Matheology § 258

[WM]

Matheology § 258

So what about Cantor’s much celebrated non-denumerable real? Where is
it? Did Cantor produce such a real number? No, he merely sketched out
the logic for a nonterminal procedure that would produce an infinitely
long digit string representing a real number that would not be in the
input stream of enumerated reals. Cantor’s procedure, and with it his
celebrated nondenumerable, infinitely long real number, will appear
with 100% certainty in the denumerable list of procedures. {{That's
the point: Every diagonal number can be distinguished at a finite
position from every other number. But if all strings are there to any
finite dephts, as is easily visualized in the Binary Tree, then there
is no chance for distinction at a finite position - and other
positions are not available.}}
There is no non-denumerable real, and every source of real numbers
is denumerable [...] Implications throughout mathematics that build
upon Cantor’s Diagonal Proof must now be carefully reconsidered.
So Who Won? Professor Leopold Kronecker was right. Irrationals are
not real {{ - at least they have no real strings of digits, and only
countably many of them can be defined in a language that can be
spoken, learned and understood}}. God made all the integers and Man
made all the rest {{and in addition something more - unfortunately.}}
[Brian L. Crissey: "Kronecker 1, Cantor 0: The End of a Hundred Years’
War"]
http://www.briancrissey.info/files/Kronecker1Cantor0.pdf

Regards, WM

[Newberry]

I was thinking that if we pin down the standard model then there will be no infinite numbers, no infinite proofs and no infinite strings. But if there are no infinite strings ... Well, it seems to me that the set of all the subsets of N has to interpreted as the set of all finite subsets. After all the elements of N are finite. It also seems to me that that this is what potential infinity means: a set is potentially infinite if all its elements are finite.

Of course, it is impossible to pin down the standard model (is it?). Any attempt to do so results in a contradiction, and then they say there is no model at all. This comes from the same folks who swear that Goedel's sentence is NOT paradoxical, hell no! Needles to say they have all the logic backwards. The standard model cannot be pinned down because Goedel's sentence IS paradoxical. Is there any way around? Hell no! (Or is there?)

[Virgil]

In article
<5da061c6-f643-419d...@a34g2000vbt.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:

> So what about Cantor零 much celebrated non-denumerable real?

Deliberate misrepresentation, as usual from WM.

It is not any one real number which is non-denumerable, it is the
collection of all of them that cannot be denumerated (put into bijection
with the naturals).

Such deliberate misrepresntataion is typical of the anti-mathematics of
WM.
--

[AMiews]

"WM" <muec...@rz.fh-augsburg.de> wrote in message
news:5da061c6-f643-419d...@a34g2000vbt.googlegroups.com...
Matheology § 258

> Irrationals are not real

e is a real number, so is Pi.

> {{ - at least they have no real strings of digits,

they are represented by strings of real digits,

> and only countably many of them can be defined in a language that can be
> spoken, learned and understood}}.

have you counted them ? How many are there ?

>God made all the integers and Man made all the rest

how can you prove that ?


> Regards, WM

[FredJeffries]

On Apr 23, 1:01 am, WM <mueck...@rz.fh-augsburg.de> wrote:

> God made all the integers and Man made all the rest

Get out of the 19th century (or 17th)

by C. R. Gallistel, Rochel Gelman and Sara Cordes, "The Cultural and
Evolutionary History of the Real Numbers" in S. Levinson, & P. Jaisson
(Eds.) "Culture and evolution"

http://hum.uchicago.edu/ck0/kennedy/classes/s09/experimentalsemantics/gallistel-etal05.pdf

<abstract>
The cultural history of the real numbers began with the positive
integers. Kronecker is
often quoted as saying, "God made the integers; all else is the work
of man," by which he
meant that the system of real numbers had been erected by
mathematicians on the
intuitively obvious foundation provided by the integers. Taken as a
statement about the
cultural history of mathematics, this is beyond dispute. But if this
is taken as a claim
about the psychological foundations of arithmetic reasoning, then we
suggest that here, as
in many other areas of psychology, introspection and intuition are
poor guides to the
inner workings of the mind.

We suggest that it is the system of real numbers that is the
psychologically
primitive system, both in the phylogenetic and the ontogenetic sense.
We review evidence
that a system for arithmetic reasoning with real numbers evolved
before language evolved.
When language evolved, it picked out from the real numbers only the
integers, thereby
making the integers the foundation of the cultural history of the
number. Secondly, we
suggest that this ancestral non-verbal real number system becomes
operative in the
prelinguistic child and makes possible the acquisition of language-
mediated counting and
language-mediated arithmetic reasoning. It is the foundation on which
an individual’s
language-mediated understanding of what numbers are and what may be
done with them
rests.
</abstract>

[Virgil]

In article <kl9jfd$c0n$1...@news.albasani.net>,

So "God" made nothing, according to WM?

>
> how can you prove that ?
>
>
> > Regards, WM

--

[WM]

On 24 Apr., 23:39, "AMiews" <inva...@invalid.com> wrote:
> "WM" <mueck...@rz.fh-augsburg.de> wrote in message

>
> news:5da061c6-f643-419d...@a34g2000vbt.googlegroups.com...
> Matheology § 258
>
> > Irrationals are not real
>
> e is a real number, so is Pi.
>
> > {{ - at least they have no real strings of digits,
>
> they are represented by strings of real digits,

Nobody can read, write or use an infinite string.
Real numbers are represented by *finite names* that allow to
*calculate a rational approximation up to any desired digit*. This is
generally called a potentially infinite sequence (of approximations)
and often mistaken as an infinite sequence. But it is not the same as
an actually infinite string that defines, by sequence of digits, a
real number without doubt.

No string of digits, without a finite formula to generate it, defines
an irrational number.

>
> > and only countably many of them can be defined in a language that can be
> > spoken, learned and understood}}.
>
> have you counted them ?  How many are there ?

Not more than aleph_0

Regards, WM

[fom]

On 4/25/2013 3:53 AM, WM wrote:
>
> Nobody can read, write or use an infinite string.
> Real numbers are represented by *finite names* that allow to
> *calculate a rational approximation up to any desired digit*.

Too bad WM does not know what mathematics is presupposed
to make such calculations that require a presumed
exact value against which error is minimized.

=============================================

WM's "logic":

http://en.wikipedia.org/wiki/The_Art_of_Being_Right#Synopsis


WM's "mathematics":

WM is an unabashed ultrafinitist who refuses to fix
a largest finite number. Each "n" in his description
depends on the subsequence of triangular numbers.

> F(n)=Sum_i(1..n)(i)
>
> 1 :=> 1
> 2 :=> 3
> 3 :=> 6
> 4 :=> 10
>
> and so on

According to Brouwerian intuitionistic reasoning,
when WM's construction reaches the point where
the sequence of triangular numbers exceeds the
ultrafinitist limit, the contradiction nullifies
the construction.

This is WM's model of mathematics:

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

until he reaches his contradiction and
it vanishes.

=====================================

The triangular numbers correspond with
the number of 'marks' representing numerals
or significant denotations occurring in any
of WM' representations of the form:

1
2, 1
3, 2, 1
...
n, ..., 3, 2, 1
...

-------------------------------------

This number of 'marks' satisfies a structural
feature of the natural numbers called a
directed set:

Defintion

A binary relation >= in a set D is said
to direct D if and only if D is nonempty
and the following three conditions are
satisfied:

DS1)

If a is an element of D, then a>=a

DS2)

If a, b, c are elements of D such
that a>=b and b>=c, then a>=c

DS3)

If a and b are elements of D, then there
exists an element c of D such that c>=a
and c>=b


So, WM's geometric reasoning for any given
n obtains a finite model domain with its
cardinality given by the associated
triangular number. The triangular number
is the "element c" of condition DS3 from
the definition.

-------------------------------------

Finally, Brouwer's explanation for finitary
reasoning is used because WM refuses to
commit to any mathematical statement with
coherent consistent usage.

Brouwer distinguishes between results with
regard to 'endless', 'halted' and
'contradictory' in his explanations

"A set is a law on the basis of
which, if repeated choices of
arbitrary natural numbers are made,
each of these choices either
generates a definite sign series,
with or without termination of the
process, or brings about the
inhibition of the process together
with the definitive annihilation
of its result."

WM cannot be an ultrafinitist and
expect others to not hold him to
task for it. In constrast to
Brouwer, he repeatedly states
that there is absolutely no
completed infinity. Therefore,
there must be a maximal natural
number for his model of
mathematics. Beyond that
number, there is no mathematics.

That is WM's belief as surmised
from his statements and reasonings
as opposed to what he says with
rhetoric.

[WM]

On 25 Apr., 12:05, fom <fomJ...@nyms.net> wrote:
> On 4/25/2013 3:53 AM, WM wrote:
>
>
>
> > Nobody can read, write or use an infinite string.
> > Real numbers are represented by *finite names* that allow to
> > *calculate a rational approximation up to any desired digit*.
>
> Too bad WM does not know what mathematics

says a total flop who failed to overcome even the lowest academic
hurdles to a professor who has been teaching theoretical physics and
mathematics at German universities for about 25 years and has
published a math best seller in the country of Leibniz, Gauss and
Cantor with the publisher of Hardy, Feyman, Oberth, Diesel ...

You must really try to imagine that situation ...
But that's the internet, in particular the domain of anonymous
cowards.

Regards, WM

[YBM]

Le 25/04/2013 13:02, WM a écrit :
> On 25 Apr., 12:05, fom <fomJ...@nyms.net> wrote:
>> On 4/25/2013 3:53 AM, WM wrote:
>>
>>
>>
>>> Nobody can read, write or use an infinite string.
>>> Real numbers are represented by *finite names* that allow to
>>> *calculate a rational approximation up to any desired digit*.
>>
>> Too bad WM does not know what mathematics
>
> says a total flop who failed to overcome even the lowest academic
> hurdles to a professor who has been teaching theoretical physics and
> mathematics at German universities for about 25 years and has
> published a math best seller in the country of Leibniz, Gauss and
> Cantor with the publisher of Hardy, Feyman, Oberth, Diesel ...

Sorry, WM, but you have proven beyond any doubt to be a complete,
pathological, pathetic crank.

The fact you're teaching in Germany is INDEED a real problem for
Germany!

Please, don't involve Leibniz, Gauss, and al. with your dementia.

[Virgil]

In article
<65d3b22a-f731-4fd7...@y2g2000vbe.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 24 Apr., 23:39, "AMiews" <inva...@invalid.com> wrote:
> > "WM" <mueck...@rz.fh-augsburg.de> wrote in message
> >
> > news:5da061c6-f643-419d...@a34g2000vbt.googlegroups.com...
> > Matheology � 258
> >
> > > Irrationals are not real
> >
> > e is a real number, so is Pi.
> >
> > > {{ - at least they have no real strings of digits,
> >
> > they are represented by strings of real digits,
>
> Nobody can read, write or use an infinite string.

