Matheology § 154: Consistency Proof!

[WM]

Matheology § 154: Consistency Proof!

The long missed solution of an outstanding problem came from a
completely unexpected side: Social science proves the consistency of
matheology by carrying out a poll.

As recently reported (see matheology § 152)
http://www.hs-augsburg.de/~mueckenh/KB/Matheology.pdf
mathematics and matheology lead to different values of the continued
fraction

1/((((((10^0)/10)+10^1)/10)+10^2)/10)+… = 0 (Cauchy)
1/((((((10^0)/10)+10^1)/10)+10^2)/10)+… > 1 (Cantor)

But 100 % of all matheologians who responded to our poll said that
this difference is not surprising since different methods have been
applied, namely the mathematical calculation invented by Cauchy and
the matheological method invented by Cantor. Although both names begin
with a C (like certainty (and even with a Ca (like can and cannot)))
the following letters are completely different.

The general opinion is that it is not surprising to find different
results when applying different methods. Even the application of the
*same* method by different people may yield different results as we
see daily in our elementary schools.

This attitude also has some consequences with respect to the human
rights. We should no longer talk of mistakes and errors in
calculations and punish pupils who deviate from the majority or main
stream, but we should only note beside the result who applied what
method and possibly also location and time because experience shows
that the result of a calculation may depend on such details.

For, he reasons pointedly: That which must not, can not be. (C.
Morgenstern)

Regards, WM

[Vurgil]

In article
<a924b8a3-c051-4e91...@d17g2000vbv.googlegroups.com>,

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

> Matheology � 154: Consistency Proof!
>
> The long missed solution of an outstanding problem came from a
> completely unexpected side: Social science proves the consistency of
> matheology by carrying out a poll.
>
> As recently reported (see matheology � 152)
> http://www.hs-augsburg.de/~mueckenh/KB/Matheology.pdf
> mathematics and matheology lead to different values of the continued
> fraction
>
> 1/((((((10^0)/10)+10^1)/10)+10^2)/10)+� = 0 (Cauchy)
> 1/((((((10^0)/10)+10^1)/10)+10^2)/10)+� > 1 (Cantor)

It is not at all clear that these expressions represent any continued
fraction at all.

Finite continued fractions look like
a_0,
or
a_0 + 1/a_1,
or
a_0 + 1/(_ 1 + 1/a_2)
or
a_0 + 1/(a_1 + 1/(a_2 + 1/a_3))
or
a_0 + 1/(a_1 + 1/(a_2 + 1/(a_3 + 1/a_4)))
and so on

where a_0 is necessarily an integer and each of the other a_i's is a
necessarily a POSITIVE integer, with the infinite case merely extending
the finite cases endlessly.


But it is not at all clear what value any of the a_i would have to have
in an expression like
"1/((((((10^0)/10)+10^1)/10)+10^2)/10)+�"

[William Hughes]

On Nov 18, 8:04 pm, Vurgil <Vur...@arg.erg> wrote:
> In article
> <a924b8a3-c051-4e91-a088-c9ee5167a...@d17g2000vbv.googlegroups.com>,

>
>  WM <mueck...@rz.fh-augsburg.de> wrote:
> > Matheology 154: Consistency Proof!
>
> > The long missed solution of an outstanding problem came from a
> > completely unexpected side: Social science proves the consistency of
> > matheology by carrying out a poll.
>
> > As recently reported (see matheology 152)
> >http://www.hs-augsburg.de/~mueckenh/KB/Matheology.pdf
> > mathematics and matheology lead to different values of the continued
> > fraction
>
> > 1/((((((10^0)/10)+10^1)/10)+10^2)/10)+ = 0 (Cauchy)
> > 1/((((((10^0)/10)+10^1)/10)+10^2)/10)+ > 1 (Cantor)
>
> It is not at all clear that these expressions represent any continued
> fraction at all.
>

One of the problems with attempting any discussion with WM
is the fact that he is unable or unwilling to define anything.
A second problem is that he will switch from a tolerably inexact
notation (decimal digits with positions) to absolute
nonsense. Nor will he agree to discuss things except using
his latest form.

The idea that he is introducing complications
because when he is clear it is obvious he is wrong is
hard to resist.

[Vurgil]

In article
<4e943067-e5ab-453a...@n5g2000yqe.googlegroups.com>,

I have long since given up resisting it.

[WM]

On 19 Nov., 01:15, William Hughes <wpihug...@gmail.com> wrote:
> On Nov 18, 8:04 pm, Vurgil <Vur...@arg.erg> wrote:
>
>
>
>
>
> > In article
> > <a924b8a3-c051-4e91-a088-c9ee5167a...@d17g2000vbv.googlegroups.com>,
>
> >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > Matheology 154: Consistency Proof!
>
> > > The long missed solution of an outstanding problem came from a
> > > completely unexpected side: Social science proves the consistency of
> > > matheology by carrying out a poll.
>
> > > As recently reported (see matheology 152)
> > >http://www.hs-augsburg.de/~mueckenh/KB/Matheology.pdf
> > > mathematics and matheology lead to different values of the continued
> > > fraction
>
> > > 1/((((((10^0)/10)+10^1)/10)+10^2)/10)+ = 0 (Cauchy)
> > > 1/((((((10^0)/10)+10^1)/10)+10^2)/10)+ > 1 (Cantor)
>
> > It is not at all clear that these expressions represent any continued
> > fraction at all.
>
> One of the problems with attempting any discussion with WM
> is the fact that he is unable or unwilling to define anything.

I am convinced that you are intelligent enough to understand above
expressions.

> The idea that he is introducing complications
> because when he is clear it is obvious he is wrong is
> hard to resist.
>

That is my impression of your reaction. But in oder to test it, here
is the complete representation of the continued fraction C:

C = (((...((((((10^0)/10)+10^1)/10)+10^2)/10)+... )+10^n)/10)+...

Now take the reciproce and find 1/C = 0 or 1/C > 1? Which one is the
correct value?

Regards, WM

[Alan Smaill]

William Hughes <wpih...@gmail.com> writes:

> The idea that he is introducing complications
> because when he is clear it is obvious he is wrong is
> hard to resist.

Smoke and Mirrors are the accoutrements of the Prophet.


--
Alan Smaill

[William Hughes]

On Nov 19, 6:28 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 19 Nov., 01:15, William Hughes <wpihug...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > On Nov 18, 8:04 pm, Vurgil <Vur...@arg.erg> wrote:
>
> > > In article
> > > <a924b8a3-c051-4e91-a088-c9ee5167a...@d17g2000vbv.googlegroups.com>,
>
> > >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > Matheology 154: Consistency Proof!
>
> > > > The long missed solution of an outstanding problem came from a
> > > > completely unexpected side: Social science proves the consistency of
> > > > matheology by carrying out a poll.
>
> > > > As recently reported (see matheology 152)
> > > >http://www.hs-augsburg.de/~mueckenh/KB/Matheology.pdf
> > > > mathematics and matheology lead to different values of the continued
> > > > fraction
>
> > > > 1/((((((10^0)/10)+10^1)/10)+10^2)/10)+ = 0 (Cauchy)
> > > > 1/((((((10^0)/10)+10^1)/10)+10^2)/10)+ > 1 (Cantor)
>
> > > It is not at all clear that these expressions represent any continued
> > > fraction at all.
>
> > One of the problems with attempting any discussion with WM
> > is the fact that he is unable or unwilling to define anything.

Another is his habit of editing out (without any indication)
bits of posts he doesn't want to deal with.

A third is the fact that while he insists that you make substantial
effort to understand his incorrect and ambiguous stuff, he makes
no effort to follow your posts.

[WM]

> no effort to follow your posts.-

A very good answer. So you can avoid any mathematical arguing. In
addition you are a very good counterfeiter. Instead of the correct
definition

1/((((((10^0)/10)+10^1)/10)+10^2)/10)+… = 0 (Cauchy)
1/((((((10^0)/10)+10^1)/10)+10^2)/10)+… > 1 (Cantor)

you quote a crippled and not understandable formula

1/((((((10^0)/10)+10^1)/10)+10^2)/10)+ = 0 (Cauchy)
1/((((((10^0)/10)+10^1)/10)+10^2)/10)+ > 1 (Cantor)

Really, it is interesting to see how you defend your position..

Regards, WM

[Vurgil]

In article
<697ac4db-a7f3-42bf...@10g2000vbu.googlegroups.com>,

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

> On 19 Nov., 01:15, William Hughes <wpihug...@gmail.com> wrote:
> > On Nov 18, 8:04 pm, Vurgil <Vur...@arg.erg> wrote:
> >
> >
> >
> >
> >
> > > In article
> > > <a924b8a3-c051-4e91-a088-c9ee5167a...@d17g2000vbv.googlegroups.com>,
> >
> > >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > Matheology 154: Consistency Proof!
> >
> > > > The long missed solution of an outstanding problem came from a
> > > > completely unexpected side: Social science proves the consistency of
> > > > matheology by carrying out a poll.
> >
> > > > As recently reported (see matheology 152)
> > > >http://www.hs-augsburg.de/~mueckenh/KB/Matheology.pdf
> > > > mathematics and matheology lead to different values of the continued
> > > > fraction
> >
> > > > 1/((((((10^0)/10)+10^1)/10)+10^2)/10)+ = 0 (Cauchy)
> > > > 1/((((((10^0)/10)+10^1)/10)+10^2)/10)+ > 1 (Cantor)
> >
> > > It is not at all clear that these expressions represent any continued
> > > fraction at all.
> >
> > One of the problems with attempting any discussion with WM
> > is the fact that he is unable or unwilling to define anything.
>
> I am convinced that you are intelligent enough to understand above
> expressions.

I, on the other hand, am not at al convinced that WM is intelligent

enough to understand above expressions.

At least not well enough to explain them to anyone else.

>
> > The idea that he is introducing complications
> > because when he is clear it is obvious he is wrong is
> > hard to resist.
> >
> That is my impression of your reaction. But in oder to test it, here
> is the complete representation of the continued fraction C:
>
> C = (((...((((((10^0)/10)+10^1)/10)+10^2)/10)+... )+10^n)/10)+...
>
> Now take the reciproce and find 1/C = 0 or 1/C > 1? Which one is the
> correct value?

