
>>> binary sequence.
>>>
Right
>>>
>>> This *implies* that there is no list containing every real number.
>>>
Actually THIS claim was/is wrong. A list containing of binary sequences at
least one representation of any real number in ]0,1[ would be a
counterexample. (The fact that all alternative representations are missing
in this list is irrelevant.)
>>
>> Cantor's original diagonal proof does not hold for binary sequences of
>> digits. [WM]
>>
Sure it does. Actually it was FORMULATES in terms of such sequences! :-)
Well, Mückenheim.
>
> It does in mathematics and in logic, regardless of how it goes in
> physics.
>
Right.
>>
>> In binary representation we have 0.111... = 1.000... [WM]
>>
Right. So what?
>
> Then use digit pairs and build the anti-diagonal effectively in base 4.
> This has been pointed out before.
>
Right.
Well, it's Mückenheim, you know.
Herb
Ross A. Finlayson
It is well known that the antidiagonal argument doesn't apply in base
two or three, in terms of real numbers with dual representation if not
the binary coded powerset.
The equivalency function ranges from .000... to .111.... That is, any
function between the integers and unit interval of real numbers would
have that range.
Because there aren't "next" numbers in the normal ordering of the real
numbers in their standard construction, it is not consistent to have
that the natural integers in their natural total order preserve order in
their bijection with the unit interval of nonnegative real numbers in
their natural total order, in the Spinozan continuum of the natural
integers, but moreso in a continuum of integers that illustrates the
line among space-filling curves enumerated pathwise along the line.
Modern standard treatments of analysis integrate over the line. They
enumerate in infinitesimal differences functional description in linear
analytical methods, in the discrete and modeled by the continuous,
except for systems modeled in the discrete, chains and processes. The
transitions among and between those models, the continuous and discrete,
are governed by basic mathematical processes.
So, the equivalency function: N/U EF, the general natural/unit
equivalency function, is not a standard real function. Its range is on
the real numbers, in terms of sharing with them their geometric
properties in use throughout topology, using not the standard
definition. EF, that equivalency function, ranges from 0 to 1 over its
domain of input the natural integers. EF is a CDF, but not a standard
real function, (vis-a-vis standard real functions, and non-real
functions that admit discontinuities).
So, as a foundation for analysis, in the methods of analysis with the
integration and so on, infinitesimals, the integral calculus, the Cauchy
condition on sums and products and other process results in convergence
of solutions, behavior about the equivalency function is modeled by
other functions, much similarly to how the Dirac delta function is not a
real function, modeled by real functions. Similarly wavelet kernels of
other transforms have general primitive infinitesimal enumerators, in
transform analysis. In that way, recast as a model of a function of
mathematical properties of the real numbers, contingent on model
contingency, as a prototypical object (in the theories of standard real
analysis) it fulfills all its properties. It's either modeled as a
function in real analysis or it's not.
So, that the list written in binary or base 3 does not have the
antidiagonal argument apply to those objects with dual representation,
in this case expansions that are paths in the tree of expansions, each
from some node onwards absolutely different from any other of the paths,
from its origin, or for any algebraic and then rational number, a finite
sequence repeated infinitely many times, where the path branches are
nodes, in the infinite balanced binary tree.
Defining the rule to define the difference of two list items among
distinguishability relations vis-a-vis all the other list elements,
including those not on the list, i.e. identified by a predicate, in
continuum analysis, greatly benefits from a linear continuum.
Regards,
Ross F.
Virgil
Herbert Newman <nomail@invalid> wrote:
> Am Sat, 04 Oct 2008 11:36:52 -0600 schrieb Virgil:
>
> >>>
> >>> Cantor's diagonal proof shows that there is no list containing every
> >>> binary sequence.
> >>>
> Right
>
> >>>
> >>> This *implies* that there is no list containing every real number.
> >>>
> Actually THIS claim was/is wrong. A list containing of binary sequences at
> least one representation of any real number in ]0,1[ would be a
> counterexample. (The fact that all alternative representations are missing
> in this list is irrelevant.)
The implication is not immediate, as one would then have to prove also
that a countable set of points having dual representations does not
affect the countability of the real line, which it does not.
But the hard part is over.