Certainly WM cannot, but there area all sorts of mathematical things
that WM cannot do but real mathematicians do quite regularly, like prove
their claims.
--

[Virgil]

In article
<04f426cb-557b-443e...@y14g2000vbk.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 25 Apr., 12:05, fom <fomJ...@nyms.net> wrote:
> > On 4/25/2013 3:53 AM, WM wrote:
> >
> >
> >
> > > Nobody can read, write or use an infinite string.
> > > Real numbers are represented by *finite names* that allow to
> > > *calculate a rational approximation up to any desired digit*.
> >

> > Too bad WM does not know what mathematics is all about
>
> says a total flop

Attacking the man instead of what he says is the strategy of a loser.
--

[Virgil]

On 4/25/2013 3:53 AM, WM wrote:
>
> Nobody can read, write or use an infinite string.
> Real numbers are represented by *finite names* that allow to
> *calculate a rational approximation up to any desired digit*.

How do such finite names as "pi", "e", "sqrt(2)". etc.,
"allow to calculate a rational approximation up to any desired digit"?
--

[Virgil]

In article
<65d3b22a-f731-4fd7...@y2g2000vbe.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:

> Nobody can read, write or use an infinite string.

One can use infinite strings, just like other sorts of infinite
sequences, by name or by other reference even if one cannot write them
out completely.
--

[WM]

On 25 Apr., 13:40, YBM <ybm...@nooos.fr.invalid> wrote:
> Le 25/04/2013 13:02, WM a écrit :
>
>
>
>
>
> > On 25 Apr., 12:05, fom <fomJ...@nyms.net> wrote:
> >> On 4/25/2013 3:53 AM, WM wrote:
>
> >>> Nobody can read, write or use an infinite string.
> >>> Real numbers are represented by *finite names* that allow to
> >>> *calculate a rational approximation up to any desired digit*.
>
> >> Too bad WM does not know what mathematics
>
> > says a total flop who failed to overcome even the lowest academic
> > hurdles to a professor who has been teaching theoretical physics and
> > mathematics at German universities for about 25 years and has
> > published a math best seller in the country of Leibniz, Gauss and
> > Cantor with the publisher of Hardy, Feyman, Oberth, Diesel ...
>
> Sorry, WM, but you have proven

Another flop with no academic merits tries to judge beyond his
horizon.

Regards, WM

[Virgil]

In article
<fe155da8-6802-463c...@a6g2000vbm.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 25 Apr., 13:40, YBM <ybm...@nooos.fr.invalid> wrote:

> > Le 25/04/2013 13:02, WM a �crit :

> >
> >
> >
> >
> >
> > > On 25 Apr., 12:05, fom <fomJ...@nyms.net> wrote:
> > >> On 4/25/2013 3:53 AM, WM wrote:
> >
> > >>> Nobody can read, write or use an infinite string.
> > >>> Real numbers are represented by *finite names* that allow to
> > >>> *calculate a rational approximation up to any desired digit*.
> >
> > >> Too bad WM does not know what mathematics
> >
> > > says a total flop who failed to overcome even the lowest academic
> > > hurdles to a professor who has been teaching theoretical physics and
> > > mathematics at German universities for about 25 years and has
> > > published a math best seller in the country of Leibniz, Gauss and
> > > Cantor with the publisher of Hardy, Feyman, Oberth, Diesel ...
> >
> > Sorry, WM, but you have proven
>
> Another flop

Only another flop by WM, who, having no grounds to attack the posting,
resorts, as usual, to the fallacy of attacking the poster.
--

[fom]

On 4/25/2013 3:53 AM, WM wrote:
>

> Nobody can read, write or use an infinite string.
> Real numbers are represented by *finite names*

The names might be COMPACT (WM really should learn
the difference), but what is presupposed by
Leibniz principle of identity of indiscernibles
is a different matter:

"All existential propositions, though true,
are not necessary, for they cannot be
proved unless an infinity of propositions
is used, i.e., unless an analysis is
carried to infinity. That is, they can
be proved only from the complete concept
of an individual, which involves infinite
existents. Thus, if I say, "Peter denies",
understanding this of a certain time, then
there is presupposed also the nature of
that time, which also involves all that
exists at that time. If I say "Peter
denies" indefinitely, abstracting from
time, then for this to be true -- whether
he has denied, or is about to deny --
it must nevertheless be proved from the
concept of Peter. But the concept of
Peter is complete, and so involves infinite
things; so one can never arrive at a
perfect proof, but one always approaches
it more and more, so that the difference
is less than any given difference."

Leibniz



WM has been asked to provide coherent systems of
logic against which to judge his statements.
Instead, he uses the axioms he denies and the
principles he rejects.


http://en.wikipedia.org/wiki/Doxastic#Types_of_reasoners

see "peculiar reasoner"

[fom]

On 4/25/2013 6:02 AM, WM wrote:
> On 25 Apr., 12:05, fom <fomJ...@nyms.net> wrote:
>> On 4/25/2013 3:53 AM, WM wrote:
>>
>>
>>
>>> Nobody can read, write or use an infinite string.
>>> Real numbers are represented by *finite names* that allow to
>>> *calculate a rational approximation up to any desired digit*.
>>
>> Too bad WM does not know what mathematics
>
> says a total flop who failed to overcome even the lowest academic
> hurdles

chuckle

=====================================================

[fom]

On 4/25/2013 6:02 AM, WM wrote:
>

> the domain of anonymous cowards.
>

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

Theorem:
AxAy(xcy -> -ycx)

Theorem:
AxAyAz((xcy /\ ycz) -> xcz)

Theorem:
Ax(-xcx)

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

Definition:
AxAy(xEQy <-> (Az(xcz <-> ycz) /\ Az(zcx <-> zcy) /\ Az(xez <-> yez) /\
Az(zex <-> zey)))

Definition:
AxAy(x=y <-> Az(xez <-> yez))

Assumption:
AxAy(Az(xcz <-> ycz) -> Az(xez <-> yez))

Assumption:
AxAy(Az(zex -> zey) -> Az(ycz -> xcz))

Assumption:
AxEyAz(zey <-> zcx)

Theorem:
AxAy(xcy -> Az(zex -> zey))

Theorem:
AxAy(Az(ycz -> xcz) -> Az(zex -> zey))

Theorem:
AxAy(xcy <-> (Az(zex -> zey) /\ Ez(zey /\ -zex)))

Theorem:
AxAy(Az(zex -> zey) -> Az(zcx -> zcy))

Theorem:
AxAy(Az(xez -> yez) -> Az(xcz -> ycz))

Theorem:
AxAy(xEQy <-> Az(xcz <-> ycz)

Theorem:
AxAy(xEQy <-> Az(zex <-> zey))

That grammatical equivalence is expressible in terms of neighborhood
filters is provable:
AxAy(xEQy <-> Az(xez <-> yez))

Theorem:
AxAy(xEQy <-> x=y)

Assumption:
AxAy((Az(ycz -> xez) /\ Ez(xez /\ -ycz)) -> Az((xez /\ -ycz) -> (Ew(xew
/\ wcy) \/ Aw(zcw -> ycw))))

Assumption:
AxAy((Ez(xcz) /\ Ez(ycz)) -> EwAz(zew -> (z=x \/ z=y)))

Definition:
Ax(x=V() <-> Ay(-(ycx <-> y=x)))

Assumption:
ExAy(-(ycx <-> y=x))

Assumption:
Ax(Ey(xcy) -> Ey(xey))

Definition:
Ax(set(x) <-> Ey(xcy))

Definition:
Ax(x=null() <-> Ay(-(xcy <-> x=y)))

Assumption:
ExAy(-(xcy <-> x=y))

Assumption:
Ax(Ey(ycx) -> Ey(yex /\ -Ez(zex /\ zey)))

Assumption:
AxEy(Az(zey <-> Ew(wex /\ zew)) /\ (Ez(xcz) -> Ez(ycz)))

Assumption:
AxEy(Az(zey <-> Aw(wex -> zew)) /\ (Ez(zcx) -> Ez(ycz)))

Definition:
AxAy(x=P(y) <-> (Ez(ycz) /\ Az(zex <-> (zcy \/ z=y))))

Assumption:
Ax(Ey(xcy) -> Ey(Az(zey <-> (zcx \/ z=x)) /\ Ez(ycz)))

Definition:
AxAy(x=S(y) <-> (Ez(ycz) /\ Az(zex <-> (zey \/ z=y))))

Assumption:
Ax(Ey(xcy) -> Ey(Az(zey <-> (zex \/ z=x)) /\ Ez(ycz)))

Assumption:
Ex(Ey(xcy) /\ null()cx /\ Ay(ycx -> Ez(zcx /\ ycz)))


Let the restricted quantifier

Ap[pEQp]

be interpreted as

Ap[pEQp](phi(p)) <-> Ap(pEQp /\ phi(p))

Then for each n and each well-formed formula phi(y, p_0, ..., p_n),

assume


=================
Ap_n[p_nEQp_n]...Ap_0[p_0EQp_0]
AxAy(
Ew(ycw) ->
(Ez((Ew(zcw) /\ (yez <-> (yex /\ phi(y, p_0, ..., p_n))))) <-> Ew(xcw))
)
=================


and assume

=================
Ap_n[p_nEQp_n]...Ap_0[p_0EQp_0]
(
AxAyAz(
(
((Ew(xcw) /\ Ew(ycw)) /\ (phi(x,y, p_0, ..., p_n)) /\
((Ew(xcw) /\ Ew(zcw)) /\ (phi(x,z, p_0, ..., p_n))
) -> (y=z)
)
->
AxAy(
Ew(ycw) ->
(Ez((Ew(zcw) /\ (yez <-> Ew(wex /\ phi(z,w, p_0, ..., p_n))))) <-> Ew(xcw))
)
)
=================

[Virgil]

In article <pJ6dnQmwNeQIfOTM...@giganews.com>,

fom <fom...@nyms.net> wrote:

> On 4/25/2013 6:02 AM, WM wrote:

> > the domain of anonymous cowards.

WM, whose habitually make claims that he cannot prove, is in no position
to make personal attacks on others.
--

[Virgil]

In article <SYidnQbCoIeYUuTM...@giganews.com>,

fom <fom...@nyms.net> wrote:

> WM has been asked to provide coherent systems of
> logic against which to judge his statements.
> Instead, he uses the axioms he denies and the
> principles he rejects.

But only improperly!
--

[AMiews]

"WM" <muec...@rz.fh-augsburg.de> wrote in message

news:65d3b22a-f731-4fd7...@y2g2000vbe.googlegroups.com...

>On 24 Apr., 23:39, "AMiews" <inva...@invalid.com> wrote:
>> "WM" <mueck...@rz.fh-augsburg.de> wrote in message
>>
>> news:5da061c6-f643-419d...@a34g2000vbt.googlegroups.com...
>> Matheology � 258
>>
>> > Irrationals are not real
>>
>> e is a real number, so is Pi.
>>
>> > {{ - at least they have no real strings of digits,
>>
>> they are represented by strings of real digits,

>Nobody can read, write or use an infinite string.

your opinion.

0.1111....... is an infinite string, however it is also represented by
1/9

>Real numbers are represented by *finite names* that allow to
>*calculate a rational approximation up to any desired digit*. This is
>generally called a potentially infinite sequence (of approximations)
>and often mistaken as an infinite sequence. But it is not the same as
>an actually infinite string that defines, by sequence of digits, a
>real number without doubt.

easily handled by assuming an error ball and show it aproches 0

>No string of digits, without a finite formula to generate it, defines
>an irrational number.

wrong, what is solution of equation that is equal to its derivative ?