Then C is NOT in the form of a continued fraction at all, at least not
of any standard type.

And WM has provided no reason to suspect that the process has any limit
at all

[Vurgil]

In article
<459031c5-2297-40d1...@e25g2000vbm.googlegroups.com>,

To which WM had no anywhere nearly as good a response.

> So you can avoid any mathematical arguing. In
> addition you are a very good counterfeiter. Instead of the correct
> definition

> 1/((((((10^0)/10)+10^1)/10)+10^2)/10)+Š = 0 (Cauchy)
> 1/((((((10^0)/10)+10^1)/10)+10^2)/10)+Š > 1 (Cantor)

> you quote a crippled and not understandable formula
> 1/((((((10^0)/10)+10^1)/10)+10^2)/10)+ = 0 (Cauchy)
> 1/((((((10^0)/10)+10^1)/10)+10^2)/10)+ > 1 (Cantor)

I do not see that either one of these minutely different forms above has
any significant advantage over the other. The left hand sides both
indicate processes that must be continued, and I see no reason to
suppose those continutations should in any way differ.

>
> Really, it is interesting to see how you defend your position..

WM, on the other hand, fails to defend his positions successfully with
remarkable consistency.

[Vurgil]

In article
<38ba87a6-bbbd-41b6...@q5g2000vbp.googlegroups.com>,

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

> On 19 Nov., 01:15, William Hughes <wpihug...@gmail.com> wrote:

> > On Nov 18, 8:04 pm, Vurgil <Vur...@arg.erg> wrote:
> >
> >
> >
> >
> >
> > > In article
> > > <a924b8a3-c051-4e91-a088-c9ee5167a...@d17g2000vbv.googlegroups.com>,
> >
> > >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > Matheology 154: Consistency Proof!
> >
> > > > The long missed solution of an outstanding problem came from a
> > > > completely unexpected side: Social science proves the consistency of
> > > > matheology by carrying out a poll.
> >
> > > > As recently reported (see matheology 152)
> > > >http://www.hs-augsburg.de/~mueckenh/KB/Matheology.pdf
> > > > mathematics and matheology lead to different values of the continued
> > > > fraction
> >
> > > > 1/((((((10^0)/10)+10^1)/10)+10^2)/10)+ = 0 (Cauchy)
> > > > 1/((((((10^0)/10)+10^1)/10)+10^2)/10)+ > 1 (Cantor)
> >
> > > It is not at all clear that these expressions represent any continued
> > > fraction at all.
> >
> > One of the problems with attempting any discussion with WM
> > is the fact that he is unable or unwilling to define anything.
>

> I am convinced that you are intelligent enough to understand above
> expressions.
>

> > The idea that he is introducing complications
> > because when he is clear it is obvious he is wrong is
> > hard to resist.
>

> That is my impression of your reaction. But in oder to test it, here
> is the complete representation of the continued fraction C:
>

> C = ((...((((((10^0)/10)+10^1)/10)+10^2)/10)+... +)10^n/10)+...

>
> Now take the reciproce and find 1/C = 0 or 1/C > 1? Which one is the
> correct value?

Any such allegedly infinite continued fraction should be representable
as a sequence of truncated continued fractions:
C_0 = a_0
C_1 = a_0 + 1/a_1
C_2 = a_0 + 1/(a_1 + 1/a_2)
C_3 = a_0 + 1/(a_1 + 1/(a_2 + 1/a_3))
and so on, with C as the limit, provided it exists.

But WM's 'C' does not seem to be capable of any such analysis, and thus
is not a continued fraction, at least in any usual sense, at all, at all.

So WM goofs AGAIN, AS USUAL!

[WM]

On 19 Nov., 23:17, Vurgil <Vur...@arg.erg> wrote:
> In article

> <459031c5-2297-40d1-b71f-1f6f24545...@e25g2000vbm.googlegroups.com>,

> > you quote a crippled and not understandable formula
> > 1/((((((10^0)/10)+10^1)/10)+10^2)/10)+ = 0 (Cauchy)
> > 1/((((((10^0)/10)+10^1)/10)+10^2)/10)+ > 1 (Cantor)
>
> I do not see that either one of these minutely different forms above has
> any significant advantage over the other.

You have again crippled my expression. Look into th original.

Regards, WM

[WM]

On 19 Nov., 23:08, Vurgil <Vur...@arg.erg> wrote:

> > That is my impression of your reaction. But in oder to test it, here
> > is the complete representation of the continued fraction C:
>
> > C = (((...((((((10^0)/10)+10^1)/10)+10^2)/10)+... )+10^n)/10)+...
>
> > Now take the reciproce and find 1/C = 0 or 1/C > 1? Which one is the
> > correct value?
>
> Then C is NOT in the form of a continued fraction at all, at least not
> of any standard type.

That is of no importance. C is a never ending, i.e. continued
fraction.

>
> And WM has provided no reason to suspect that the process has any limit

> at all-

1/C is a real number. The question remain: Which one is it? 0,
according to mathematics, or >1 , according to set theory.

Regards, WM

[Virgil]

In article
<917673f3-6c48-42b0...@n8g2000vbb.googlegroups.com>,

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

> On 19 Nov., 23:08, Vurgil <Vur...@arg.erg> wrote:
>
> > > That is my impression of your reaction. But in oder to test it, here
> > > is the complete representation of the continued fraction C:
> >
> > > C = (((...((((((10^0)/10)+10^1)/10)+10^2)/10)+... )+10^n)/10)+...
> >
> > > Now take the reciproce and find 1/C = 0 or 1/C > 1? Which one is the
> > > correct value?
> >
> > Then C is NOT in the form of a continued fraction at all, at least not
> > of any standard type.
>
> That is of no importance. C is a never ending, i.e. continued
> fraction.

"Never ending fraction" and "Continued fraction" are quote different in
standard terminology. Of one thing continued fractions can, and often
do, end. it is only when they are for irrational numbers that they do
not end.

> >
> > And WM has provided no reason to suspect that the process has any limit
> > at all-
>
> 1/C is a real number. The question remain: Which one is it? 0,
> according to mathematics, or >1 , according to set theory.

Every unending continued fraction, at least in the usual sense of
continued fractions, may "converge" to a real number, but that does not
apply to the expression you presented, at least until you can show that
it matches the usual definition of a continued fraction, which it does
not appear to do.
So that your claim that it must somehow "converge" to some real number
requires a proof that you have clearly not provided.
--

[Virgil]

In article
<1bee78f8-3d99-4090...@y6g2000vbb.googlegroups.com>,

Your ability to express yourself has always been crippled by your own
incapacities. And as I merely copied and pasted what others wrote, any
such alleged errors are not mine.
--

[William Hughes]

For a teacher of calculus you have a very
naive view of limits. It is not enough to
give a an infinite process and declare a limit.
Eg, Process I

1,2,3,4,...
2,1,3,4,...
1,2,3,4,...
2,1,3,4,...
1,2,3,4,...
2,1,3,4,...
...

There is no limit. So you can't say "In the limit
3 does not move"

You also talk about "the" limit as if there were
only one.

Consider

2,3,2,3,...

Two possible limits
No limit (elements are real numbers)
{ {}, {{}} } (elements are sets of ordinals)

[WM]

On 21 Nov., 14:10, William Hughes <wpihug...@gmail.com> wrote:
> It is not enough to
> give a an infinite process and declare a limit.

But it is enough to to consider the following sequence
> >> > 01.
> >> > 0.1
> >> > 010.1
> >> > 01.01
> >> > 0101.01
> >> > 010.101
> >> > 01010.101
> >> > 0101.0101
> >> > ...
to see that it has the (improper) limit
(((…((((((10^0)/10)+10^1)/10)+10^2)/10)+… )+10^n)/10)+…
infinite.
.
And it is as easy to see that in set theory the set of indices left of
the decimal point
> >> > 0_2 1_1 .
> >> > 0_2 . 1_1
> >> > 0_4 1_3 0_2 . 1_1
> >> > 0_4 1_3 . 0_2 1_1
> >> > 0_6 1_5 0_4 1_3 . 0_2 1_1
> >> > 0_6 1_5 0_4 . 1_3 0_2 1_1
> >> > 0_8 1_7 0_6 1_5 0_4 . 1_3 0_2 1_1
> >> > 0_8 1_7 0_6 1_5 . 0_4 1_3 0_2 1_1
> >> > ...
has limit { }.

And it is further easy to know, that decimal fractions as conceived by
Simon Stevinus cannot contain numerals without indices, even if no one
does see them because they were not explicitly written.

Regards, WM

[William Hughes]

Your basic problem remains. You continue to talk
about "the" limit as if there was only one.

[WM]

On 21 Nov., 16:24, William Hughes <wpihug...@gmail.com> wrote:
> Your basic problem remains.   You continue to talk
> about "the" limit as if there was only one.

The real sequence has a limit. And if you dislike infinity as an
improper limit, then take the reciprocals. They have *the* limit 0.
This sequence is independent of anything else but its terms or its
definition.

Set theory shows that *this sequence* has a limit without indices on
the left hand side, and hence has another limit (< 1) or no limit. Or
the reciprocals have a limit > 0. This result does in no way depend on
anything else but set theory being incompatible with mathematics.

Regards, WM

[William Hughes]

On Nov 21, 11:37 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 21 Nov., 16:24, William Hughes <wpihug...@gmail.com> wrote:
>
> > Your basic problem remains.   You continue to talk
> > about "the" limit as if there was only one.
>
> The real sequence has a limit. And if you dislike infinity as an
> improper limit, then take the reciprocals. They have *the* limit 0.

Correct. Note that this limit is a real number.

> This sequence is independent of anything else but its terms or its
> definition.
>


> Set theory shows that *this sequence* has a limit without indices on

You do not define *this sequence*. If you mean the sequence of real
numbers you are incorrect. Set theory does say that there is
a limit of the set of digits to the left of the decimal place.
This limit is a set.

> the left hand side, and hence has another limit (< 1) or no limit.

The limit is {}. {} is not a real number. {} does not have a
reciprocal

Two different limits which are not the same. No reason for them
to be the same. No contradiction.