Virgil
"Ross A. Finlayson" <r...@tiki-lounge.com.invalid> wrote:
> It is well known that the antidiagonal argument doesn't apply in base
> two or three, in terms of real numbers with dual representation if not
> the binary coded powerset.
The Cantor proof applies to both binary and trinary infinite strings,
and given that the number of dually representable numbers is countable,
we can make a new list including both reps for each real with two of
them in addition to its original contents, and the anti-diagonal for the
new list will differ from all of them, including both reps of every real
with two of them.
So much for that.
Ross A. Finlayson
No, you can't say the antidiagonal isn't on the list, because of dual
representation and the lack of how many antidiagonal rules there can be.
Basically in these low numeric bases (and base ifinity) the rules aren't
given that to apply a list element in its n'th place only in terms of
n'th digit in the expansion, because the antidiagonal in that base is
constructed from those elements, different from the n'th term of the
n'th list item. There isn't enough variation to be different from the
diagonal term in the antidiagonal term. Basically you can't list all
the rules.
Considering for example the iterations of a convergent sequence, in
defining recurrent products, that the convergent product isn't the same
as infinite product, is embedded in the notation, where generally it's
convenient to consider that it actually is.
Regards,
Ross F.
Virgil
"Ross A. Finlayson" <r...@tiki-lounge.com.invalid> wrote:
> Virgil wrote:
> > In article <gc8cnk$jjd$1...@aioe.org>,
> > "Ross A. Finlayson" <r...@tiki-lounge.com.invalid> wrote:
> >
> >
> >> It is well known that the antidiagonal argument doesn't apply in base
> >> two or three, in terms of real numbers with dual representation if not
> >> the binary coded powerset.
> >
> > The Cantor proof applies to both binary and trinary infinite strings,
> > and given that the number of dually representable numbers is countable,
> > we can make a new list including both reps for each real with two of
> > them in addition to its original contents, and the anti-diagonal for the
> > new list will differ from all of them, including both reps of every real
> > with two of them.
> >
> > So much for that.
>
> No, you can't say the antidiagonal isn't on the list, because of dual
> representation and the lack of how many antidiagonal rules there can be.
I can if the list is revamped to include all non-unique representations
(NURs).
There are only countably many NURs, since all such represent rationals
which are themselves countable.
Given any list of representations we may create a new list with all the
originals and ALL NURs. And anti-diagonals to this new list cannot be
either. Specifically, they cannot be NURs.
Newberry
I think you folks should set forth the parameters of your discussion.
Are we trying to prove ZF inconsistent, or are we saying that the
diagonal argumnt is implausible/counterintuitive?
Ross A. Finlayson
Hello,
About the antidiagonal argument in base 2 and 3, it's not an "diagonal"
argument any more in that "I can construct from the diagonal and a rule
an item not on the list" it is a "maintaining forward indices over list
items, a rule can be given and given that segment of the expansion of an
item, I can construct an item not on the list, defining the . It's not
the diagonal of the expansion, as a list of items, as a matrix of items,
the matrix diagonal, it's not a diagonal argument anymore.
So, there's no antidiagonal argument in base 2 or 3, it's not a diagonal
argument. Similarly it has long been said that the base 10 antidiagonal
argument (diagonal argument) needs repair. Instead of just noticing
it's not a normal argument, i.e. that works in any base, it's also not
strictly an antidiagonal argument any more. To organize it into the
matrix structure from base 2, you're showing that that it can be
deconstructed to that base two structure, it's not the antidiagonal
argument any more, with the dual representation on the list, rather the
rule's structure as the radices vary in the expansion for the
reversibility in the maintenance in the structure of the decomposition
coefficients, in the mutual structure of any separation. That also
leads to exclusion results.
So, there is to be the decomposition of the finite combinatorics case.
Consider something like 2^N, which written here would generally means
the powerset of the integers, (except) where N is finite. Then there is
N!, N factorial, the factorial product of the factors, in this case the
factors being the n-set {1, ..., N}. Then, 2^N is the number of subsets
of a set, N! the number of combinations, those are very particular
values in finite combinatorics, while in cardinal analysis of the
trans-finite there is no meaningful difference between them in
arithmetic derived across those metrics across the trans-finite.