>
>> > and only countably many of them can be defined in a language that can
>> > be
>> > spoken, learned and understood}}.
>
>> have you counted them ? How many are there ?
>
>Not more than aleph_0
>

so you have not counted them, and do not know how many their are. Isnt that
the same as putting a bound on numbers as above?

>Regards, WM

[WM]

On 27 Apr., 01:03, "AMiews" <inva...@invalid.com> wrote:
> "WM" <mueck...@rz.fh-augsburg.de> wrote in message
>
> news:65d3b22a-f731-4fd7...@y2g2000vbe.googlegroups.com...
>
> >On 24 Apr., 23:39, "AMiews" <inva...@invalid.com> wrote:
> >> "WM" <mueck...@rz.fh-augsburg.de> wrote in message
>
> >>news:5da061c6-f643-419d...@a34g2000vbt.googlegroups.com...
> >> Matheology § 258
>
> >> > Irrationals are not real

> >> > {{ - at least they have no real strings of digits,
>
> >> they are represented by strings of real digits,
> >Nobody can read, write or use an infinite string.
>
> your opinion.
>
>     0.1111.......  is an infinite string,

No, it fits in a single line and contains less than 20 symbols.

> however it is also represented by
> 1/9

That is the finite word representing the same as that word you wrote
above.

> >No string of digits, without a finite formula to generate it, defines
> >an irrational number.
>
> wrong, what is solution of equation that is equal to its derivative ?

You just wrote a finite formula.

>
>
>
> >> > and only countably many of them can be defined in a language that can
> >> > be
> >> > spoken, learned and understood}}.
>
> >> have you counted them ? How many are there ?
>
> >Not more than aleph_0
>
> so you have not counted them, and do not know how many their are.  Isnt that
> the same as putting a bound on numbers as above?

There is no bound. But if someone claims that smallest infinity is
called aleph_0, then it is easy to prove that there are not more than
aleph_0 finite words.

Regards, WM

[Virgil]

In article
<f6869ed0-d430-4a6c...@g9g2000vbl.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 27 Apr., 01:03, "AMiews" <inva...@invalid.com> wrote:

> >

> > � � 0.1111....... �is an infinite string,

>
> No, it fits in a single line and contains less than 20 symbols.

It is the name of an infinite string.

>
> > however it is also represented by
> > 1/9
>
> That is the finite word representing the same as that word you wrote
> above.

Since even WM concedes that it is a "word" or name, the issue is settled.
--

[WM]

On 27 Apr., 10:26, Virgil <vir...@ligriv.com> wrote:
> In article
> <f6869ed0-d430-4a6c-b30e-f0ca02e68...@g9g2000vbl.googlegroups.com>,

>
>  WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 27 Apr., 01:03, "AMiews" <inva...@invalid.com> wrote:
>
> > >     0.1111.......  is an infinite string,
>
> > No, it fits in a single line and contains less than 20 symbols.
>
> It is the name of an infinite string.
>

Of course, but it is not an infinite string.
Infinite strings do not exist in the internet, because computers and
their personnel are much more rational than matheologians.
Names of infinite strings are countable.

Regards, WM

[AMiews]

"WM" <muec...@rz.fh-augsburg.de> wrote in message

news:591dbeaa-2493-4b0c...@a14g2000vbm.googlegroups.com...

wrong again. WM you going in circles, infinite circles as they never end.

[Virgil]

In article
<591dbeaa-2493-4b0c...@a14g2000vbm.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 27 Apr., 10:26, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <f6869ed0-d430-4a6c-b30e-f0ca02e68...@g9g2000vbl.googlegroups.com>,
> >
> > �WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 27 Apr., 01:03, "AMiews" <inva...@invalid.com> wrote:
> >
> > > > � � 0.1111....... �is an infinite string,
> >
> > > No, it fits in a single line and contains less than 20 symbols.
> >
> > It is the name of an infinite string.
> >
> Of course, but it is not an infinite string.

No one ever works with actual numbers in mathematics,
they only work with names or numerals for numbers.

So why is working only with names a problem?
s.

> Infinite strings do not exist in the internet

They do as named objects, as do numbers.
--

[WM]

On 27 Apr., 21:51, Virgil <vir...@ligriv.com> wrote:

>
> No one ever works with actual numbers in mathematics,
> they only work with names or numerals for numbers.

Therefore no one can prove uncountability. All the names that allow to
call a number belong to a countable set.

>
> So why is working only with names a problem?

That is not a problem in mathematics. It is a problem for
matheologians.

>
> > Infinite strings do not exist in the internet
>
> They do as named objects, as do numbers.

Yes, but not more than countably many.

Regards, WM

[Virgil]

In article
<7058d749-ce72-4a0e...@s4g2000vbr.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 27 Apr., 21:51, Virgil <vir...@ligriv.com> wrote:
>
> >
> > No one ever works with actual numbers in mathematics,
> > they only work with names or numerals for numbers.
>
> Therefore no one can prove uncountability.

If one had to get hold of actual numbers to do mathematics, there could
be no mathematics at all.

And it is the axiom system for the field of real numbers which implies
uncountability, not the naming of numbers.

> >
> > So why is working only with names a problem?
>
> That is not a problem in mathematics. It is a problem for
> matheologians.

A type that exists only in WM's imagination, though he applies the term
broadly to the vast majority of those whom everyone else calls
mathematicians.

> >
> > > Infinite strings do not exist in the internet
> >
> > They do as named objects, as do numbers.
>
> Yes, but not more than countably many.

The evidence for uncountability does not rely on being able to name
uncountably many individuals.

There are more things in heaven and earth, WM ,than are dreamt of in
your philosophy.
--

[Ross A. Finlayson]

On Apr 27, 3:12 pm, Virgil <vir...@ligriv.com> wrote:
> In article

> <7058d749-ce72-4a0e-9dd0-3d82f6554...@s4g2000vbr.googlegroups.com>,

But, didn't you just dream of them in your philosophy? Or, is your
theory incomplete, or inconsistent?

Having just put a name on all of them, congratulations: there's
more. Basically Burali-Forti: Ord is irregular.

Well-order the reals, via Fefermann V = L, the universe as
constructible has for each element: that's its own name. Are the
reals a set?

The evidence for uncountability relies largely on constructive
proofs. And, the arguments for uncountability of the reals don't
apply to EF the natural/unit equivalency function.

Arguments for uncountability of the reals don't apply to EF: putting
the elements of the unit interval in a row, while satisfying notions
such as continuity, has range R_[0,1].

Bring forth applications of transfinite cardinals. EF has application
as the unit line segment.


Regards,

Ross Finlayson

[Virgil]

In article
<7de487c4-ec1b-4e98...@g5g2000pbp.googlegroups.com>,

Every real mathematician does, but WM dreams that they aren't even there
to be dreamt of!

Which is the whole point! WM is lacking what it takes to be a real
mathematician.
--

[Rupert]

On Tuesday, April 23, 2013 10:01:58 AM UTC+2, WM wrote:
> Matheology § 258
>
>
>
> So what about Cantor’s much celebrated non-denumerable real? Where is
>
> it? Did Cantor produce such a real number?

"Non-denumerable real" is meaningless. Cantor proved that the set of reals is non-denumerable.

[Ross A. Finlayson]

No, he proved that any mapping from the naturals that isn't constant
monotone, or sweep, is not onto.

So: draw a line, from point to point. Congratulations, you think
that's impossible.

Regards,

Ross Finlayson

[Newberry]

Oh yeah? Many of you folks claimed that Cantor's proof was constructive, which to my mind means that he exhibited a real number that was not on 1-2-1 list. So which one is it?

[Rupert]

Give me the list and I'll give you the number.

[Virgil]

On Tuesday, April 23, 2013 10:01:58 AM UTC+2, WM wrote:

> So what about Cantor�s much celebrated non-denumerable real? Where is
> it? Did Cantor produce such a real number?

WM once again displays his profound misunderstanding and
misrepresentation of Cantor's position.

It is only a set whole large of reals which can achieve uncountability,
not a single real, as WM implies above.

What Cantor showed was that certain sets, including the members of any
real interval of positive length, cannot be put into one-to-one
correspondence with the set of naturals without leaving some of the
non-naturals being left over unaccounted for.
--

[Zeit Geist]

This is from the article which WM quotes.

Let us demonstrate that Cantor’s non-denumerable “real” is not real in another way. Let us assume, as Cantor did, that an enumeration E of the reals exists and that an infinitely precise, new decimal expansion can be created by changing the nth digit of the nth enumerated real. Let us designate this new real number as g, which will not be in E, hence the reals will not be denumerable.
Let us then create a second parallel output stream called h, created in an identical manner as g, with one small alteration: one randomly selected digit of h will be replaced by a random digit, so, with probability 0.9, g ≠ h. Now g ∈R and h ∈R,
and R is an ordered field, so g and h must be able to be ordered by comparing digits as they are generated, or they are not real. Let g(n) denote the nth decimal place of g, and let h(n) denote the nth decimal place of h. €


My thoughts here are that when the random digit is selected, ( his terms ), it's place
is certain. By that, I mean that it is not determined to be a m-th place, but it is certain
that it must be some n-th place.

I realize that the concepts of determined and certain need to be clarified.
However, I beleive they are sufficient for my arguments.
I will alaberate further if asked.

The same argument goes for choosing a random number for g(n).

Basically, I believe when he says, "randomly selects", we would say,
"arbitrarily choose".

The author continues:
If g(n)=h(n)for 0≤n≤k,but g(k+1)<h(k+1)
then we can conclude that g < h. Similarly, if g(n) = h(n)
for 0≤n≤k , but g(k+1)>h(k+1), then we can
conclude that g > h. €€
But to establish g = h, it must be true that g(n) = h(n) €€
for 0≤n≤∞.Thusinthe10%ofthetimethat g=hand €
k is unknown, the output streams will generate digits forever
wi€thout resolving the order of g and h. So these two “reals” g
and h €cannot be ordered, so g€∉ R and h ∉ R. Thus
infinitely precise digit expansions are not real. Irrationals are
not real.

He says, now basically, that since we can't know for sure
at which point the equality of g and h fails, or even if it fails
at all; then tricotomy fails and g and h are not "real" real numbers.

However, we do know that I'd the randomly select digit is equal to
the value of the digit at the randomly selected place in g,
then g = h.
OTOH, if the randomly selected digit is not equal to the value of the
digit at the randomly selected place in g, them g ~= h, where the
equality fails in accordance to the values of the digit in the randomly
place in g and the randomly selected digit.
Therefore, g and h can be ordered in any case.
The probabilities are irrelevant.

Thus, the proof is crap.

Any thoughts WM?

PS The randomly selecected and yet unknown number reminds
of WM's unfindable line that contain all natural numbers.

Are there any apples on this thread?, ZG

[Rupert]

You are confusing "Given certain information, we would not be in a position to determine in a finite number of steps whether or not g=h" with "It is indeterminate whether or not g=h". These are not the same thing.

[Zeit Geist]

So what you are saying is that in that
10% of the time we can't find whether
they are equal in a finite amount of
steps, even though we have a matheamtical
apparatus to determine the fact?

[WM]

(1) Cantor proves that any list of reals does not contain the first n
digits (d_1, ..., d_n) of the diagonal d in the first lines (l_1, ...,
l_n).

For every n in |N. Nothing less, nothing more!

(A1) By assuming that the list can be finished, he concluded that d is
not in the list.