[WM]

On 21 Nov., 16:54, William Hughes <wpihug...@gmail.com> wrote:
> On Nov 21, 11:37 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > On 21 Nov., 16:24, William Hughes <wpihug...@gmail.com> wrote:
>
> > > Your basic problem remains.   You continue to talk
> > > about "the" limit as if there was only one.
>
> > The real sequence has a limit. And if you dislike infinity as an
> > improper limit, then take the reciprocals. They have *the* limit 0.
>
> Correct.  Note that this limit is a real number.
>
> > This sequence is independent of anything else but its terms or its
> > definition.
>
> > Set theory shows that *this sequence* has a limit without indices on
>
> You do not define *this sequence*.

This is exactly the same sequence. There is nothing further to define.

> If you mean the sequence of real
> numbers you are incorrect.

I mean just this sequence of real numbers. And I am correct. Set
theory does not leave any digit left of the decimal point in the
limit.

>  Set theory does say that there is
> a limit of the set of digits to the left of the decimal place.
> This limit is a set.

And this limit excludes the existence of any digit, which implies that
there is no digit.

>
> > the left hand side, and hence has another limit (< 1) or no limit.
>
> The limit is {}.  {} is not a real number.  {} does not have a
> reciprocal

But the numbers allowed by an empty set of decimal left to the point
has a reciprocal, namely a value larger than 1.

>
> Two different limits which are not the same.

But one of the limits excludes the other one. And that is a
contradiction.

You can also conclude that two different calculations of the same
stuff may lead to two different results because the calculations are
different. But mathematics does not tolerate that. That would kill
math.

Regards, WM

[William Hughes]

On Nov 21, 12:00 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 21 Nov., 16:54, William Hughes <wpihug...@gmail.com> wrote:

<snip>

> > The limit is {}.  {} is not a real number.  {} does not have a
> > reciprocal
>
> But the numbers allowed by an empty set of decimal left to the point
> has a reciprocal, namely a value larger than 1.

Absolute nonsense. There are no numbers "allowed by an empty set".
How can a set consisting of no numbers have a reciprocal?

[WM]

Not nonsense but as usual you have not understaood.
There are not numerals left of the decimal point, but there may be
numerals right of the decimal point. So there is a reciprocal of
0.abc... between 1 and oo.

But that is not so important. Important and mathematical is only this:

Every infinite sequence of real numbers either has no limit or has a
limit in the real numbers or the improper limit oo. In any case there
are never two or more limits! If existing, it can be calculated
according to Cauchy. If set theory supplies a tool, then the limit can
be calculated according to Cantor too. Or we can find some
restrictions in this way.

Here we find a funny result like that: Cauchy states, that there is a
house. Cantor says that there are no stones. WH says that there is no
contradiction.

Of course everybody can claim what he likes. It is not very new. There
are some matheologians who claim that "there" are numbers which nobody
can name. Compared to that, your statement is only moderately
unmathematical. But all people whom I have met, who are very
intelligent but not yet brainwashed by matheology, support my
position. That is very satisfactory for me.

Regards, WM

[William Hughes]

On Nov 21, 1:20 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 21 Nov., 17:41, William Hughes <wpihug...@gmail.com> wrote:
>
> > On Nov 21, 12:00 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > On 21 Nov., 16:54, William Hughes <wpihug...@gmail.com> wrote:
>
> > <snip>
>
> > > > The limit is {}.  {} is not a real number.  {} does not have a
> > > > reciprocal
>
> > > But the numbers allowed by an empty set of decimal left to the point
> > > has a reciprocal, namely a value larger than 1.
>
> > Absolute nonsense.  There are no numbers "allowed by an empty set".
> > How can a set consisting of no numbers have a reciprocal?
>
> Not nonsense but as usual you have not understaood.
> There are not numerals left of the decimal point, but there may be
> numerals right of the decimal point.

Nope. The limit of the set of digits to the left of the decimal
point is not a set of digits to the right of the decimal.
If we change the limit to the set of digits to the left or right of
the decimal point we still get {}. {} is not a real number
and does not have a reciprocal.

> So there is a reciprocal of
> 0.abc... between 1 and oo.
>
> But that is not so important. Important and mathematical is only this:
>
> Every infinite sequence of real numbers either has no limit or has a
> limit in the real numbers or the improper limit oo. In any case there
> are never two or more limits!

Piffle. You really know nothing about limits do you.

> If existing, it can be calculated
> according to Cauchy. If set theory supplies a tool, then the limit can
> be calculated according to Cantor too.

Piffle.

> Or we can find some
> restrictions in this way.

Possibly, but we need more than handwaving.

[WM]

On 21 Nov., 18:43, William Hughes <wpihug...@gmail.com> wrote:
> On Nov 21, 1:20 pm, WM <mueck...@rz.fh-augsburg.de> wrote:

> Nope.  The limit of the set of digits to the left of the decimal
> point is not a set of digits to the right of the decimal.

Of course it is not, but it does not prohibit that there are digits on
the right.

> If we change the limit to the set of digits to the left or right of
> the decimal point we still get {}.  {} is not a real number
> and does not have a reciprocal.

We cannot conclude from set theory that the digits on the right of the
decimal point vanish.

>
> > Every infinite sequence of real numbers either has no limit or has a
> > limit in the real numbers or the improper limit oo. In any case there
> > are never two or more limits!
>
> Piffle. You really know nothing about limits do you.

In my book on analysis I write: a sequence may have many accumulation
points, If there is only one accumulation point, we call it the limit
of the sequence. (But I did not invent that definition.)

>
> > If existing, it can be calculated
> > according to Cauchy. If set theory supplies a tool, then the limit can
> > be calculated according to Cantor too.
>
> Piffle.

A good argument. Possibly your last one.

>
> > Or we can find some
> > restrictions in this way.
>
> Possibly, but we need more than handwaving.

Is the limit { } of the set of digits based upon handwawing or is it
the only possible result of set theory? If the latter is true: Do you
agree that we can state: Set theory is not suitable to determine
restrictions for limits of sequences?

Regards, WM

[William Hughes]

On Nov 21, 1:57 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 21 Nov., 18:43, William Hughes <wpihug...@gmail.com> wrote:
>
> > On Nov 21, 1:20 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> > Nope.  The limit of the set of digits to the left of the decimal
> > point is not a set of digits to the right of the decimal.
>
> Of course it is not, but it does not prohibit that there are digits on
> the right.
>
> > If we change the limit to the set of digits to the left or right of
> > the decimal point we still get {}.  {} is not a real number
> > and does not have a reciprocal.
>
> We cannot conclude from set theory that the digits on the right of the
> decimal point vanish.

Yes we can.

>
>
>
> > > Every infinite sequence of real numbers either has no limit or has a
> > > limit in the real numbers or the improper limit oo. In any case there
> > > are never two or more limits!
>
> > Piffle. You really know nothing about limits do you.
>
> In my book on analysis I write: a sequence may have many accumulation
> points, If there is only one accumulation point, we call it the limit
> of the sequence. (But I did not invent that definition.)

Anyone who is writing a book on analysis should understand that
accumulation point depends on the topology used.
Sure, one usually use the "standard" topology
derived from the standard metric, and one may even
use language suggesting that this is the only possible
topology, but you still have to know there are
other possibilities.

[WM]

On 21 Nov., 19:23, William Hughes <wpihug...@gmail.com> wrote:
> On Nov 21, 1:57 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > On 21 Nov., 18:43, William Hughes <wpihug...@gmail.com> wrote:
>
> > > On Nov 21, 1:20 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> > > Nope.  The limit of the set of digits to the left of the decimal
> > > point is not a set of digits to the right of the decimal.
>
> > Of course it is not, but it does not prohibit that there are digits on
> > the right.
>
> > > If we change the limit to the set of digits to the left or right of
> > > the decimal point we still get {}.  {} is not a real number
> > > and does not have a reciprocal.
>
> > We cannot conclude from set theory that the digits on the right of the
> > decimal point vanish.
>
> Yes we can.

No. Set theory does not destruct the set of all natural numbers. Set
theory only shows that the set can be enumerated by the set of all
even numbers, .i.e., all inidces disappear left and gather right.

>
>
>
> > > > Every infinite sequence of real numbers either has no limit or has a
> > > > limit in the real numbers or the improper limit oo. In any case there
> > > > are never two or more limits!
>
> > > Piffle. You really know nothing about limits do you.
>
> > In my book on analysis I write: a sequence may have many accumulation
> > points, If there is only one accumulation point, we call it the limit
> > of the sequence. (But I did not invent that definition.)
>
> Anyone who is writing a book on analysis should understand that
> accumulation point depends on the topology used.

If the book concerns real analysis only, then the topology is clear.
By the way, this topology is explained in the book.

> Sure, one usually use the "standard" topology
> derived from the standard metric, and one may even
> use language suggesting that this is the only possible
> topology, but you still have to know there are
> other possibilities.

Of course, how shouldn't I. Perhaps I will be going to write another
book on other topologies in the future. Why not? That is not an
esoteric lore. But my sequence is a sequence of real numbers and it
has one and only one real number (or infinity) as its (improper)
limit. And just for this very case set theory shows that the limit is
less than 1. No, it is not attempted to calculate any other limit. We
consider only the one and only possible limit of the real sequence in
the real numbers.

Therefore all your struggling is in vain. Perhaps a sect of
matheologians will survive for a while and die out only very slowly
(their last words being: There is no contradiction!) but every sober
mind outside of that sect will recognize that it is impossible to have
a house and at the same place to find no bricks (or other material).

Regards, WM

[William Hughes]

On Nov 21, 3:13 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 21 Nov., 19:23, William Hughes <wpihug...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > On Nov 21, 1:57 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > On 21 Nov., 18:43, William Hughes <wpihug...@gmail.com> wrote:
>
> > > > On Nov 21, 1:20 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > Nope.  The limit of the set of digits to the left of the decimal
> > > > point is not a set of digits to the right of the decimal.
>
> > > Of course it is not, but it does not prohibit that there are digits on
> > > the right.
>
> > > > If we change the limit to the set of digits to the left or right of
> > > > the decimal point we still get {}.  {} is not a real number
> > > > and does not have a reciprocal.
>
> > > We cannot conclude from set theory that the digits on the right of the
> > > decimal point vanish.
>
> > Yes we can.
>
> No. Set theory does not destruct the set of all natural numbers. Set
> theory only shows that the set can be enumerated by the set of all
> even numbers, .i.e., all inidces disappear left and gather right.