N! = (N * N-1 * ... * 1) = (1 * 2 * ... * N)
2^N = (2 * 2 * ... * 2) N many times
N^2 = N squared
In finite combinatorics, there really is accurate analysis in the
concrete mathematics with the cycle and subset numbers with the
permutation groups and so on. There are the Pochhammer conventions with
the rising and falling indices in the generation of well-known numerical
series.
So, as the complexity of the list in its structure increases and
decreases, the means of describing a function in the least particular
form for a particular list for a given structure, giving basically a
function to return that function given the structure of the list, where
the form is applied so provided, the structural form of the function in
the _later_ use of the function to demonstrate the evidence, that
proviso is as well proviso of the best search program of a list for an
element not in the list, in search probability algorithms. That is
where, the list structure is formed so that particular rules are used in
the initialization of objects so that the fall along the hash lines that
determine the addressing. In making the list's structure defining its
contents and given some particular field of the input, the rule maker
wants to fashion the best rule that will most correctly and quickly show
for an item that it's not on the list.
That is about, making a list, and even modifying the list contents, with
the list maker having a rule to translate list entries in their
representation given the list offset. This is in finite combinatorics,
where it's a finite width expansion, in the finite length list, on
computers generally fixed width binary strings for each item of the list
which is addressible. Then, the idea is to maintain a rule for an
antihash so that in terms of computing an antidiagonal for the list, the
antihash basically maintains among the categories the range of the
values that aren't each and aren't every item in the list (container, in
the finite). Then, in computing a rule, the antidiagonal starts large
and diminishes as there are results, and then there are positive and
negative keys for the membership applications. It's posible to maintain
the finite ranges in an antidiagonal rule, where in the infinite case,
the processes to determine all the elements in the range not in the
list, of the process of the generation of a function that is an
"antidiagonal" rule, in the infinite case have that the ranges are
mostly infinite.
Here the idea is to encode as much information as possible, given the
structure of an object, the structure of a container of those objects,
assigning each object a number or label, and the objects by their label
to the negative search function, find, encode as much information as
possible in the shortest possible "antidiagonal" function, that given
access to various features of the structure, indicates an item's
presence or absence in the list. The idea is basically to look at the
considered process in the finite to determine what primitives among the
infinite have these particular behaviors, when in the infinite there is
a logic in the reverse that accords the proviso of an inexhaustible
source of examples, in the finite those are combinatorially enumerated.
Say for example it's an list with a thousand entries, of two bit codes.
In building the implementation of the body of the function to generate
a (two-bit) item not on that list, where the particular implementation
that generates an antidiagonal is the antidiagonal function which has as
parameters the n'th element of an expansion as string representation for
the n'th list in the enumeration of the list labels. The binary
antidiagonal is defined quite directly in terms of algorithmic
primitives in coordinate matrix access.
That's getting besides the point that in the general finite lists of
finite width expansions and also generally in terms of enumerable
structure, the particular structure of the "antidiagonal" from the
"anti-diagonal argument in base 10 with the carefulness about the
.999... = 1.000..." is actually seen as only a single example of a
family of functions, computable functions with applications, that given
as parameters the structure of the objects and list (and thus objects
and list) generates the range of items unmapped by the function from the
union of the inputs to the support space of the function. In the
finite there is much more process in the structure of the implementation
of a nonmembership function, that returns a subset of the output range
that is not in the container.
So, the antidiagonal argument is actually the proviso of a nonmembership
function. In the trans-finite, because the structure of the list
elements as expansions through all the finite has those particular
structures of those expansions in terms of there thus being a matrix or
set of well-ordered pairs from NxN and N_b, actually trios NxN, N_b, and
as well n for the antidiagonal rule to construct the expansion in that
place of the antidiagonal element's expansion, or the contract that it
is called in order of evaluation of the list elements. (An antidiagonal
function in the infinite might definitely be irrespective of evaluation
order, exhaustively in the finite evaluation might vary from insertion
order, in block matrix structure of the container).
Regards,
Ross F.
WM
That is just what I said. The only treatment that failed to consider
that point was Cantor's own of 1892.
Regrads, WM
Virgil
"Ross A. Finlayson" <r...@tiki-lounge.com.invalid> wrote:
> About the antidiagonal argument in base 2 and 3, it's not an "diagonal"
> argument any more in that "I can construct from the diagonal and a rule
> an item not on the list" it is a "maintaining forward indices over list
> items, a rule can be given and given that segment of the expansion of an
> item, I can construct an item not on the list, defining the . It's not
> the diagonal of the expansion, as a list of items, as a matrix of items,
> the matrix diagonal, it's not a diagonal argument anymore.