(2) This can be countered by the fact that every list containing all
rational numbers (which is possible, since they are countable)
contains in the lines beyond l_n infinitely many numbers starting with
(d_1, ..., d_n).

For every n in |N.

This is not in contradiction with what Cantor proved, but there is no
reason to believe that his assumption should have more weight than the
contrary assumption:

(A2) The list cannot be finished. (2) remains true in all cases.

Regards, WM

[WM]

That is a very remarkable statement. It leads to the following
questions:

How many lists can be *given*, i.e., completely defined?
How many digit replacement rules can be defined?
What is the cardinality of the product of both sets?

Regards, WM

[WM]

On 28 Apr., 07:58, Zeit Geist <tucsond...@me.com> wrote:

> However, we do know that I'd the randomly select digit is equal to
> the value of the digit at the randomly selected place in g,
> then g = h.
> OTOH, if the randomly selected digit is not equal to the value of the
> digit at the randomly selected place in g, them g ~= h, where the
> equality fails in accordance to the values of the digit in the randomly
> place in g and the randomly selected digit.
> Therefore, g and h can be ordered in any case.

A sequence of randomly selected digits will never led to an irrational
number, since every digit belongs to a finite initial segment of a
sequence of digits.
>
> Any thoughts WM?

See my answers to Rupert. All lists that ever can be defined in the
universe belong to a countable set (finite by material constraints but
at most countably infinite by logical constraints (countability of all
words)). So do the substitution rules. No proof of uncountability in
sight.

Regards, WM

[Bergholt Stuttley Johnson]

WM wrote:
> What is the cardinality of the product of both sets?

That is easy.
http://www.math.ksu.edu/~nagy/real-an/ap-b-card.pdf
Theorem B.3 (ii)

[Rupert]

Cantor's theorem is a theorem of RCA_0. As a corollary of this, the real number not in the denumerable sequence of real numbers may be given as a recursive function of the denumerable sequence of real numbers. You can make sense of the notion of a recursive function even when the argument may range over a set which has uncountable cardinality. For example, suppose I am given an "oracle machine" which will tell me the n-th digit of the binary expansion of the m-th number in the list. Then I can construct a Turing machine which will give me the n-th digit of the real number not in the list, provided that this Turing machine is allowed to make a call to the oracle machine when it needs to.

If the list itself can be recursively specified, then there is an effective procedure for obtaining a recursive specification of the real not in the list.

There are aleph-null different ways in which one could give the real number not in the list as a recursive function of the list.

The number of possible lists is c, the number of possible ways in which the real number can be given as a recursive function of the list is aleph-null.

If you restrict your attention to recursively specificable lists then the number of possible lists will only be aleph-null. But in that case the real number obtained will necessarily be a recursive real, so there will only be aleph-null possibilities for the real number obtained.

[fom]

Before WM attempts to introduce "science" into his
justifications for these questions, realize that
actual scientific theory is not involved. WM's
arguments underlying these questions are a philosophical
position nicely summarized by Chris Menzel:

http://plato.stanford.edu/entries/actualism/

It also has a link for criticisms of the position:

http://plato.stanford.edu/entries/actualism/


----------------------------------------------------

WM's "logic":

http://en.wikipedia.org/wiki/The_Art_of_Being_Right#Synopsis


WM's "mathematics":

WM is an unabashed ultrafinitist who refuses to fix
a largest finite number. Each "n" in his description
depends on the subsequence of triangular numbers.

> F(n)=Sum_i(1..n)(i)
>
> 1 :=> 1
> 2 :=> 3
> 3 :=> 6
> 4 :=> 10
>
> and so on

According to Brouwerian intuitionistic reasoning,
when WM's construction reaches the point where
the sequence of triangular numbers exceeds the
ultrafinitist limit, the contradiction nullifies
the construction.

This is WM's model of mathematics:

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

until he reaches his contradiction and
it vanishes.

=====================================

The triangular numbers correspond with
the number of 'marks' representing numerals
or significant denotations occurring in any
of WM' representations of the form:

1
2, 1
3, 2, 1
...
n, ..., 3, 2, 1
...

-------------------------------------

This number of 'marks' satisfies a structural
feature of the natural numbers called a
directed set:

Defintion

A binary relation >= in a set D is said
to direct D if and only if D is nonempty
and the following three conditions are
satisfied:

DS1)

If a is an element of D, then a>=a

DS2)

If a, b, c are elements of D such
that a>=b and b>=c, then a>=c

DS3)

If a and b are elements of D, then there
exists an element c of D such that c>=a
and c>=b


So, WM's geometric reasoning for any given
n obtains a finite model domain with its
cardinality given by the associated
triangular number. The triangular number
is the "element c" of condition DS3 from
the definition.

-------------------------------------

Finally, Brouwer's explanation for finitary
reasoning is used because WM refuses to
commit to any mathematical statement with
coherent consistent usage.

Brouwer distinguishes between results with
regard to 'endless', 'halted' and
'contradictory' in his explanations

"A set is a law on the basis of
which, if repeated choices of
arbitrary natural numbers are made,
each of these choices either
generates a definite sign series,
with or without termination of the
process, or brings about the
inhibition of the process together
with the definitive annihilation
of its result."

WM cannot be an ultrafinitist and
expect others to not hold him to
task for it. In constrast to
Brouwer, he repeatedly states
that there is absolutely no
completed infinity. Therefore,
there must be a maximal natural
number for his model of
mathematics. Beyond that
number, there is no mathematics.

That is WM's belief as surmised
from his statements and reasonings
as opposed to what he says with
rhetoric.

[fom]

On 4/28/2013 3:06 AM, WM wrote:
> On 28 Apr., 02:32, Rupert <rupertmccal...@yahoo.com> wrote:
>> On Tuesday, April 23, 2013 10:01:58 AM UTC+2, WM wrote:

>>> Matheology � 258
>>
>>> So what about Cantor�s much celebrated non-denumerable real? Where is

>>
>>> it? Did Cantor produce such a real number?
>>
>> "Non-denumerable real" is meaningless. Cantor proved that the set of reals is non-denumerable.
>
> (1) Cantor proves that any list of reals does not contain the first n
> digits (d_1, ..., d_n) of the diagonal d in the first lines (l_1, ...,
> l_n).
>
> For every n in |N. Nothing less, nothing more!
>
> (A1) By assuming that the list can be finished, he concluded that d is
> not in the list.
>

By assuming that the Euclidean algorithm for
long division can be finished, every quotient
has an exact value.

It has been demonstrated that WM cannot tell the
difference between singular terms and plural terms
in his own grammar. There is no reason to believe
WM can comprehend it in mathematical statements.

=============================================

[fom]

On 4/28/2013 3:06 AM, WM wrote:

> (2) This can be countered by the fact that every list containing all
> rational numbers (which is possible, since they are countable)

WM is not allowed to invoke completed
infinities in his arguments.

On the one hand, he denies them. On the
other hand, his use of them for the purpose
of "proving contradiction" is invalid since
he makes no such argument within defined
parameters of standard mathematical logic.

==============================================

[fom]

On 4/28/2013 3:06 AM, WM wrote:
>
>

> This is not in contradiction with what Cantor proved, but there is no
> reason to believe that his assumption should have more weight than the
> contrary assumption:
>
> (A2) The list cannot be finished. (2) remains true in all cases.
>

Cantor's argument is misrepresented once
again.

The argument only applies where a claim
is made concerning the completion of a
list. It does not claim a complete listing
of real numbers.

==============================================

[Rupert]

We could be in a position where we cannot specify in advance a finite upper bound on the number of steps it will take to verify that they are equal. The usual stance of mathematicians these days is not to worry about the method of verification, but just the mathematical reality.

[Rupert]

On Sunday, April 28, 2013 10:06:59 AM UTC+2, WM wrote:
> On 28 Apr., 02:32, Rupert <rupertmccal...@yahoo.com> wrote:
>
> > On Tuesday, April 23, 2013 10:01:58 AM UTC+2, WM wrote:
>
> > > Matheology § 258
>
> >
>
> > > So what about Cantor’s much celebrated non-denumerable real? Where is
>
> >
>
> > > it? Did Cantor produce such a real number?
>
> >
>
> > "Non-denumerable real" is meaningless. Cantor proved that the set of reals is non-denumerable.
>
>
>
> (1) Cantor proves that any list of reals does not contain the first n
>
> digits (d_1, ..., d_n) of the diagonal d in the first lines (l_1, ...,
>
> l_n).
>
>
>
> For every n in |N. Nothing less, nothing more!
>

No. He proved that, given any list of reals, one can construct a real which differs from the diagonal in every digit, and is not in the list.

[WM]

On 28 Apr., 12:14, Rupert <rupertmccal...@yahoo.com> wrote:
> On Sunday, April 28, 2013 10:10:58 AM UTC+2, WM wrote:
> > On 28 Apr., 06:00, Rupert <rupertmccal...@yahoo.com> wrote:
>
> > > On Sunday, April 28, 2013 5:35:19 AM UTC+2, Newberry wrote:
>
> > > > Oh yeah? Many of you folks claimed that Cantor's proof was constructive, which to my mind means that he exhibited a real number that was not on 1-2-1 list. So which one is it?
>
> > > Give me the list and I'll give you the number.
>
> > That is a very remarkable statement. It leads to the following
>
> > questions:
>
> > How many lists can be *given*, i.e., completely defined?
>
> > How many digit replacement rules can be defined?
>
> > What is the cardinality of the product of both sets?
>

> Cantor's theorem is a theorem of RCA_0. As a corollary of this, the real number not in the denumerable sequence of real numbers may be given as a recursive function of the denumerable sequence of real numbers. You can make sense of the notion of a recursive function even when the argument may range over a set which has uncountable cardinality. For example, suppose I am given an "oracle machine" which will tell me the n-th digit of the binary expansion of the m-th number in the list. Then I can construct a Turing machine which will give me the n-th digit of the real number not in the list, provided that this Turing machine is allowed to make a call to the oracle machine when it needs to.

And I can tell you inifinitely many numbers with the same criterion
(namely having the first n digits in common with that number that is
not in the list).

>
> If the list itself can be recursively specified, then there is an effective procedure for obtaining a recursive specification of the real not in the list.

If the list contains all rational numbers, then there is no chance to
find a sequence of digits that is not in the list.

>
> There are aleph-null different ways in which one could give the real number not in the list as a recursive function of the list.

Correct, aleph_0 - and not more. So there are at most aleph_0 numbers
that are not in the list and yet can be given.

>
> The number of possible lists is c, the number of possible ways in which the real number can be given as a recursive function of the list is aleph-null.

So the number of all diagonals ever to be constructed is countable.
>
> If you restrict your attention to recursively specificable lists then the number of possible lists will only be aleph-null. But in that case the real number obtained will necessarily be a recursive real, so there will only be aleph-null possibilities for the real number obtained.-

All other "real" numbers are not real in the sense that they ever
could be applied in any way in mathematics as individuals. They can
never act as diagonal numbers, they can never enter a calculation and
they can never appear as a result or as an intermediate result. In
brief: They do not exist in real mathematics and elsewhere.