Absolute nonsense. This is not even wrong.

[Virgil]

In article
<168832cc-10d3-4040...@ib4g2000vbb.googlegroups.com>,

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

> And it is as easy to see that in set theory the set of indices left of
> the decimal point
> > >> > 0_2 1_1 .
> > >> > 0_2 . 1_1
> > >> > 0_4 1_3 0_2 . 1_1
> > >> > 0_4 1_3 . 0_2 1_1
> > >> > 0_6 1_5 0_4 1_3 . 0_2 1_1
> > >> > 0_6 1_5 0_4 . 1_3 0_2 1_1
> > >> > 0_8 1_7 0_6 1_5 0_4 . 1_3 0_2 1_1
> > >> > 0_8 1_7 0_6 1_5 . 0_4 1_3 0_2 1_1
> > >> > ...
> has limit { }.

While I see a point between other expressions , I do not see anything
that could be properly interpreted as a DECIMAL point.

Or does WM regard such things as 1_3 and 1_5 as decimal digits?

Furthermore in standard mathematics

02 11 . has 2 "indices left of the point"
...
08 17 06 15 . 04 13 02 11 has 4 "indices left of the point"

And the limit of number of "positions" would be infinite on either side
of that point.

On the other hand, the limit of the set on the left minus the set on the
right would be empty.

Though
--

[Virgil]

In article
<50dd0545-5eeb-421a...@n5g2000vbk.googlegroups.com>,

It had far more to do with WM being incompatible with mathematics.
--

[Virgil]

In article
<8029d26e-ccf5-430d...@p22g2000vby.googlegroups.com>,

The "numbers allowed by an empty set of decimal left to the point" are
no numbers as all.

And numbers, plural, unless they are all equal. do not have "a"
reciprocal, they have reciprocals plural.

> >
> > Two different limits which are not the same.
>
> But one of the limits excludes the other one. And that is a
> contradiction.

Neither excludes the other anywhere but in Wolkenmuekenheim.

>
> You can also conclude that two different calculations of the same
> stuff may lead to two different results because the calculations are
> different. But mathematics does not tolerate that. That would kill
> math.

Not any where nearly as fatally as adopting WMytheology would kill it,
--

[Virgil]

In article
<1ae6a5ac-811c-4ed4...@e25g2000vbm.googlegroups.com>,

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

> On 21 Nov., 17:41, William Hughes <wpihug...@gmail.com> wrote:
> > On Nov 21, 12:00 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > On 21 Nov., 16:54, William Hughes <wpihug...@gmail.com> wrote:
> >
> > <snip>
> >
> > > > The limit is {}.  {} is not a real number.  {} does not have a
> > > > reciprocal
> >
> > > But the numbers allowed by an empty set of decimal left to the point
> > > has a reciprocal, namely a value larger than 1.
> >
> > Absolute nonsense.  There are no numbers "allowed by an empty set".
> > How can a set consisting of no numbers have a reciprocal?
>
> Not nonsense but as usual you have not understaood.
> There are not numerals left of the decimal point, but there may be
> numerals right of the decimal point. So there is a reciprocal of
> 0.abc... between 1 and oo.

Since the so-called "digits" on either side of what you miscall a
"decimal" point eventually exceed every finite quantity, whatever else
that point may be, it is not a decimal point.

At least not outside of Wolkenmuekenheim!

>
> But that is not so important. Important and mathematical is only this:
>
> Every infinite sequence of real numbers either has no limit or has a
> limit in the real numbers or the improper limit oo. In any case there
> are never two or more limits! If existing, it can be calculated
> according to Cauchy. If set theory supplies a tool, then the limit can
> be calculated according to Cantor too. Or we can find some
> restrictions in this way.
>
> Here we find a funny result like that: Cauchy states, that there is a
> house. Cantor says that there are no stones. WH says that there is no
> contradiction.

Most houses outside of Wolkenmuekenheim are built of wood or brick or
both so do not contradict anything but WMatheology.

>
> Of course everybody can claim what he likes. It is not very new. There
> are some matheologians who claim that "there" are numbers which nobody
> can name.

In fact anyone and everyone accepting any standard model of the real
numbers system, and most non-standard models, knows that!
--

[Virgil]

In article
<a2670703-576c-4073...@l18g2000vbv.googlegroups.com>,

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

> On 21 Nov., 18:43, William Hughes <wpihug...@gmail.com> wrote:
> > On Nov 21, 1:20 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > Nope.  The limit of the set of digits to the left of the decimal
> > point is not a set of digits to the right of the decimal.
>
> Of course it is not, but it does not prohibit that there are digits on
> the right.
>
> > If we change the limit to the set of digits to the left or right of
> > the decimal point we still get {}.  {} is not a real number
> > and does not have a reciprocal.
>
> We cannot conclude from set theory that the digits on the right of the
> decimal point vanish.
> >
> > > Every infinite sequence of real numbers either has no limit or has a
> > > limit in the real numbers or the improper limit oo. In any case there
> > > are never two or more limits!
> >
> > Piffle. You really know nothing about limits do you.
>
> In my book on analysis I write: a sequence may have many accumulation
> points, If there is only one accumulation point, we call it the limit
> of the sequence. (But I did not invent that definition.)

Just as well, as what you do invent is either not mathematics at all or
merely very bad mathematics.
--

[Virgil]

In article
<ff10fc05-5690-48cb...@bq2g2000vbb.googlegroups.com>,

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

> > Sure, one usually use the "standard" topology
> > derived from the standard metric, and one may even
> > use language suggesting that this is the only possible
> > topology, but you still have to know there are
> > other possibilities.
>
> Of course, how shouldn't I.

There is so much of standard mathematics that you do not know and so
much of what you think you know which isn't standard mathematics, that
we are never quite sure of what you think you know.
--

[WM]

On 21 Nov., 21:14, William Hughes <wpihug...@gmail.com> wrote:

>
> > > > We cannot conclude from set theory that the digits on the right of the
> > > > decimal point vanish.
>
> > > Yes we can.
>
> > No. Set theory does not destruct the set of all natural numbers. Set
> > theory only shows that the set can be enumerated by the set of all
> > even numbers, .i.e., all inidces disappear left and gather right.
>

> Absolute nonsense.  This is not even wrong.-

The typical answer of a man who has not understood. But my explanation
may really be a bit too hard for you. So let us stay on a lower level
and simply continue:
> > > Yes we can.

Can we etsimate by means of set theory how many digits *left* to the
decimal point will be present in the limit (as calculated by
analytical means) of the real sequence

> > 01.
> > 0.1
> > 010.1
> > 01.01
> > 0101.01
> > 010.101
> > 01010.101
> > 0101.0101
> > ...

?

Regards, WM

[WM]

On 21 Nov., 22:01, Virgil <vir...@ligriv.com> wrote:
> In article
> <168832cc-10d3-4040-b710-1337ecfbc...@ib4g2000vbb.googlegroups.com>,

>
>  WM <mueck...@rz.fh-augsburg.de> wrote:
> > And it is as easy to see that in set theory the set of indices left of
> > the decimal point
> > > >> > 0_2 1_1 .
> > > >> > 0_2 . 1_1
> > > >> > 0_4 1_3 0_2 . 1_1
> > > >> > 0_4 1_3 . 0_2 1_1
> > > >> > 0_6 1_5 0_4 1_3 . 0_2 1_1
> > > >> > 0_6 1_5 0_4 . 1_3 0_2 1_1
> > > >> > 0_8 1_7 0_6 1_5 0_4 . 1_3 0_2 1_1
> > > >> > 0_8 1_7 0_6 1_5 . 0_4 1_3 0_2 1_1
> > > >> > ...
> > has limit { }.
>
> While I see a point between other expressions , I do not see anything
> that could be properly interpreted as  a DECIMAL point.

This point is the decimal point.

>
> Or does WM regard such things as 1_3 and 1_5 as decimal digits?

The digit here is 1, the indexes are 3 and 5.

>
> And the limit of number of "positions"  would be infinite on either side
> of that point.

In mathematics this is true. In set theory it is false.

Regards, WM

[William Hughes]

Yes, The set of digits left of the decimal point is the
empty set. Simplest argument. Start with

100.000...
10.000...
1.000...
0.1000...
0.01000...
...


The 1 does not exist in the limit. This 1 corresponds to
the digit with index 5. We conclude that for
each index the digit corresponding to the digit does
not exist in the limit. Thus the set of digits in the limit
is the empty set. Thus, in the limit, the set of digits to
the left of the decimal point is the empty set.
Note we can write this the other way round

0.01000...
0.1000...
1.000...
10.000...
100.000...
...

The 1 does not exist in the limit (where could it be?)
So having the limit of the series be oo does not change things

[WM]

On 22 Nov., 16:27, William Hughes <wpihug...@gmail.com> wrote:
> On Nov 22, 3:30 am, WM <mueck...@rz.fh-augsburg.de> wrote:

> > Can we estimate by means of set theory how many digits *left* to the

> > decimal point will be present in the limit (as calculated by
> > analytical means) of the real sequence
>
> > > > 01.
> > > > 0.1
> > > > 010.1
> > > > 01.01
> > > > 0101.01
> > > > 010.101
> > > > 01010.101
> > > > 0101.0101
> > > > ...
>
> > ?

> Yes, The set of digits left of the decimal point is the
> empty set.

This is in contradiction to analysis (although analysis is said to be
based upon set theory). Just my point.

>  Simplest argument.  Start with
>
> 100.000...
> 10.000...
> 1.000...
> 0.1000...
> 0.01000...
> ...
>
> The 1 does not exist in the limit.  This 1 corresponds to
> the digit with index 5.  We conclude that for
> each index the digit corresponding to the digit does
> not exist in the limit.  Thus the set of digits in the limit
> is the empty set.  Thus, in the limit, the set of digits to
> the left of the decimal point is the empty set.

What has this problem to do with my question? I explicitly used
alternating sequences 010101... (moving from left to right, - so your
next example is completetly off topic). Analysis gives a result. Set
theory gives another result which is incompatible with analysis.