No one claims that the "anti-diagonal" works with "diagonals".
>
> So, there's no antidiagonal argument in base 2 or 3, it's not a diagonal
> argument.
Given any list of nontrivial infinite strings (mapping from N to some
set of at least two characters) even with the choice of characters
limited to less than 4, one can still have a rule which produces for any
such list a string of those same characters which is not listed in that
list, even if those strings are interpreted as numerals to some fixed
base
Anything else is irrelevant to Cantor's so-called "diagonal" proof.
And not all Ross' psychedelic poetizing prohibits that proof.
Virgil
<546abbb7-eb8c-4fb3...@l76g2000hse.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
And as Cantor was not talking about numerals (representations of
numbers), the "point" was totally irrelevant then.
MoeBlee
> > But as the whole concept of actually existing infinite sets is self-
> > contradicting, this will not help either.
>
> > Regards, WM
>
> I think you folks should set forth the parameters of your discussion.
> Are we trying to prove ZF inconsistent, or are we saying that the
> diagonal argumnt is implausible/counterintuitive?
With WM? Good luck!
MoeBlee
Herbert Newman
>>
>> Give an example of any treatment of Cantor's theorem in a textbook
>> that fails to take into account that multiple decimal representations
>> can represent the same real. There are no such treatments.
>>
> That is just what I said. The only treatment that failed to consider
> that point was Cantor's own of 1892.
>
Huh?! Since he didn't consider lists of _real numbers_ in his paper from
1891 (!) why on earth SHOULD he consider that point there?
He writes:
"In the paper entitled 'On a property of a set [Inbegriff] of all real
algebraic numbers' (Journ. Math. Bd. 77, S. 258), there appeared, probably
for the first time, a proof of the proposition that there is an infinite
manifold, which cannot be put into a one-one correlation with the totality
[Gesamtheit] of all finite whole numbers 1, 2, 3, …, n, …, or, as I am used
to saying, which do not have the power (Mächtigkeit) if the number series
1, 2, 3, …, n, …. [...]
However, there is a proof of this proposition that is much simpler, and
which does not depend on considering the irrational numbers."
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
And he proceeds:
"Namely, let m and w be two different characters, and consider a set
[Inbegriff] M of elements
E = (x_1, x_2, … , x_n, …)
which depend on infinitely many coordinates x_1, x_2, … , x_n, …, and where
each of the coordinates is either m or w. Let M be the totality
[Gesamtheit] of all elements E.
To the elements of M belong e.g. the following three:
E^I = (m, m, m, m, … ),
E^II = (w, w, w, w, … ),
E^III = (m, w, m, w, … ).
I maintain now that such a manifold [Mannigfaltigkeit] M does not have the
power of the series 1, 2, 3, …, n, …."
Finally he formulates his famous proof (based on the diagonal argument).
Herb
Herbert Newman
>>>
>>> But as the whole concept of actually existing infinite sets is self-
>>> contradicting, this will not help either. [WM]
>>>
>> I think you folks should set forth the parameters of your discussion.
>> Are we trying to prove ZF inconsistent, or are we saying that the
>> diagonal argument is implausible/counterintuitive?
>>
Well, *we* (i.e. non-cranks) certainly don't say anything like that.
>
> With WM? Good luck!
>
:-)
Herb
herbzet
Yes, good.
Here's a link to the paper and an english translation
of "Uber ein elementare Frage der Mannigfaltigkeitslehre"
("On an Elementary Question of Set Theory")
http://uk.geocities.com/fr...@btinternet.com/cantor/diagarg.htm
--
hz
Dik T. Winter
> On 4 Okt., 14:12, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
> > >I know that this point is very relevant to Cantor's conclusion. Every
> > >modern textbook excludes replacement of 9 by 0 or so.
> >
> > Give an example of any treatment of Cantor's theorem in a textbook
> > that fails to take into account that multiple decimal representations
> > can represent the same real. There are no such treatements.
>
> That is just what I said. The only treatment that failed to consider
> that point was Cantor's own of 1892.