Regards, WM

[WM]

On 28 Apr., 15:25, fom <fomJ...@nyms.net> wrote:
> On 4/28/2013 3:06 AM, WM wrote:
>
>
>
> > This is not in contradiction with what Cantor proved, but there is no
> > reason to believe that his assumption should have more weight than the
> > contrary assumption:
>
> > (A2) The list cannot be finished. (2) remains true in all cases.
>
> Cantor's argument is misrepresented once
> again.
>
> The argument only applies where a claim
> is made concerning the completion of a
> list.  It does not claim a complete listing
> of real numbers.

Of course that is not claimed. Otherwise Cantor would contradict
himself.
Claimed is completeness of the list with respect to the natural
numbers enumerating its lines

First think, then write! And, please, stop spamming this group with
your boring repetitions of what you think that I think. Nobody would
read that.

Regards, WM

[WM]

> No. He proved that, given any list of reals, one can construct a real which differs from the diagonal in every digit, and is not in the list.-

His proof assumes that every line of the list can be looked at (or put
into a CPU of a computer) such that no line remains. But that
condition is wrong, since for every line there are infinitely many
following. If all rationals are in the list, then we have:

Forall n: (d_1, ..., d_n) is repeated infinitely often in the list.

Regards, WM

[fom]

On 4/27/2013 3:30 PM, WM wrote:
> On 27 Apr., 21:51, Virgil <vir...@ligriv.com> wrote:
>
>>
>> No one ever works with actual numbers in mathematics,
>> they only work with names or numerals for numbers.
>

> Therefore no one can prove uncountability. All the names that allow to
> call a number belong to a countable set.

http://plato.stanford.edu/entries/actualism/

plato.stanford.edu/entries/actualism/actualist-problems.html

===========================================



WM's "logic":

http://en.wikipedia.org/wiki/The_Art_of_Being_Right#Synopsis

WM's "mathematics":

[Rupert]

On Sunday, April 28, 2013 3:50:11 PM UTC+2, WM wrote:
> On 28 Apr., 12:14, Rupert <rupertmccal...@yahoo.com> wrote:
>
> > On Sunday, April 28, 2013 10:10:58 AM UTC+2, WM wrote:
>
> > > On 28 Apr., 06:00, Rupert <rupertmccal...@yahoo.com> wrote:
>
> >
>
> > > > On Sunday, April 28, 2013 5:35:19 AM UTC+2, Newberry wrote:
>
> >
>
> > > > > Oh yeah? Many of you folks claimed that Cantor's proof was constructive, which to my mind means that he exhibited a real number that was not on 1-2-1 list. So which one is it?
>
> >
>
> > > > Give me the list and I'll give you the number.
>
> >
>
> > > That is a very remarkable statement. It leads to the following
>
> >
>
> > > questions:
>
> >
>
> > > How many lists can be *given*, i.e., completely defined?
>
> >
>
> > > How many digit replacement rules can be defined?
>
> >
>
> > > What is the cardinality of the product of both sets?
>
> >
>
> > Cantor's theorem is a theorem of RCA_0. As a corollary of this, the real number not in the denumerable sequence of real numbers may be given as a recursive function of the denumerable sequence of real numbers. You can make sense of the notion of a recursive function even when the argument may range over a set which has uncountable cardinality. For example, suppose I am given an "oracle machine" which will tell me the n-th digit of the binary expansion of the m-th number in the list. Then I can construct a Turing machine which will give me the n-th digit of the real number not in the list, provided that this Turing machine is allowed to make a call to the oracle machine when it needs to.
>
>
>
> And I can tell you inifinitely many numbers with the same criterion
>
> (namely having the first n digits in common with that number that is
>
> not in the list).
>

I don't know what criterion you're referring to here. These would be different numbers.

>
>
> >
>
> > If the list itself can be recursively specified, then there is an effective procedure for obtaining a recursive specification of the real not in the list.
>
>
>
> If the list contains all rational numbers, then there is no chance to
>
> find a sequence of digits that is not in the list.
>

Of course there is.

> >
>
> > There are aleph-null different ways in which one could give the real number not in the list as a recursive function of the list.
>
>
>
> Correct, aleph_0 - and not more. So there are at most aleph_0 numbers
>
> that are not in the list and yet can be given.
>

No, that does not follow. There are aleph-null different ways to define a recursive function from lists of reals to reals which always give you a real not in the list. But, for any given particular list, it will always be the case that there are more than aleph-null reals not in the list, because not all of them have to be given by such recursive functions.

> >
>
> > The number of possible lists is c, the number of possible ways in which the real number can be given as a recursive function of the list is aleph-null.
>
>
>
> So the number of all diagonals ever to be constructed is countable.
>

Depends what you mean by "diagonals".

The point is that the set of all reals is not countable.

> >
>
> > If you restrict your attention to recursively specificable lists then the number of possible lists will only be aleph-null. But in that case the real number obtained will necessarily be a recursive real, so there will only be aleph-null possibilities for the real number obtained.-
>
>
>
> All other "real" numbers are not real in the sense that they ever
>
> could be applied in any way in mathematics as individuals.

That's false, there are plenty of non-computable reals which are still arithmetically definable, for example. And whether a real number can be defined in a particular language is neither here nor there so far as the "reality" of the real number is concerned.

> They can
>
> never act as diagonal numbers, they can never enter a calculation and
>
> they can never appear as a result or as an intermediate result. In
>
> brief: They do not exist in real mathematics and elsewhere.
>

The conclusion that they do not exist is not warranted.

>
>
> Regards, WM

[fom]

On 4/28/2013 8:50 AM, WM wrote:
>
>
> All other "real" numbers are not real in the sense that they ever
> could be applied in any way in mathematics as individuals. They can
> never act as diagonal numbers, they can never enter a calculation and
> they can never appear as a result or as an intermediate result. In
> brief: They do not exist in real mathematics and elsewhere.
>

http://plato.stanford.edu/entries/actualism/

plato.stanford.edu/entries/actualism/actualist-problems.html

============================================

[WM]

On 28 Apr., 12:14, Rupert <rupertmccal...@yahoo.com> wrote:

I oversaw this:

> The number of possible lists is c,

No. What is a possible list? That is a list that can be defined in
some way. But since you cannot define an infinitude of lines by an
infinitude of definitions, you need a finite definition for a list.
Only then it can be possibly used, investigated, and discussed.
Therefore the number of possible lists in mathematics is aleph_0.

Regards, WM

[fom]

On 4/28/2013 8:57 AM, WM wrote:
> On 28 Apr., 15:25, fom <fomJ...@nyms.net> wrote:
>> On 4/28/2013 3:06 AM, WM wrote:
>>
>>
>>
>>> This is not in contradiction with what Cantor proved, but there is no
>>> reason to believe that his assumption should have more weight than the
>>> contrary assumption:
>>
>>> (A2) The list cannot be finished. (2) remains true in all cases.
>>
>> Cantor's argument is misrepresented once
>> again.
>>
>> The argument only applies where a claim
>> is made concerning the completion of a
>> list. It does not claim a complete listing
>> of real numbers.
>
> Of course that is not claimed. Otherwise Cantor would contradict
> himself.
> Claimed is completeness of the list with respect to the natural
> numbers enumerating its lines
>

Once again, Cantor's argument is misrepresented.

The argument is a scheme that applies where a claim is made
concerning the completion of a list WITH RESPECT TO A SINGULAR
NOTION OF INFINITY SUCH AS WM HAS ATTEMPTED TO PROVE WITH HIS
CRITICISMS OF ONE-TO-ONE CORRESPONDENCE AS A DETERMINER OF
CARDINAL EQUIVALENCE.

> First think, then write! And, please, stop spamming this group with
> your boring repetitions of what you think that I think. Nobody would
> read that.

I do not think about what you think. I know what you say.
I know how you behave. And I know that actions speak louder
than words.

Until I get bored, you will be faced with the facts of what
you say and do.

==============================================================

WM's "logic":

http://en.wikipedia.org/wiki/The_Art_of_Being_Right#Synopsis

[fom]

On 4/28/2013 9:03 AM, WM wrote:
>
> His proof assumes that every line of the list can be looked at (or put
> into a CPU of a computer) such that no line remains.

Once again, Cantor's argument scheme is
misrepresented by WM.

WM's "logic":

http://en.wikipedia.org/wiki/The_Art_of_Being_Right#Synopsis

> But that
> condition is wrong, since for every line there are infinitely many
> following. If all rationals are in the list, then we have:
>
> Forall n: (d_1, ..., d_n) is repeated infinitely often in the list.
>

Irrelevant.

[Rupert]

On Sunday, April 28, 2013 4:29:27 PM UTC+2, WM wrote:
> On 28 Apr., 12:14, Rupert <rupertmccal...@yahoo.com> wrote:
>
>
>
>
>
> I oversaw this:
>
>
>
> > The number of possible lists is c,
>
>
>
> No. What is a possible list? That is a list that can be defined in
>
> some way.

Wrong.

[fom]

http://plato.stanford.edu/entries/actualism/

plato.stanford.edu/entries/actualism/actualist-problems.html

================================================

[fom]

On 4/28/2013 8:57 AM, WM wrote:
>

> stop spamming this group with
> your boring repetitions of what you think that I think. Nobody would
> read that.

I find your remarks humorous.

First, you are the one who has called me a parrot,
a deplorable slave, a coward, etc.

So rot in hell.

When I gave you the opportunity to address a
properly formulated non-standard theory, you were
the one who behaved like a coward.

In fact, with every opportunity given to you in the
past by anyone you have behaved similarly.

You are the one spamming an unmoderated newsgroup
with your agenda. You are the one insulting
people. And, although not anonymous, you are the
one with a professorship who cannot get these
views of yours published in a respectable peer
reviewed journal.

When confronted by someone concerning your "theory"
and any reason for paying attention to it, I had been
directed to your "papers" on arXiv. I expect that they
will be removed once my complaint to the moderators is
received. They have a stated policy concerning the
publication of relevant, scientific documents. It
will be surprising if your nonsense survives review.

I know that no one who pays regular attention
to your nonsense threads will have any reason
to read the postings I make. On the other hand,
it is important that a casual observer recognize
how you engage in mathematical discourse is not
an appropriate form of argumentation.

So rot in hell.

[Rupert]

Even if, for example, you were to work in RCA_0+"Every real is computable", Cantor's theorem would still hold. It would tell you that given any computable list of computable reals there exists a computable real not in the list.

[Nam Nguyen]

On 28/04/2013 7:27 AM, Rupert wrote:

> The usual stance of mathematicians these days is not to worry about the method of verification, but just the mathematical reality.
>

You surely contradict what Shoenfield taught in his Mathematical Logic.

--
----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI
----------------------------------------------------

[Nam Nguyen]

On 28/04/2013 9:58 AM, Nam Nguyen wrote:
> On 28/04/2013 7:27 AM, Rupert wrote:
>
>> The usual stance of mathematicians these days is not to worry about
>> the method of verification, but just the mathematical reality.
>>
>
> You surely contradict what Shoenfield taught in his Mathematical Logic.

In it he said:

"The conspicuous feature of mathematics, as opposed to other sciences,
is the use of proofs instead of observations." (pg. 1).

[Rupert]

On Sunday, April 28, 2013 5:58:33 PM UTC+2, Nam Nguyen wrote:
> On 28/04/2013 7:27 AM, Rupert wrote:
>
>
>
> > The usual stance of mathematicians these days is not to worry about the method of verification, but just the mathematical reality.
>
> >
>
>
>
> You surely contradict what Shoenfield taught in his Mathematical Logic.
>

You think so? Could you possibly give a citation to support this claim?