Regards, WM

[William Hughes]

It answers it.

<I explicitly used
> alternating sequences 010101...

And I deal with the simpler case first
and then with the more complicated alternating
digit case. Note

This 1 corresponds to
the digit with index 5.  We conclude that for

each index the digit corresponding to the [index] does

not exist in the limit.

This covers the alternating digit case and a lot of other
cases.

[WM]

No, yours is a much more difficult case. That's why I conceived the
simplest case.

Regards, WM

[William Hughes]

Note that I was able to handle your
"simple" case using induction.

I consider the following case easy.
If you disagree maybe you can say
why?


Consider the sequence of real numbers

1.0
10.0
100.0
...

The limit is oo (unbounded)

According to set theory, the number of 1's in the limit
is 0. (The limit of the set of positions at which we
have a 1 is the empty set).

Your contention: This is a contradiction. You cannot get oo
with only 0's

My contention: The two limits are different and there is
no contradiction.

[WM]

On 22 Nov., 20:22, William Hughes <wpihug...@gmail.com> wrote:
> Note that I was able to handle your
> "simple" case using induction.
>
> I consider the following case easy.
> If you disagree maybe you can say
> why?
>
> Consider the sequence of real numbers
>
> 1.0
> 10.0
> 100.0
> ...
>
> The limit is oo (unbounded)
>
> According to set theory, the number of 1's in the limit
> is 0.  (The limit of the set of positions at which we
> have a 1 is the empty set).

Why should the 1 disappear completely? But let's assume it.

According to analysis the number of 1's in the limit is 1 and the
number of zeros left to the point is infinite. Proof by failure: Try
to establish in analysis the limit oo without 1 by the sequence
0
00
000
...

>
> Your contention:  This is a contradiction.  You cannot get oo
>                   with only 0's

My contention is same as before: Obviously set theory cannot reproduce
the results of analysis.

>
> My contention:   The two limits are different and there is
>                  no contradiction.

You try to justify one error by another one. Set theory cannot
reproduce mathematics. That fact is not mended by constructing another
failure. But you see it better here

> > 01.
> > 0.1
> > 010.1
> > 01.01
> > 0101.01
> > 010.101
> > 01010.101
> > 0101.0101
> > ...

where set theory yields no digit and analysis yields inifinitely many
digits left to the point.

Taking the result of set theory seriously, we get a limit less than 1.
And in your example we can get rid of a 1 and have a limit oo with
only zeros. Do you think to defend set theory by that arguing? Should
really somebody take it seriously? And apply it anywhere? I don't
believe so.

Regards, WM

[William Hughes]

On Nov 22, 4:54 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 22 Nov., 20:22, William Hughes <wpihug...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > Note that I was able to handle your
> > "simple" case using induction.
>
> > I consider the following case easy.
> > If you disagree maybe you can say
> > why?
>
> > Consider the sequence of real numbers
>
> > 1.0
> > 10.0
> > 100.0
> > ...
>
> > The limit is oo (unbounded)
>
> > According to set theory, the number of 1's in the limit
> > is 0.  (The limit of the set of positions at which we
> > have a 1 is the empty set).
>
> Why should the 1 disappear completely?

The limit of the set of positions at which we

have a 1 is the empty set. If there is a 1 it has to
be a one without a position.

> But let's assume it.
>
> According to analysis the number of 1's in the limit is 1 and the
> number of zeros left to the point is infinite.

Nonsense. There is no such thing as a number with an
infinite number of zeros left of the decimal point.

According to analysis the sequence grows without bound.

[WM]

On 22 Nov., 22:09, William Hughes <wpihug...@gmail.com> wrote:
> On Nov 22, 4:54 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
>
>
>
>
> > On 22 Nov., 20:22, William Hughes <wpihug...@gmail.com> wrote:
>
> > > Note that I was able to handle your
> > > "simple" case using induction.
>
> > > I consider the following case easy.
> > > If you disagree maybe you can say
> > > why?
>
> > > Consider the sequence of real numbers
>
> > > 1.0
> > > 10.0
> > > 100.0
> > > ...
>
> > > The limit is oo (unbounded)
>
> > > According to set theory, the number of 1's in the limit
> > > is 0.  (The limit of the set of positions at which we
> > > have a 1 is the empty set).
>
> > Why should the 1 disappear completely?
>
> The limit of the set of positions at which we
> have a 1 is the empty set.  If there is a 1 it has to
> be a one without a position.

Iff infinity can be finished.

>
> > But let's assume it.
>
> > According to analysis the number of 1's in the limit is 1 and the
> > number of zeros left to the point is infinite.
>
> Nonsense.  There is no such thing as a number with an
> infinite number of zeros left of the decimal point.

Infinity is certainly not a number with a finite number of zeros left
of the decimal point. In analysis oo has more digits than any finite
number. Do you know the definition: For every n in |N : n < oo.
>
> According to analysis the sequence grows without bound.-

Correct. For every finite number of digits you find another digit.
That means there are more digits than any finite number of digits. But
the first one, namely the 1, certainly never disappears.
(And not even in set theory, because according to set theory there are
infinitely many *finite* positions in the limit. Now you try to rob
the 1 of its finite place? Do you think that the limit has more than
the aleph_0 finite indexes?)

Regards, WM

[William Hughes]

On Nov 22, 5:22 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 22 Nov., 22:09, William Hughes <wpihug...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > On Nov 22, 4:54 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > On 22 Nov., 20:22, William Hughes <wpihug...@gmail.com> wrote:
>
> > > > Note that I was able to handle your
> > > > "simple" case using induction.
>
> > > > I consider the following case easy.
> > > > If you disagree maybe you can say
> > > > why?
>
> > > > Consider the sequence of real numbers
>
> > > > 1.0
> > > > 10.0
> > > > 100.0
> > > > ...
>
> > > > The limit is oo (unbounded)
>
> > > > According to set theory, the number of 1's in the limit
> > > > is 0.  (The limit of the set of positions at which we
> > > > have a 1 is the empty set).
>
> > > Why should the 1 disappear completely?
>
> > The limit of the set of positions at which we
> > have a 1 is the empty set.  If there is a 1 it has to
> > be a one without a position.
>
> Iff infinity can be finished.

Nope. We can establish that each integer
has the property that it is not the index
of a position with a 1 without using
completed infinity.

P(n): n is not the index of a position with a 1.

P(1) is true.
If P(n) is true then P(n+1) is true.

For each natural number n, P(n) is true.

[Virgil]

In article
<05781c68-6213-47d1...@l18g2000vbv.googlegroups.com>,

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

> On 21 Nov., 21:14, William Hughes <wpihug...@gmail.com> wrote:
>
> >
> > > > > We cannot conclude from set theory that the digits on the right of the
> > > > > decimal point vanish.
> >
> > > > Yes we can.
> >
> > > No. Set theory does not destruct the set of all natural numbers. Set
> > > theory only shows that the set can be enumerated by the set of all
> > > even numbers, .i.e., all inidces disappear left and gather right.
> >
> > Absolute nonsense.  This is not even wrong.-
>
> The typical answer of a man who has not understood. But my explanation
> may really be a bit too hard for you. So let us stay on a lower level
> and simply continue:
> > > > Yes we can.
>
> Can we etsimate by means of set theory how many digits *left* to the
> decimal point will be present in the limit (as calculated by
> analytical means) of the real sequence

That question does not make any sense as written.
--

[Virgil]

In article
<76d8019f-77b8-4a3f...@u9g2000vbm.googlegroups.com>,

But if so, they are dealing with different questions.
--

[Virgil]

In article
<addf68fc-fef6-4d32...@e25g2000vbm.googlegroups.com>,

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

> No, yours is a much more difficult case. That's why I conceived the
> simplest case.

WM does not conceive at all that there are TWO SEPARATE SEQUENCES
involved, which quite properly behave differently.

The real number case involves a single strictly increasing and unbounded
sequence of distinct real numbers, which involves one form of limiting
process.

The set of digits case involves two disjoint sequences of sets of
digits, each of which involves an entirely different form of limiting
process than does the sequence of real numbers case.

That WM tries to conflate these different interpretations is, as usual,
something that only works in Wolkenmuekenheim.
--

[Virgil]

In article
<aeb6ee03-6e59-4a6e...@ez26g2000vbb.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:

> According to analysis

WM is not competent to speak either for analysis or for set theory,
at least anywhere outside of his Wolkenmuekenheim .
--

[Virgil]

In article
<bb0725b1-94b8-426e...@q5g2000vbp.googlegroups.com>,

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

> Infinity is certainly not a number with a finite number of zeros left
> of the decimal point. In analysis oo has more digits than any finite
> number. Do you know the definition: For every n in |N : n < oo.

Numbers do not have digits, only numerals, which are not numbers but
only names of numbers, have digits.

And "oo", whether regarded as a numeral or the name of a number, has no
digits at all.

Until WM can distinguish between numerals and numbers, he should stop
trying to dictate how they must behave.
--

[WM]

> For each natural number n, P(n) is true.-

Great! What about the digits in the sequence
1.
12.
123.
...

Of course, this sequence is a little bit more sophisticated than
yours. In order to avoid complications with numbers > 9 consider the
indices only. Here induction shows:
n is not the index of a position with a finite index.
According to set theory all digits vanish in the infinite.
Set theory is really a theory of spirit.
But analysis runs differently.
And say, why is the anti-diagonal determined in such a boring way? We
could make it vanish completely. Then Cantorian disciples would come
to rest.

Regards, WM

[WM]

On 23 Nov., 03:20, Virgil <vir...@ligriv.com> wrote:

> > Analysis gives a result. Set
> > theory gives another result which is incompatible with analysis.
>
> But if so, they are dealing with different questions.

They are not dealing with different questions but with one and the
same: What is the number of digits left to the point in the limit of
the infinite real sequence determining this limit uniquely.

> > 01.
> > 0.1
> > 010.1
> > 01.01
> > 0101.01
> > 010.101
> > 01010.101
> > 0101.0101
> > ...

They are applying different techniques. And they are obtaining
different results. In other branches of mathematics such a case would
be called a contradiction. Of course in set theory, we have Orwell's
cattle of sheep shouting down all other voices "There is no
contradiction, there is no contradiction, ..."