In that paper 9 and 0 do not occur, so it is irrelevant. The paper
was *not* about numbers that the sequences could possibly represent.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Dik T. Winter
> On Sun, 5 Oct 2008 13:47:03 -0700 (PDT) WM wrote:
> > That is just what I said. The only treatment that failed to consider
> > that point was Cantor's own of 1892.
> "In the paper entitled 'On a property of a set [Inbegriff] of all real
> algebraic numbers' (Journ. Math. Bd. 77, S. 258), there appeared, probably
> for the first time, a proof of the proposition that there is an infinite
> manifold, which cannot be put into a one-one correlation with the totality
> used to saying, which do not have the power (Mächtigkeit) if the number
>
> However, there is a proof of this proposition that is much simpler, and
> which does not depend on considering the irrational numbers."
> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
You will not convince WM with this because he argues that the words "this
proposition" above refers to the precise proposition proven in the other
paper, not the proposition that is actually stated just above that
sentence.
WM
> In article
> <546abbb7-eb8c-4fb3-a394-b3e576c59...@l76g2000hse.googlegroups.com>,
He talked about real numbers and intervals of real numbers in that
paper.
Regards, WM
WM
The diagonal argument is false, aleph_0 and with it ZF is
inconsistent.
Regards, WM
Herbert Newman
>>
>> And as Cantor was not talking about numerals (representations of
>> numbers), the "point" was totally irrelevant then.
>>
> He talked about real numbers and intervals of real numbers in that
> paper. [WM]
>
No. He just briefly mentioned them while referring to an earlier paper.
Actually, he wrote (in the paper under discussion):
"In the paper entitled 'On a property of a set [Inbegriff] of all real
algebraic numbers' (Journ. Math. Bd. 77, S. 258), there appeared, probably
for the first time, a proof of the proposition that there is an infinite
manifold, which cannot be put into a one-one correlation with the totality
[Gesamtheit] of all finite whole numbers 1, 2, 3, ..., n, ..., or, as I am
used to saying, which do not have the power (Mächtigkeit) if the number
series 1, 2, 3, ..., n, .... [...]
However, there is a proof of this proposition that is much simpler, and
which does not depend on considering the irrational numbers."
Note that the tile of the paper is
"Über ein elementare Frage der Mannigfaltigkeitslehre"
Herb
Herbert Newman
>
> You will not convince WM with this because he argues that the words "this
> proposition" ["jenem Satze"] above refers to the precise proposition proven
> in the other paper, not the proposition that is actually stated just above
> that sentence.
>
Yes, I know. Though this just shows that WM does not even understand German
language. Cantor uses the term "jenem" (instead of "diesem") - which (in
German) indicates that he's referring to the proposition
"...that there is an infinite manifold, which cannot be put into a one-one
correlation with the totality [Gesamtheit] of all finite whole numbers 1,
2, 3, ..., n, ..., or, as I am used to saying, which do not have the power
(Mächtigkeit) if the number series 1, 2, 3, ..., n, ... ."
And even if in doubt this SHOULD be clear by the fact that Cantor proceeds
with:
"Namely, let m and w be two different characters, and consider a set
[Inbegriff] M of elements
E = (x_1, x_2, ..., x_n, ...)
which depend on infinitely many coordinates x_1, x_2, ... , x_n, ..., and
where each of the coordinates is either m or w. Let M be the totality
[Gesamtheit] of all elements E.
To the elements of M belong e.g. the following three:
E^I = (m, m, m, m, ...),
E^II = (w, w, w, w, ...),
E^III = (m, w, m, w, ...).
I maintain now that such a manifold [Mannigfaltigkeit] M does not have the
power of the series 1, 2, 3, ..., n, ... ."
followed by the actual proof.
So it's indeed CLEAR (to any non-crank that is) that Cantor is referring to
the mentioned proposition when claiming:
"However, there is a proof of this proposition that is much simpler, and
which does not depend on considering the irrational numbers."
Herb
Virgil
<a12831ff-ca0f-4018...@y29g2000hsf.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
But not in that argument. And once the existence of uncountable sets was
established, as it was there, it is relatively trivial to extend that
proof to the set of reals.
>
> Regards, WM
Virgil
<681a3c5d-bef7-4882...@x41g2000hsb.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
These are mere presumptions, and WM has not been able to prove them to
the satisfaction of the referees of any mathematical journal.