[Rupert]

On Sunday, April 28, 2013 6:10:07 PM UTC+2, Nam Nguyen wrote:
> On 28/04/2013 9:58 AM, Nam Nguyen wrote:
>
> > On 28/04/2013 7:27 AM, Rupert wrote:
>
> >
>
> >> The usual stance of mathematicians these days is not to worry about
>
> >> the method of verification, but just the mathematical reality.
>
> >>
>
> >
>
> > You surely contradict what Shoenfield taught in his Mathematical Logic.
>
>
>
> In it he said:
>
>
>
> "The conspicuous feature of mathematics, as opposed to other sciences,
>
> is the use of proofs instead of observations." (pg. 1).
>

That in no way contradicts what I wrote.

[WM]

> Wrong.-

So you believe in things that cannot be defined?
Your choice.
But not mathematics.
In mathematics we use only things the existence of which can be
proved.

Regards, WM

[Nam Nguyen]

It does. How do you know what is in any reality, _without observation_ ?

Also, what is the difference between his "proofs" and your "method of
verification"?

[WM]

> Even if, for example, you were to work in RCA_0+"Every real is computable", Cantor's theorem would still hold. It would tell you that given any computable list of computable reals there exists a computable real not in the list.-

Better work in mathematics.
There the following theorem holds for every list that contains all
rationals numbers:
forall n in |N: The sequence (d_1, ..., d_n) is in infinitely many
entries of the list beyond line n.

Don't forget: For every natural number!
Don't forget: No real number can be defined by its infinite sequence
of digits. In every case a finite definition (like 1/9 or 0.111... or
sqrt2 or e or pi) is required.
You can distinguish a finite definition from an infinite sequence?

Regards, WM

[WM]

On 28 Apr., 16:37, fom <fomJ...@nyms.net> wrote:
> On 4/28/2013 9:03 AM, WM wrote:

> > for every line there are infinitely many
> > following. If all rationals are in the list, then we have:
>
> > Forall n: (d_1, ..., d_n) is repeated infinitely often in the list.
>
> Irrelevant.

Why? I find it at least very interesting.

Regards, WM

[Rupert]

On Sunday, April 28, 2013 6:45:32 PM UTC+2, WM wrote:
> On 28 Apr., 16:39, Rupert <rupertmccal...@yahoo.com> wrote:
>
> > On Sunday, April 28, 2013 4:29:27 PM UTC+2, WM wrote:
>
> > > On 28 Apr., 12:14, Rupert <rupertmccal...@yahoo.com> wrote:
>
> >
>
> > > I oversaw this:
>
> >
>
> > > > The number of possible lists is c,
>
> >
>
> > > No. What is a possible list? That is a list that can be defined in
>
> >
>
> > > some way.
>
> >
>
> > Wrong.-
>
>
>
> So you believe in things that cannot be defined?

Yes.

>
> Your choice.
>
> But not mathematics.
>

Of course it is mathematics. That is the way that mathematics is practiced these days.

Also, as pointed out elsewhere, even in the theory RCA_0+"every real is computable", Cantor's theorem still holds. In that context it says that given any computable list of computable reals there is a computable real not in the list.

> In mathematics we use only things the existence of which can be
>
> proved.
>

That's changing the subject.

If I assume the axiom of choice then I can prove the existence of a choice set for the cosets of Q in the additive group of R, that's not the same as saying that I can define it.

And I can prove the existence of reals which are undefinable in any given countable language.

[Rupert]

On Sunday, April 28, 2013 6:51:28 PM UTC+2, WM wrote:
> On 28 Apr., 17:12, Rupert <rupertmccal...@yahoo.com> wrote:
>
> > On Sunday, April 28, 2013 4:40:18 PM UTC+2, fom wrote:
>
> > > On 4/28/2013 9:29 AM, WM wrote:
>
> >
>
> > > > On 28 Apr., 12:14, Rupert <rupertmccal...@yahoo.com> wrote:
>
> >
>
> > > > I oversaw this:
>
> >
>
> > > >> The number of possible lists is c,
>
> >
>
> > > > No. What is a possible list? That is a list that can be defined in
>
> >
>
> > > > some way. But since you cannot define an infinitude of lines by an
>
> >
>
> > > > infinitude of definitions, you need a finite definition for a list.
>
> >
>
> > > > Only then it can be possibly used, investigated, and discussed.
>
> >
>
> > > > Therefore the number of possible lists in mathematics is aleph_0.
>
> >
>
> > >http://plato.stanford.edu/entries/actualism/
>
> >
>
> > > plato.stanford.edu/entries/actualism/actualist-problems.html
>
> >
>
> > > ================================================
>
> >
>
> > > WM's "logic":
>
> >
>
> > >http://en.wikipedia.org/wiki/The_Art_of_Being_Right#Synopsis
>
> >
>
> > Even if, for example, you were to work in RCA_0+"Every real is computable", Cantor's theorem would still hold. It would tell you that given any computable list of computable reals there exists a computable real not in the list.-
>
>
>
> Better work in mathematics.
>

Well, that's usually taken to be ZFC, and Cantor's theorem certainly holds there.

> There the following theorem holds for every list that contains all
>
> rationals numbers:
>
> forall n in |N: The sequence (d_1, ..., d_n) is in infinitely many
>
> entries of the list beyond line n.
>

Sure, so what? Big deal.

>
>
> Don't forget: For every natural number!
>
> Don't forget: No real number can be defined by its infinite sequence
>
> of digits. In every case a finite definition (like 1/9 or 0.111... or
>
> sqrt2 or e or pi) is required.
>

If it's definable at all, yeah. So what?

> You can distinguish a finite definition from an infinite sequence?
>

Yup.

>
>
> Regards, WM

[Rupert]

On Sunday, April 28, 2013 6:48:27 PM UTC+2, Nam Nguyen wrote:
> On 28/04/2013 10:18 AM, Rupert wrote:
>
> > On Sunday, April 28, 2013 6:10:07 PM UTC+2, Nam Nguyen wrote:
>
> >> On 28/04/2013 9:58 AM, Nam Nguyen wrote:
>
> >>
>
> >>> On 28/04/2013 7:27 AM, Rupert wrote:
>
> >>
>
> >>>
>
> >>
>
> >>>> The usual stance of mathematicians these days is not to worry about
>
> >>
>
> >>>> the method of verification, but just the mathematical reality.
>
> >>
>
> >>>>
>
> >>
>
> >>>
>
> >>
>
> >>> You surely contradict what Shoenfield taught in his Mathematical Logic.
>
> >>
>
> >>
>
> >>
>
> >> In it he said:
>
> >>
>
> >>
>
> >>
>
> >> "The conspicuous feature of mathematics, as opposed to other sciences,
>
> >>
>
> >> is the use of proofs instead of observations." (pg. 1).
>
> >>
>
> >
>
> > That in no way contradicts what I wrote.
>
>
>
> It does. How do you know what is in any reality, _without observation_ ?
>

In mathematics, proof is the only means we have of knowing the nature of reality. But that in no way contradicts what I wrote. Because in modern mathematics we assume that the reality has a determinate character regardless of whether or not we can observe it, or what the method of observation is. For example, it is legitimate to speak of "the first natural number n such that there is a sequence of twenty 7's in a row in the decimal expansion of pi starting at the n-th digit after the decimal point, if it exists", regardless of whether or not we have any feasible means of determining what this natural number is. Or we can speak of "the ordinal alpha such that 2^(aleph_null)=aleph-alpha", even though it is impossible to determine on the basis of the axioms of set theory what this ordinal is.

>
>
> Also, what is the difference between his "proofs" and your "method of
>
> verification"?
>

Quite a lot, because I was talking about comparing two real numbers solely by means of inspecting finite fragments of their decimal expansions, whereas using the full power of what can be proved from an axiom system for mathematics might well go a long way beyond that.

[Newberry]

On Saturday, April 27, 2013 9:00:49 PM UTC-7, Rupert wrote:
> On Sunday, April 28, 2013 5:35:19 AM UTC+2, Newberry wrote:
>

> > On Saturday, April 27, 2013 5:32:31 PM UTC-7, Rupert wrote:
>
> >
>
> > > On Tuesday, April 23, 2013 10:01:58 AM UTC+2, WM wrote:
>
> >
>
> > >
>
> >
>
> > > > Matheology § 258
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > So what about Cantor’s much celebrated non-denumerable real? Where is
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > it? Did Cantor produce such a real number?
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > "Non-denumerable real" is meaningless. Cantor proved that the set of reals is non-denumerable.
>
> >
>
> >
>
> >
>

> > Oh yeah? Many of you folks claimed that Cantor's proof was constructive, which to my mind means that he exhibited a real number that was not on 1-2-1 list. So which one is it?
>
>
>
> Give me the list and I'll give you the number.

Is the proof constructive or not?

[fom]

I have no doubt.

There is legitimate mathematical insight associated
with your constructions. That you identify absolute
infinity with denumerability does make the Cantor
argument inapplicable in the sense that the Cantor
argument requires that a presumed totality be given.

As noted before, however, there are appropriate
mathematical investigations into systems based on
those characteristics. And the same is true with
regard to philosophical investigations. You simply
pursue an agenda that disrespects all.

[Ross A. Finlayson]

Consider the natural/unit equivalency function (so-called exactly for
this reason), with domain N and co-domain R_[0,1]. This is n/d for n
from zero to d and d goes to infinity.

f_d(n) = n/d, 0 <= n <= d
f(n) = lim_d->oo lim_n=0^d n/d

The expansions of the elements have this feature: for each n, the
first n many elements of f(n) as .000[][][]... are zero, and, in
binary, the only value different from each has value one. The
resulting anti-"diagonal" has expansion .111... = 1, and lim_n->d n/d
= 1.

Then, besides that this function has properties as being a CDF, there
are considerations about representations in expansions with any given
integer radix, vis-a-vis the particular properties for the radix being
0, 1, or infinity. The elements of its range satisfy being elements
of the complete ordered field, just not that the operations are
interchangeable except through the projection onto R_[0,1], with an
extended definition of R (as sees initial development here on this
forum). As well there are other properties of this function of
interest for their novelty, with regards to the real numbers of the
natural continuum and their foundation.

So, there's an enumeration of elements of the unit interval, and:
with the elements of its range dense in the unit interval and
otherwise meeting continuity and being suitable as a prototype unit
for measure (and of a metric, etc.), the arguments for uncountability
(anti-diagonal, nested intervals, continued fractions as well) don't
apply to it.

Regards,

Ross Finlayson

[FredJeffries]

The Final Jeopardy category is "Logical Fallacies".

Your Final Jeopardy answer is:
Use of a statement irrelevant to the argument but which the speaker
finds "at least very interesting".

You have ten seconds in which to write down your question. Cue music.

[Virgil]

In article <G7ydnSPQ7IkUseDM...@giganews.com>,

fom <fom...@nyms.net> wrote:

> On 4/27/2013 3:30 PM, WM wrote:
> > On 27 Apr., 21:51, Virgil <vir...@ligriv.com> wrote:
> >
> >>
> >> No one ever works with actual numbers in mathematics,
> >> they only work with names or numerals for numbers.
> >
> > Therefore no one can prove uncountability.