Regards, WM

[WM]

On 23 Nov., 03:33, Virgil <vir...@ligriv.com> wrote:
> In article

> <addf68fc-fef6-4d32-9962-158449bcb...@e25g2000vbm.googlegroups.com>,

>
>  WM <mueck...@rz.fh-augsburg.de> wrote:
> > No, yours is a much more difficult case. That's why I conceived the
> > simplest case.
>
> WM does not conceive at all that there are TWO SEPARATE SEQUENCES
> involved, which quite properly behave differently.
>
> The real number case involves a single strictly increasing and unbounded
> sequence of distinct real numbers, which involves one form of limiting
> process.
>
> The set of digits case involves two disjoint sequences of sets of
> digits, each of which

is required to establish and construct the real numbers of analysis.
Exactly the numerals, the bricks of the numbers are determined.
However, there are none, according to set theory.

If set theory could not do that job, what would its application be
good for at all?

Regards, WM

[Virgil]

In article
<56887e9f-687a-4adc...@f17g2000vbz.googlegroups.com>,

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

> On 23 Nov., 03:33, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <addf68fc-fef6-4d32-9962-158449bcb...@e25g2000vbm.googlegroups.com>,
> >
> > �WM <mueck...@rz.fh-augsburg.de> wrote:
> > > No, yours is a much more difficult case. That's why I conceived the
> > > simplest case.
> >
> > WM does not conceive at all that there are TWO SEPARATE SEQUENCES
> > involved, which quite properly behave differently.
> >
> > The real number case involves a single strictly increasing and unbounded
> > sequence of distinct real numbers, which involves one form of limiting
> > process.
> >
> > The set of digits case involves two disjoint sequences of sets of
> > digits, each of which
>
> is required to establish and construct the real numbers of analysis.

False! One can establish the existence and properties of the real number
field without ever relying on a single digit.

> Exactly the numerals, the bricks of the numbers are determined.
> However, there are none, according to set theory.

Set theories such as ZFC say nothing about the existence of numerals as
such, neither pro nor con. But there is nothing about numerals that is
incompatible wih any settheory that I know of.

That WM does not know this is another mark of his overall mathematical
incompetence.

> If set theory could not do that job, what would its application be
> good for at all?

Since WM can not do that job either, what is he good for at all?
--

[Virgil]

In article
<cb8c0679-6c2d-4f07...@k20g2000vbj.googlegroups.com>,

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

> On 23 Nov., 03:20, Virgil <vir...@ligriv.com> wrote:
>
> > > Analysis gives a result. Set
> > > theory gives another result which is incompatible with analysis.
> >
> > But if so, they are dealing with different questions.
>
> They are not dealing with different questions but with one and the
> same: What is the number of digits left to the point in the limit of
> the infinite real sequence determining this limit uniquely.
>
> > > 01.
> > > 0.1
> > > 010.1
> > > 01.01
> > > 0101.01
> > > 010.101
> > > 01010.101
> > > 0101.0101
> > > ...

The numbers of digit positions to the left of the points appears to be
given by the sequence 2, 1, 3, 2, 4, 3, 5, 4, ...
And that sequence has no limit at all. but is merely unbounded above.


>
--

[Virgil]

In article
<caaeacd6-40ef-4d48...@k6g2000vbr.googlegroups.com>,

Your "sequence" is not well defined, since there is no obvious successor
to 1232456789.
--

[WM]

On 23 Nov., 08:15, Virgil <vir...@ligriv.com> wrote:
> In article

> <56887e9f-687a-4adc-9860-5699c4e90...@f17g2000vbz.googlegroups.com>,

>
>
>
>
>
>  WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 23 Nov., 03:33, Virgil <vir...@ligriv.com> wrote:
> > > In article
> > > <addf68fc-fef6-4d32-9962-158449bcb...@e25g2000vbm.googlegroups.com>,
>
> > >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > No, yours is a much more difficult case. That's why I conceived the
> > > > simplest case.
>
> > > WM does not conceive at all that there are TWO SEPARATE SEQUENCES
> > > involved, which quite properly behave differently.
>
> > > The real number case involves a single strictly increasing and unbounded
> > > sequence of distinct real numbers, which involves one form of limiting
> > > process.
>
> > > The set of digits case involves two disjoint sequences of sets of
> > > digits, each of which
>
> > is required to establish and construct the real numbers of analysis.
>
> False! One can establish the existence and properties of the real number
> field without ever relying on a single digit.

But one need not do so. At least many people use decimal numbers.

> > Exactly the numerals, the bricks of the numbers are determined.
> > However, there are none, according to set theory.
>
> Set theories such as ZFC say nothing about the existence of numerals as
> such, neither pro nor con. But there is nothing about numerals that is
> incompatible wih any settheory that I know of.

There is no contradiction, there is no contradiction, there is no
contradiction, ...

Regards, WM

[WM]

On 23 Nov., 08:22, Virgil <vir...@ligriv.com> wrote:
> In article

> <cb8c0679-6c2d-4f07-8fcb-2144eaf39...@k20g2000vbj.googlegroups.com>,

That's why in analysis the number of digits is unbounded (and larger
than 0).

Regards, WM

[WM]

On 23 Nov., 08:25, Virgil <vir...@ligriv.com> wrote:
> In article

> <caaeacd6-40ef-4d48-8efc-c1932fe1f...@k6g2000vbr.googlegroups.com>,

That's the good news: The successor question is irrelevant. Matheology
fails with every possible successor. But of course,

Ther's no con-
tra-dic-tion!
Ther's no con-
tra-dic-tion!
Ther's no con-
tra-dic-tion!
...

Regards, WM

[William Hughes]

So to summarize:

Analysis:
limit in real numbers: unbounded
(oo in extended reals)

limit of set of 1's: not estimated

Set Theory
limit in real numbers: not estimated
limit of set of 1's: {}

Contradiction: None (limits are determined
by finite terms)

[WM]

On 23 Nov., 13:27, William Hughes <wpihug...@gmail.com> wrote:

> So to summarize:
>
> Analysis:
>    limit in real numbers: unbounded
>                          (oo in extended reals)
>
>    limit of set of 1's:  not estimated

Analysis says there are infinitely many 1's distinguished by their
positions which are an abbreviation for the exponents of 10.

Remember: 111 = 1*10^0 + 1*10^1 + 1*10^2
...111 = 1*10^0 + 1*10^1 + 1*10^2 + ...
This is a sum over all natural numbers (and 0). Therefore the
positions with 1's in the decimal representation are infinitely many
(or aleph_0).

If this was disputed, then you could also dispute that the cardinality
of |N is "estimated" as aleph_0.

>
> Set Theory
>    limit in real numbers: not estimated
>    limit of set of 1's: {}
>

Contradiction:  One - and that is sufficient. But if required I could
present a lot more.

Regrads, WM

[William Hughes]

On Nov 23, 10:11 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 23 Nov., 13:27, William Hughes <wpihug...@gmail.com> wrote:
>
> > So to summarize:
>
> > Analysis:
> >    limit in real numbers: unbounded
> >                          (oo in extended reals)
>
> >    limit of set of 1's:  not estimated
>

<snip stuff not about the limit>

[WM]

On 23 Nov., 15:16, William Hughes <wpihug...@gmail.com> wrote:
> On Nov 23, 10:11 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > On 23 Nov., 13:27, William Hughes <wpihug...@gmail.com> wrote:
>
> > > So to summarize:
>
> > > Analysis:
> > >    limit in real numbers: unbounded
> > >                          (oo in extended reals)
>
> > >    limit of set of 1's:

oo = Limit[n-->oo] SUM[k=0 to n] 10^k
= 1*10^0 + 1*10^1 + 1*10^2 + 1*10^3 + ...
= ...111 =

So much about the limit.

And here something you may have missed this morning. You invtroduced
the sequence
1
10
100
...
But it seems you are no longer interested in your argument?

> P(n): n is not the index of a position with a 1.

> P(1) is true.
> If P(n) is true then P(n+1) is true.

> For each natural number n, P(n) is true.-

Great! What about the digits in the sequence
1.
12.
123.
...

In order to avoid complications with numbers > 9 consider the
indices only. Here induction shows:
n is not the index of a position with a finite index.
According to set theory all digits vanish in the infinite.
Set theory is really a theory of spirit.

But analysis runs differently (see above).
And say, why is the anti-diagonal yet determined in such a boring way?

[William Hughes]

On Nov 23, 10:32 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 23 Nov., 15:16, William Hughes <wpihug...@gmail.com> wrote:
>
> > On Nov 23, 10:11 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > On 23 Nov., 13:27, William Hughes <wpihug...@gmail.com> wrote:
>
> > > > So to summarize:
>
> > > > Analysis:
> > > >    limit in real numbers: unbounded
> > > >                          (oo in extended reals)
>
> > > >    limit of set of 1's:
>
> oo =  Limit[n-->oo] SUM[k=0 to n] 10^k
>      = 1*10^0 + 1*10^1 + 1*10^2 + 1*10^3 + ...
>      = ...111 =
>

this piece of nonsense has nothing to do
with the limit of the sequence

1
10
100
...

which is oo = Limit[n-->oo] 10^k
and it not represented by any numeral.

[WM]

On 23 Nov., 15:42, William Hughes <wpihug...@gmail.com> wrote:
> On Nov 23, 10:32 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
>
>
>
>
> > On 23 Nov., 15:16, William Hughes <wpihug...@gmail.com> wrote:
>
> > > On Nov 23, 10:11 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > > On 23 Nov., 13:27, William Hughes <wpihug...@gmail.com> wrote:
>
> > > > > So to summarize:
>
> > > > > Analysis:
> > > > >    limit in real numbers: unbounded
> > > > >                          (oo in extended reals)
>
> > > > >    limit of set of 1's:
>
> > oo =  Limit[n-->oo] SUM[k=0 to n] 10^k
> >      = 1*10^0 + 1*10^1 + 1*10^2 + 1*10^3 + ...
> >      = ...111
>

> this piece of nonsense has nothing to do
> with the limit of the sequence
>
> 1
> 10
> 100
> ...
>
> which is oo = Limit[n-->oo] 10^k
> and it not represented by any numeral.