That presumes that there cannot be unnamed numbers.

But that presumption is no part of real mathematics, however much
WMytheology presumes it in his Wolkenmuekenheim. .
--

[WM]

On 28 Apr., 19:18, Rupert <rupertmccal...@yahoo.com> wrote:

>
> > So you believe in things that cannot be defined?
>
> Yes.
>

That is a feature you share with theology.

>
> > Your choice.
>
> > But not mathematics.
>
> Of course it is mathematics. That is the way that mathematics is practiced these days.

No, it is matheology. Three centuries ago, there were university
chairs for astrology, today there are such for matheology disguised as
set theory.

No! Things that only exist in minds but cannot exist in minds, do not
belong to mathematics.

>
> Also, as pointed out elsewhere, even in the theory RCA_0+"every real is computable", Cantor's theorem still holds. In that context it says that given any computable list of computable reals there is a computable real not in the list.
>
> > In mathematics we use only things the existence of which can be
>
> > proved.
>
> That's changing the subject.

No, that's mathematics.

>
> If I assume the axiom of choice then I can prove the existence of a choice set for the cosets of Q in the additive group of R, that's not the same as saying that I can define it.
>
> And I can prove the existence of reals which are undefinable in any given countable language.

If I assume some nonsense, I can proof some nonsense. Nevertheless it
is nonsense and remains so.

Please answer a question. What do you think is meaning of this fact:
For every n in |N: every list containing all rational numbers
contains in the lines beyond line l_n infinitely many numbers starting
with (d_1, ..., d_n).

Regards, WM

[WM]

On 28 Apr., 19:20, Rupert <rupertmccal...@yahoo.com> wrote:

>
> > There the following theorem holds for every list that contains all
>
> > rationals numbers:
>
> > forall n in |N: The sequence (d_1, ..., d_n) is in infinitely many
>
> > entries of the list beyond line n.
>
> Sure, so what? Big deal.

This means: Cantor's proof does not say anything about the presence of
the diagonal number in such a list. In contrast we see that every
finite initial segment of the diagonal number is in the list. In
addition to believing in unthinkable thoughts and unusable numbers you
must believe that the diagonal number is more than all its finite
initial segments. But you know that you cannot define a real number by
more than all its finite initial segments.

Isn't that a bit too much nonsense that you have learned in your
"mathematics" lessons?

>
>
>
> > Don't forget: For every natural number!
>
> > Don't forget: No real number can be defined by its infinite sequence
>
> > of digits. In every case a finite definition (like 1/9 or 0.111... or
>
> > sqrt2 or e or pi) is required.
>
> If it's definable at all, yeah. So what?
>
> > You can distinguish a finite definition from an infinite sequence?
>
> Yup.

Then you know that above I listed only finite definitions and that it
is impossible to use a real number in any other way.

Regards, WM

[Virgil]

In article
<73f8bec4-2ba7-467c...@a14g2000vbm.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:

> On 28 Apr., 06:00, Rupert <rupertmccal...@yahoo.com> wrote:
> > On Sunday, April 28, 2013 5:35:19 AM UTC+2, Newberry wrote:
>
> > > Oh yeah? Many of you folks claimed that Cantor's proof was constructive,
> > > which to my mind means that he exhibited a real number that was not on
> > > 1-2-1 list. So which one is it?
> >
> > Give me the list and I'll give you the number.
>

> That is a very remarkable statement. It leads to the following
> questions:
>
> How many lists can be *given*, i.e., completely defined?

That is a problem only for those claiming that all reals can be
listed/counted.

If no such lists of all the objects in a set can exist then any set of
objects that cannot be so listed is an uncountable set by definition.
> How many digit replacement rules can be defined?

One such rule is enough, and at least one can be defined.
--

[WM]

On 28 Apr., 20:33, fom <fomJ...@nyms.net> wrote:
> On 4/28/2013 12:13 PM, WM wrote:
>
> > On 28 Apr., 16:37, fom <fomJ...@nyms.net> wrote:
> >> On 4/28/2013 9:03 AM, WM wrote:
>
> >>> for every line there are infinitely many
> >>> following. If all rationals are in the list, then we have:
>
> >>> Forall n: (d_1, ..., d_n) is repeated infinitely often in the list.
>
> >> Irrelevant.
>
> > Why? I find it at least very interesting.
>
> I have no doubt.
>
> There is legitimate mathematical insight associated
> with your constructions.  That you identify absolute
> infinity with denumerability does make the Cantor
> argument inapplicable in the sense that the Cantor
> argument requires that a presumed totality be given.
>

The Cantor argument presumes a countable totality. I accept this
assumption and show that the conclusion onto an uncountable infinity
does not follow.

Every Cantor-list that contains all rational numbers contains also all
FISs of the diagonal. In mathematics, the diagonal is nothing outside
of the sequence of all its FIS. The limit another thing. But the limit
is not in the list and is not the diagonal of the list.

Regards, WM

[WM]

On 28 Apr., 21:13, FredJeffries <fredjeffr...@gmail.com> wrote:
> On Apr 28, 10:13 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > On 28 Apr., 16:37, fom <fomJ...@nyms.net> wrote:
>
> > > On 4/28/2013 9:03 AM, WM wrote:
> > > > for every line there are infinitely many
> > > > following. If all rationals are in the list, then we have:
>
> > > > Forall n: (d_1, ..., d_n) is repeated infinitely often in the list.
>
> > > Irrelevant.
>
> > Why? I find it at least very interesting.
>
> The Final Jeopardy category is "Logical Fallacies".

One logical fallacy is certainly the assumption that the limit of the
finite segments of the diagonal is defined by digits.

Limits of rational sequences are not rational and are not defined by
digits.

Regards, WM

[WM]

On 28 Apr., 21:54, Virgil <vir...@ligriv.com> wrote:
> In article <G7ydnSPQ7IkUseDMnZ2dnUVZ_t6dn...@giganews.com>,

>
>  fom <fomJ...@nyms.net> wrote:
> > On 4/27/2013 3:30 PM, WM wrote:
> > > On 27 Apr., 21:51, Virgil <vir...@ligriv.com> wrote:
>
> > >> No one ever works with actual numbers in mathematics,
> > >> they only work with names or numerals for numbers.
>
> > > Therefore no one can prove uncountability.
>
> That presumes that there cannot be unnamed numbers.

There cannot be unnamed names.
Numbers are ideas. There cannot exist ideas that nobody can think.

Regards, WM

[Virgil]

In article
<e0c78922-cd14-4aaa...@s9g2000vba.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 28 Apr., 12:14, Rupert <rupertmccal...@yahoo.com> wrote:
>
>
> I oversaw this:
>
> > The number of possible lists is c,
>
> No. What is a possible list? That is a list that can be defined in
> some way. But since you cannot define an infinitude of lines by an
> infinitude of definitions, you need a finite definition for a list.
> Only then it can be possibly used, investigated, and discussed.
> Therefore the number of possible lists in mathematics is aleph_0.

Then there are actually infinite sets and the walls of Wolkenmuekenheim
have fallen before the horns of reason.

If the cardinality of the set of all lists is aleph_0, as conceded
above, what is the cardinality of the set of all sets of lists?

It must be at least as large, just from counting its one element sets.
Can WM prove that it is no larger?
That would require WM to do something like listing all sets of such
lists.
--

[Virgil]

In article
<36602922-d48d-435c...@k8g2000vbz.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:

> Better work in mathematics.

Why should they have to when WM can't?

> There the following theorem holds for every list that contains all
> rationals numbers:
> forall n in |N: The sequence (d_1, ..., d_n) is in infinitely many
> entries of the list beyond line n.

What are those d_n? if they are rationals taken in order from some list
of all rationals, then none of them need ever reappear beyond line n of
that list.

>

> Don't forget: No real number can be defined by its infinite sequence
> of digits.

Actually, every real is definable by an infinite sequence of digits
along with a sign and a radix point position in the sequence and the
radix base value. The only problem is in finding which sequence.

But if a finite definition produces an infinite sequence of digits which
in turn defines a real, then that infinite sequence defines that real,
too.


"Ignorance is preferable to error, and he is less remote from the truth
who believes nothing than he who believes what is wrong."

-- Thomas Jefferson
--

[Virgil]

In article
<ea8a13ca-4dd7-4458...@r7g2000vbw.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:




> In mathematics we

Already a lie.
--

[Virgil]

In article
<c7029193-6ca4-4593...@r4g2000vbf.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:


> > Cantor's theorem is a theorem of RCA_0. As a corollary of this, the real
> > number not in the denumerable sequence of real numbers may be given as a
> > recursive function of the denumerable sequence of real numbers. You can
> > make sense of the notion of a recursive function even when the argument may
> > range over a set which has uncountable cardinality. For example, suppose I
> > am given an "oracle machine" which will tell me the n-th digit of the
> > binary expansion of the m-th number in the list. Then I can construct a
> > Turing machine which will give me the n-th digit of the real number not in
> > the list, provided that this Turing machine is allowed to make a call to
> > the oracle machine when it needs to.
>
> And I can tell you inifinitely many numbers with the same criterion
> (namely having the first n digits in common with that number that is
> not in the list).

Then WM must have access to infinitely many numbers, which makes WM a
liar!
>
> >
> > If the list itself can be recursively specified, then there is an effective
> > procedure for obtaining a recursive specification of the real not in the
> > list.
>
> If the list contains all rational numbers, then there is no chance to
> find a sequence of digits that is not in the list.

Then WM is claiming that such numbers as
Sum_{n in |N} 1/10^(n!)
are all rational?
> >
> > There are aleph-null different ways in which one could give the real number
> > not in the list as a recursive function of the list.
>
> Correct, aleph_0 - and not more. So there are at most aleph_0 numbers
> that are not in the list and yet can be given.

But even to the most complete of such listings, as many more numbers
exist unlisted, so no list can be a complete list.
> >
> > The number of possible lists is c, the number of possible ways in which the
> > real number can be given as a recursive function of the list is aleph-null.
>
> So the number of all diagonals ever to be constructed is countable.

The number of NON-members of any such list is always at least one
so no such list is ever a complete listing of all reals.

Thus there is no counting process which can ever count all of them.

>
> All other "real" numbers are not real in the sense that they ever
> could be applied in any way in mathematics as individuals.

They are 'real' in the sense that the least upper bound and greatest
lower bound properties of the 'real numbers' do not work without them.



>They can
> never act as diagonal numbers

Nonsense. There is no sequence of digits that cannot serve as an
antidiagonal to a sutable sequences of reals.


, they can never enter a calculation and
> they can never appear as a result or as an intermediate result. In
> brief: They do not exist in real mathematics and elsewhere.


If they do not exist, then neither do least upper bounds or greatest
lower bounds, and thus no real number systems either.

Talk about baby with bathwater-ism!!!
--

[Dan]

> There cannot be unnamed names.
> Numbers are ideas. There cannot exist ideas that nobody can think.
>
> Regards, WM

There always exists thoughts you haven't (at least yet) thought of .
The set of all thoughts is more than you can think at once (in one
thought) .
The set of all countable sequences is more than you can count at
once.

Give me a list of reals and I shall give you a number not on the list.
Any 'sequence of thoughts' you can give me still constitutes a 'single
thought' from you .
But the totality of possible thoughts is not within the gasp of a
'single thought' , thus , not amenable to sequence in thought.

Indeed , there cannot exist ideas that nobody can think , but the 'Set
of all ideas' is more than anybody can think .
Though does not follow a 'thinkable pattern' even if , the 'elements
of thought' , by definition , do .

"The limits of the soul you would not find out, though you should
traverse every way, so deep a
logos does it have" -Heraclitus

[Virgil]

In article
<78ed6b32-fc1b-4d66...@a6g2000vbm.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:

> On 28 Apr., 02:32, Rupert <rupertmccal...@yahoo.com> wrote:
> > On Tuesday, April 23, 2013 10:01:58 AM UTC+2, WM wrote:
> > > Matheology � 258
> >
> > > So what about Cantor�s much celebrated non-denumerable real? Where is
> >
> > > it? Did Cantor produce such a real number?
> >
> > "Non-denumerable real" is meaningless. Cantor proved that the set of reals
> > is non-denumerable.
>

> (1) Cantor proves that any list of reals does not contain the first n
> digits (d_1, ..., d_n) of the diagonal d in the first lines (l_1, ...,
> l_n).
>
> For every n in |N. Nothing less, nothing more!

Cantor proved that and a lot more, all of which remains is valid
everywhere other than in WMytheology.
>
> (A1) By assuming that the list can be finished, he concluded that d is
> not in the list.

Is there any list_position/digit_positiion for which the anti-diagonal
is not defined?
>
> (2) This can be countered


Not outside Wolkenmuekenheim.
>
> This is not in contradiction with what Cantor proved, but there is no
> reason to believe that his assumption should have more weight than the
> contrary assumption:
>
> (A2) The list cannot be finished. (2) remains true in all cases.

If the list cannot never be finished the the set from which it is being
constructed is uncountable already.

Note that, outside Wolkenmuekenheim, countability implies listability:
A set is, by definition, at least outside Wolkenmuekenheim, countable
ONLY IF there is a surjection from |N to that set!
And is uncountable when there cannot be any such surjection!

Thus any set, for which no such surjection can exist, regardless of the
reason, is uncountable by definition!
--

[Virgil]

In article
<3b7e1a24-cee9-4617...@e13g2000vbn.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:

> On 28 Apr., 15:29, Rupert <rupertmccal...@yahoo.com> wrote:

> > On Sunday, April 28, 2013 10:06:59 AM UTC+2, WM wrote:
> > > On 28 Apr., 02:32, Rupert <rupertmccal...@yahoo.com> wrote:
> >
> > > > On Tuesday, April 23, 2013 10:01:58 AM UTC+2, WM wrote:
> >
> > > > > Matheology � 258
> >
> > > > > So what about Cantor�s much celebrated non-denumerable real? Where is
> >
> > > > > it? Did Cantor produce such a real number?
> >
> > > > "Non-denumerable real" is meaningless. Cantor proved that the set of
> > > > reals is non-denumerable.
> >
> > > (1) Cantor proves that any list of reals does not contain the first n
> >
> > > digits (d_1, ..., d_n) of the diagonal d in the first lines (l_1, ...,
> >
> > > l_n).
> >
> > > For every n in |N. Nothing less, nothing more!
> >

> > No. He proved that, given any list of reals, one can construct a real which
> > differs from the diagonal in every digit, and is not in the list.-
>
> His proof assumes that every line of the list can be looked at (or put
> into a CPU of a computer) such that no line remains.

The very definition of countability requires that a set be listable.
A set which is not listable in any way is also not countable.

Except in WMytheology, where all definitions bend to WM's will.


> But that
> condition is wrong

Only in WMytheology!

It is still the condition required by standard mathematics,
so holds everywhere outside the thrall of WMytheology.
--

[Virgil]

In article
<415ae8d6-b07f-4eb4...@k8g2000vbz.googlegroups.com>,

The myths that interest WM are irrelevant outside his WMytheology.
--

[Virgil]

In article
<f2222cbe-a6c2-44e0...@l2g2000vbp.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:

> On 28 Apr., 15:25, fom <fomJ...@nyms.net> wrote:

> > On 4/28/2013 3:06 AM, WM wrote:
> >
> >
> >
> > > This is not in contradiction with what Cantor proved, but there is no
> > > reason to believe that his assumption should have more weight than the
> > > contrary assumption:
> >
> > > (A2) The list cannot be finished. (2) remains true in all cases.
> >

> > Cantor's argument is misrepresented once
> > again.
> >
> > The argument only applies where a claim
> > is made concerning the completion of a
> > list. �It does not claim a complete listing
> > of real numbers.
>
> Of course that is not claimed. Otherwise Cantor would contradict
> himself.
> Claimed is completeness of the list with respect to the natural
> numbers enumerating its lines

Cantor does not have to claim existence of any 'complete' list of reals.

Those who claim that the set of reals is countable are the ones who need
to prove a 'complete' listing of the reals in [0,1] CAN exist.

Cantor need only need show that ANY list of members of the closed
interval [0,1], and many incomplete such lists do exist, is necessarily
an incomplete listing of the reals in [0,1].

Which Cantor does! Neatly and unambiguously!

And all the obfuscations that WM throws up do not obscure the validity
of that proof.
--

[Virgil]

In article
<65e1befa-5b6d-4cc4...@16g2000vbx.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:


>
> A sequence of randomly selected digits will never led to an irrational
> number

If those digits are concatenated into ever longer decimal, the
probability that the limit number is irrational is 1 and the probability
that that limit is rational is zero,
> >
> > Any thoughts WM?
>
> See my answers to Rupert. All lists that ever can be defined in the
> universe belong to a countable set (finite by material constraints but
> at most countably infinite by logical constraints (countability of all
> words)). So do the substitution rules. No proof of uncountability in
> sight.

To prove countability of a set S, one must prove a surjection from |N
to S.

So to show that the reals are countable, WM has to prove existence of a
surjection from |N to |R.

Cantor proved that no such surjection is possible.

And none of WM 's incessant counter arguments have established a
surjection from |N to |R.

So Cantor wins!
Again!!
As Usual!!!

And in the conflict between Mathematics and WMytheology,
Mathematics Wins!
Again!!
As Usual !!!
--

[Zeit Geist]

On Sunday, April 28, 2013 1:19:21 AM UTC-7, WM wrote:
> On 28 Apr., 07:58, Zeit Geist wrote:
>
>
>
> > However, we do know that I'd the randomly select digit is equal to
>
> > the value of the digit at the randomly selected place in g,
>
> > then g = h.
>
> > OTOH, if the randomly selected digit is not equal to the value of the
>
> > digit at the randomly selected place in g, them g ~= h, where the
>
> > equality fails in accordance to the values of the digit in the randomly
>
> > place in g and the randomly selected digit.
>
> > Therefore, g and h can be ordered in any case.

>
>
>
> A sequence of randomly selected digits will never led to an irrational
>

> number, since every digit belongs to a finite initial segment of a
>
> sequence of digits.
>
I agree, if you randomly select a digit for each place one at a time,
you will never fill every place.

However, we can prove in mathematics, by the use of logic,
that every real number, meaning every Cauchy sequence,
has at least one decimal expansion. We can also show that
every decimal expansion ( finite or not ) corresponds to
exactly one real number. Hence, these randomly selected
numbers are of no use.

I contend a string of randomly selected digit is not a real number.

>
> > Any thoughts WM?
>
>
>
> See my answers to Rupert. All lists that ever can be defined in the
>
> universe belong to a countable set (finite by material constraints but
>
> at most countably infinite by logical constraints (countability of all
>
> words)). So do the substitution rules. No proof of uncountability in
>
> sight.
>

Yes, again I agree, the set of all list must be countable. However,
this does not prevent us from defining sets whose cardinality is
uncountable. For example, we can define R, the set of real numbers,
as a complete ordered field. I can also define each and every rational
number, as well as, each and every algebraic irrational number
in a set that is at most countably infinite. We could also define that
entire set in a finite set of definitions. Either way, we still have
a countably infinite set of definitions leftover.
So this no hinderence to a proof of the uncountability of R.
As a matter of fact, a model of ZFC can even be contained in
A countable set. That is, if a model of ZFC exists.
>
> Regards, WM

Just for the record I only want RED apples, ZG

[Virgil]

In article
<39bbcfdd-31b7-41a1...@p10g2000vbn.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 28 Apr., 21:54, Virgil <vir...@ligriv.com> wrote:
> > In article <G7ydnSPQ7IkUseDMnZ2dnUVZ_t6dn...@giganews.com>,
> >
> >  fom <fomJ...@nyms.net> wrote:
> > > On 4/27/2013 3:30 PM, WM wrote:
> > > > On 27 Apr., 21:51, Virgil <vir...@ligriv.com> wrote:
> >
> > > >> No one ever works with actual numbers in mathematics,
> > > >> they only work with names or numerals for numbers.
> >
> > > > Therefore no one can prove uncountability.
> >
> > That presumes that there cannot be unnamed numbers.
>
> There cannot be unnamed names.

Does a name like "James" itself have a name?
Maybe naming names is required in German, but not in English, and not in
mathematics.

> Numbers are ideas. There cannot exist ideas that nobody can think.

There are lots of good ideas that WM cannot think, and far too many bad
ones that only WM ever thinks.
--

[Virgil]

In article
<81c1a997-9c56-494e...@m1g2000vbe.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> Limits of rational sequences are not rational

The limit of the rational sequence f(n) = 1/n, is certainly rational.

For every rational, q, with |q| < 1, the sequence
1, 1+q, 1+q+q^2, 1 + q + q^2 + q^3, ..., has a rational limit.

Wm must be a lousy mathematician to claim that

"Limits of rational sequences are not rational"

--

[Zeit Geist]

On Sunday, April 28, 2013 6:27:40 AM UTC-7, Rupert wrote:
> On Sunday, April 28, 2013 8:44:06 AM UTC+2, Zeit Geist wrote:
>

>
>
> We could be in a position where we cannot specify in advance a finite upper bound on the number of steps it will take to verify that they are equal. The usual stance of mathematicians these days is not to worry about the method of verification, but just the mathematical reality.

Out of curiosity, is that your stance, not to worry about method of verification.
I would, personally, agree. If we have a proof, in an presumed consistent setting, then that
is sufficient. I don't think every conjecture needs to be verified case by case, to find
it's truth value in a given system. Indeed, this impossible in most cases in mathematics.
Furthermore, if a certain apperatus is unable to verify a conjecure, then that is a limitation
of the apparatus, not the mathematical system.

Just for the record, I know little of Recursion theory or Computation theory.
Maybe the reason for my agreement with the stance above.
I can follow the simple terms, but when you state consequences, I have to
take you word for it.

Thanks.

I think I have solid line on some apples, ZG

[Virgil]

In article
<84572f0e-7d39-4e50...@e9g2000vbg.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> The Cantor argument presumes a countable totality.

The very definition of countability presumes a countable totality, or
more specifically, a surjection from |N to any set being claimed to be
countable.

Suppose WM claims that the set S, of binary sequences,
functions from |N to {m,w}, is countable.

The WM is claiming that there is a function, say f:|N -> S, which is
surjection.

So a claim of cuntability requires existence of a surjection and a proof
of uncountability only requires that any such caim be falsified.

Which Cantor did, quite nicely!

And not all of WM's wriggling can undo it.
--