That is of little interest! You want to excuse one mistake by another
one.

The sequence
1
11
111
...
with its limit

oo = Limit[n-->oo] SUM[k=0 to n] 10^k
= 1*10^0 + 1*10^1 + 1*10^2 + 1*10^3 + ...
= ...111

is represented by digits. So is my original sequence. The number of
digits is oo when calculated by analysis, but 0 when calculated by set
theory. This is one contradiction. And in mathematics one
contradiction is sufficient to run a proof by contradiction.

Regards, WM

[William Hughes]

On Nov 23, 11:56 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 23 Nov., 15:42, William Hughes <wpihug...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > On Nov 23, 10:32 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > On 23 Nov., 15:16, William Hughes <wpihug...@gmail.com> wrote:
>
> > > > On Nov 23, 10:11 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > > > On 23 Nov., 13:27, William Hughes <wpihug...@gmail.com> wrote:
>
> > > > > > So to summarize:
>
> > > > > > Analysis:
> > > > > >    limit in real numbers: unbounded
> > > > > >                          (oo in extended reals)
>
> > > > > >    limit of set of 1's:
>
>
> > > oo =  Limit[n-->oo] SUM[k=0 to n] 10^k
> > >      = 1*10^0 + 1*10^1 + 1*10^2 + 1*10^3 + ...
> > >      = ...111
>
> > this piece of nonsense has nothing to do
> > with the limit of the sequence
>
> > 1
> > 10
> > 100
> > ...
>
> > which is oo = Limit[n-->oo] 10^k
> > and it not represented by any numeral.
>
> That is of little interest!

On the contrary, the fact that the analytic *limit*
cannot be described in terms of digits is
the point.

[WM]

> the point.-

No, that is not a point. The analytic limit can be calculated.
Analysis infers from the limit the number of required digits, namely
oo. This is all correct. Set theory cannot describe the limit and not
describe the number of digits.

That is the point!

Regards, WM

[WM]

A further explanation may be appropriate: In your example set theory
is not in contradiction with analysis. Both theories prove the
existence of infinitely many digits and both cannot fix the position
of the leading 1.

My example shows a contradiction. Here only analysis proves the
existence of infinitely many digits left to the point while set theory
proves no digit left.

That's why your example is completely off topic.

Regards, WM

[William Hughes]

Yes

> Analysis infers from the limit the number of required digits,

Piffle.

[WM]

On 23 Nov., 18:30, William Hughes <wpihug...@gmail.com> wrote:

> > > On the contrary, the fact that the analytic *limit*
> > > cannot be described in terms of digits is
> > > the point.-
>
> > No, that is not a point. The analytic limit can be calculated.
> Yes
> > Analysis infers from the limit the number of required digits,
>

> Piffle.-
Look here: Limit[n-->oo] SUM[k=0 to n] a_k*10^k = oo.
The set { a_k | k in |N } has cardinality aleph_0, according to
analysis, for every infinite sequence with limit oo.

Regards, WM

[William Hughes]

On Nov 23, 1:54 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 23 Nov., 18:30, William Hughes <wpihug...@gmail.com> wrote:
>
> > > > On the contrary, the fact that the analytic *limit*
> > > > cannot be described in terms of digits is
> > > > the point.-
>
> > > No, that is not a point. The analytic limit can be calculated.
> > Yes
> > > Analysis infers from the limit the number of required digits,
>
> > Piffle.-

Nope adding irrelevant stuff does not help.
It's still Piffle to say

[WM]

On 23 Nov., 19:12, William Hughes <wpihug...@gmail.com> wrote:
> On Nov 23, 1:54 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > On 23 Nov., 18:30, William Hughes <wpihug...@gmail.com> wrote:
>
> > > > > On the contrary, the fact that the analytic *limit*
> > > > > cannot be described in terms of digits is
> > > > > the point.-
>
> > > > No, that is not a point. The analytic limit can be calculated.
> > > Yes
> > > > Analysis infers from the limit the number of required digits,
>
> > > Piffle.-
>
> Nope adding irrelevant stuff does not help.

Why then did you do so with your sequence 1, 10, 100, ...?

> It's still Piffle to say
>
>      Analysis infers from the limit the number of required digits

It is obvious that you don't like analysis. Try to understand the
function called logarithm with base 10. The number of digits is [lgx]
+ 1.

Regards, WM

[William Hughes]

Saying oo has an infinite number of digits is nonsense
(even though saying log_10(oo)+1 = oo is not nonsense).
Your claim that oo requires digits is Piffle.

[WM]

On 23 Nov., 19:51, William Hughes <wpihug...@gmail.com> wrote:
> On Nov 23, 2:23 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > On 23 Nov., 19:12, William Hughes <wpihug...@gmail.com> wrote:
> > > It's still Piffle to say
>
> > >      Analysis infers from the limit the number of required digits
>
> > It is obvious that you don't like analysis. Try to understand the
> > function called logarithm with base 10. The number of digits is [lgx]
> > + 1.
>
> Saying oo has an infinite number of digits is nonsense

It is not nonsense since set theory has "improved" analysis.

> (even though saying log_10(oo)+1 = oo is not nonsense).

Correct.
And moreover we know that [lgx] + 1 gives the number of digits of x.

Limit[n-->oo] SUM[k=0 to n] a_k*10^k = oo.

The set { a_k | k in |N } has cardinality aleph_0.

> Your claim that oo requires digits is Piffle.

I do not say that oo requires digits. But oo can be expressed by
digits. And to say that there are aleph_0 digits required is not
nonsense as long as there are people who believe in the existence of
infinitely many elements of |N.

But may you believe it or not: Can you imagine to admit a
contradiction of set theory and mathematics, if it could be proven
that analysis shows the existence of infinitely many digits left to
the point in the limit of my sequence?

Regards, WM

[William Hughes]

On Nov 23, 3:38 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 23 Nov., 19:51, William Hughes <wpihug...@gmail.com> wrote:
>

<snip>

> > Saying oo has an infinite number of digits is nonsense
>
> It is not nonsense

Well, looks like communication
has come to a halt.

You are correct, if analysis requires
that oo be described by digits
then there is a contradiction in mathematics.
I see no way of convincing you that oo
cannot be expressed by digits, nor have
you given any evidence other
that vague handwaving that it can.

[Virgil]

In article
<db19ad29-d6fd-4adc...@w1g2000vbx.googlegroups.com>,

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

> On 23 Nov., 13:27, William Hughes <wpihug...@gmail.com> wrote:
>
> > So to summarize:
> >
> > Analysis:
> >    limit in real numbers: unbounded
> >                          (oo in extended reals)
> >
> >    limit of set of 1's:  not estimated
>
> Analysis says there are infinitely many 1's distinguished by their
> positions which are an abbreviation for the exponents of 10.

How can WM claim that analysis says there are infinitely many 1's when,
at least in WM's eyes, analysis does not accept that there can be
infinitely many of anything.
>
--

[Virgil]

In article
<97bb704d-eccc-4f22...@y6g2000vbb.googlegroups.com>,

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

> According to set theory all digits vanish in the infinite.

It is not that things vanish, it is whether they end up as members of a
given set or its compliment.

And if every member of a set is removed from that set, it ends up almost
as empty as WM's head.
--

[Virgil]

In article
<e976b2c0-c990-414f...@n5g2000vbk.googlegroups.com>,

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

> >
> > which is oo = Limit[n-->oo] 10^k
> > and it not represented by any numeral.
>
> That is of little interest! You want to excuse one mistake by another
> one.

It is of interest to every one who, unlike WM, can think straight about
things.

>
> The sequence
> 1
> 11
> 111
> ...
> with its limit
> oo = Limit[n-->oo] SUM[k=0 to n] 10^k
> = 1*10^0 + 1*10^1 + 1*10^2 + 1*10^3 + ...
> = ...111
> is represented by digits. So is my original sequence.

Your original "sequence" eventually replaces each non-zero digit to the
left of the radix point with a zero digit, which then remains in that
digit position thereafter, so WM's sequence ends up with all zeros to
the left of the radix point.

If WM disputes this, let him identify any digit position for which this
does not happen as I have described!

> The number of
> digits is oo when calculated by analysis, but 0 when calculated by set
> theory.

The number of non-zero digits to the left of a radix point in a limit
requiring "infinitely many digits" cannot be calculated by analysis, as
the result in analysis is not a number, and thus as a number cannot
exist at all.

> This is one contradiction. And in mathematics one
> contradiction is sufficient to run a proof by contradiction.

"RUN"?


WM makes the most interesting spelling errors!

Or maybe that is just the way he misunderstands proofs.
--

[Virgil]

In article
<d4e6c3c7-2e14-4ae2...@ez26g2000vbb.googlegroups.com>,

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

> On 23 Nov., 17:09, William Hughes <wpihug...@gmail.com> wrote:
> > On Nov 23, 11:56 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > On the contrary, the fact that the analytic *limit*
> > cannot be described in terms of digits is
> > the point.
>
> A further explanation may be appropriate: In your example set theory
> is not in contradiction with analysis. Both theories prove the
> existence of infinitely many digits and both cannot fix the position
> of the leading 1.

As usual, WM is misrepresenting things.

Both true set theory and true analysis sow that there is no such thing
as a "leading 1" for any limit of the sequencing described.

>
> My example shows a contradiction.

WM's examples almost always contradict standard mathematics, but why WM
should think that that is a virtue is unknown.
--

[Virgil]

In article
<2ceaeaba-fad1-4520...@u9g2000vbm.googlegroups.com>,

Agreed, but that it can be expressed as a real number seems to be a
delusion that WM cannot shake. Even if one extends the reals by a one or
two point compactification so that there is a value in the extended
reals, that value is not representable as a digit string.

> Analysis infers from the limit the number of required digits, namely
> oo.

Which 'number' of digits preceding a radix point is not allowed.

So WM is off daydreaming in his Wolkenmuekenheim again!
--

[WM]

On 23 Nov., 20:54, William Hughes <wpihug...@gmail.com> wrote:

> Well, looks like communication
> has come to a halt.
>
> You are correct, if analysis requires
> that oo be described by digits
> then there is a contradiction in mathematics.
> I see no way of convincing you that oo
> cannot be expressed by digits, nor have
> you given any evidence other
> that vague handwaving that it can.

Clear mathematical formulas are not handwaving.

An analytical expression of infinity is this

Limit[n-->oo] SUM[k=0 to n] a_k*10^k = oo

An expression by digits means nothing but leaving the powers of 10
aside and writing from right to left, i.e., the abbreviation
..., a_k, ..., a_3, a_2, a_1, a_0
where for every k there exists a digit
a_(k+m) =/= 0, with m in |N.

This gives abbreviations like oo = ...111 = ...999
However, like sin(0) = sin(2pi) = sin(4pi) = 0 = 0+0
there is no problem with multi representation.

in any case the limit of my sequence

> > 01.
> > 0.1
> > 010.1
> > 01.01
> > 0101.01
> > 010.101
> > 01010.101
> > 0101.0101
> > ...

has infinitely many digits right to the point as well as left to the
point.

So it is incorrect to say that communication has come to a halt but it
is correct to say that you have no further arguments to defend your
position but do not want to admit that for reasons that I don't know.

Regards, WM

[Virgil]

In article
<9e3efe24-bbc4-4664...@w7g2000vbb.googlegroups.com>,

Then WM must be claiming that oo can be expressed as a sequence of
digits.

Which I will not believe until I see it done,
and done WITHOUT abbreviation or ellipsis.
--

[Virgil]

In article
<fb41db64-83af-4ce4...@ib4g2000vbb.googlegroups.com>,

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

> On 23 Nov., 19:12, William Hughes <wpihug...@gmail.com> wrote:
> > On Nov 23, 1:54 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > On 23 Nov., 18:30, William Hughes <wpihug...@gmail.com> wrote:
> >
> > > > > > On the contrary, the fact that the analytic *limit*
> > > > > > cannot be described in terms of digits is
> > > > > > the point.-
> >
> > > > > No, that is not a point. The analytic limit can be calculated.
> > > > Yes
> > > > > Analysis infers from the limit the number of required digits,
> >
> > > > Piffle.-
> >
> > Nope adding irrelevant stuff does not help.
>
> Why then did you do so with your sequence 1, 10, 100, ...?

To show you why!

>
> > It's still Piffle to say
> >
> >      Analysis infers from the limit the number of required digits

That analysis infers an impossibility, unless WM can actually show us an
unabbreviated string of the required numbers of digits.

>
> It is obvious that you don't like analysis.

We like it is its original form a good deal better than in the corrupted
form WM keeps trying to sell to us.
--

[WM]

On 23 Nov., 22:19, WM <mueck...@rz.fh-augsburg.de> wrote:

> in any case the limit of my sequence
> > > 01.
> > > 0.1
> > > 010.1
> > > 01.01
> > > 0101.01
> > > 010.101
> > > 01010.101
> > > 0101.0101
> > > ...
>
> has infinitely many digits right to the point as well as left to the
> point.

This can be proved by attaching pairs 01 of digits always from the
left and from the right side in a symmetrical way:

01.
01.01
0101.01
0101.0101
...

The overall behaviour of the digits (not of the indices) is the same
as that of the original sequence.

Regards, WM

[WM]

On 23 Nov., 22:27, Virgil <vir...@ligriv.com> wrote:

> That analysis infers an impossibility, unless WM can actually show us an
> unabbreviated string of the required numbers of digits.

Take the unabbreviated string of digits of 1/9 = 0.111... which,
according to your matheology, does exist, and reflect it at the
decimal point.

Regards, WM

[Virgil]

In article
<cc2e0bbd-0188-47d8...@n5g2000vbk.googlegroups.com>,

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

> On 23 Nov., 19:51, William Hughes <wpihug...@gmail.com> wrote:
> > On Nov 23, 2:23 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > On 23 Nov., 19:12, William Hughes <wpihug...@gmail.com> wrote:
> > > > It's still Piffle to say
> >
> > > >      Analysis infers from the limit the number of required digits
> >
> > > It is obvious that you don't like analysis. Try to understand the
> > > function called logarithm with base 10. The number of digits is [lgx]
> > > + 1.
> >
> > Saying oo has an infinite number of digits is nonsense
>
> It is not nonsense since set theory has "improved" analysis.

Set theory has certainly not corrupted analysis to anywhere nearly the
extent that WM is trying to corrupt it.

>
> > (even though saying log_10(oo)+1 = oo is not nonsense).
>
> Correct.
> And moreover we know that [lgx] + 1 gives the number of digits of x.

Then, according to WM, ln(-1) + 1 = 1 + pi*i must be the number of
digits in -1.

>
> But may you believe it or not: Can you imagine to admit a
> contradiction of set theory and mathematics, if it could be proven
> that analysis shows the existence of infinitely many digits left to
> the point in the limit of my sequence?

Since analysis, properly done, shows no such thing, your question is
moot.

Analysis can show that the limit VALUE is oo in the extended reals, but
does not presume to claim that there is a decimal, or any other place
value based numeral, representing that limit value.

At least outside of Wolkenmuekenheim!
--

[Virgil]

In article
<7e92096f-b31a-41fd...@o30g2000vbu.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:

> On 23 Nov., 08:25, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <caaeacd6-40ef-4d48-8efc-c1932fe1f...@k6g2000vbr.googlegroups.com>,
> >
> >
> >
> >
> >
> >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 22 Nov., 22:38, William Hughes <wpihug...@gmail.com> wrote:
> > > > On Nov 22, 5:22 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > > > On 22 Nov., 22:09, William Hughes <wpihug...@gmail.com> wrote:
> >
> > > > > > On Nov 22, 4:54 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > > > > > On 22 Nov., 20:22, William Hughes <wpihug...@gmail.com> wrote:
> >
> > > > > > > > Note that I was able to handle your
> > > > > > > > "simple" case using induction.
> >
> > > > > > > > I consider the following case easy.
> > > > > > > > If you disagree maybe you can say
> > > > > > > > why?
> >
> > > > > > > > Consider the sequence of real numbers
> >
> > > > > > > > 1.0
> > > > > > > > 10.0
> > > > > > > > 100.0
> > > > > > > > ...
> >
> > > > > > > > The limit is oo (unbounded)
> >
> > > > > > > > According to set theory, the number of 1's in the limit
> > > > > > > > is 0. (The limit of the set of positions at which we
> > > > > > > > have a 1 is the empty set).
> >
> > > > > > > Why should the 1 disappear completely?
> >
> > > > > > The limit of the set of positions at which we
> > > > > > have a 1 is the empty set. If there is a 1 it has to
> > > > > > be a one without a position.
> >
> > > > > Iff infinity can be finished.
> >
> > > > Nope. We can establish that each integer
> > > > has the property that it is not the index
> > > > of a position with a 1 without using
> > > > completed infinity.

> >
> > > > P(n): n is not the index of a position with a 1.
> >
> > > > P(1) is true.
> > > > If P(n) is true then P(n+1) is true.
> >
> > > > For each natural number n, P(n) is true.-
> >
> > > Great! What about the digits in the sequence
> > > 1.
> > > 12.
> > > 123.
> > > ...
> >

> > Your "sequence" is not well defined, since there is no obvious successor
> > to 1232456789.
>
> That's the good news: The successor question is irrelevant. Matheology
> fails with every possible successor. But of course,
>
> Ther's no con-
> tra-dic-tion!
> Ther's no con-
> tra-dic-tion!
> Ther's no con-
> tra-dic-tion!
> ...
Other than in your Wolkenmuekenheim, quite true!
--

[Virgil]

In article
<585c28bc-8437-412d...@p17g2000vbn.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:

> On 23 Nov., 08:22, Virgil <vir...@ligriv.com> wrote:
> > In article

> > <cb8c0679-6c2d-4f07-8fcb-2144eaf39...@k20g2000vbj.googlegroups.com>,
> >
> >
> >
> >
> >
> >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 23 Nov., 03:20, Virgil <vir...@ligriv.com> wrote:
> >
> > > > > Analysis gives a result. Set
> > > > > theory gives another result which is incompatible with analysis.
> >
> > > > But if so, they are dealing with different questions.
> >
> > > They are not dealing with different questions but with one and the
> > > same: What is the number of digits left to the point in the limit of
> > > the infinite real sequence determining this limit uniquely.

> >
> > > > > 01.
> > > > > 0.1
> > > > > 010.1
> > > > > 01.01
> > > > > 0101.01
> > > > > 010.101
> > > > > 01010.101
> > > > > 0101.0101
> > > > > ...
> >

> > The numbers of digit positions to the left of the points appears to be
> > given by the sequence 2, 1,  3, 2,  4, 3,  5, 4, ...
> > And that sequence has no limit at all. but is merely unbounded above.
>
> That's why in analysis the number of digits is unbounded (and larger
> than 0).

The trouble is that there is no such decimal numeral, or other radix
based numeral, which is unbounded. Every such numeral must have a first
digit and only finitely many preceding its radix point.

At least in standard mathematics.

What WM may allow to go on in his Wolkenmuekenheim has no effect on what
is allowed in standard mathematics.
--

[Virgil]

In article
<54a009e6-07fd-48d1...@u9g2000vbm.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:

> On 23 Nov., 08:15, Virgil <vir...@ligriv.com> wrote:
> > In article

> > <56887e9f-687a-4adc-9860-5699c4e90...@f17g2000vbz.googlegroups.com>,
> >
> >
> >
> >
> >
> >  WM <mueck...@rz.fh-augsburg.de> wrote:

> > > On 23 Nov., 03:33, Virgil <vir...@ligriv.com> wrote:
> > > > In article

> > > > <addf68fc-fef6-4d32-9962-158449bcb...@e25g2000vbm.googlegroups.com>,
> >
> > > >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > > No, yours is a much more difficult case. That's why I conceived the
> > > > > simplest case.
> >
> > > > WM does not conceive at all that there are TWO SEPARATE SEQUENCES
> > > > involved, which quite properly behave differently.
> >
> > > > The real number case involves a single strictly increasing and unbounded
> > > > sequence of distinct real numbers, which involves one form of limiting
> > > > process.
> >
> > > > The set of digits case involves two disjoint sequences of sets of
> > > > digits, each of which
> >
> > > is required to establish and construct the real numbers of analysis.
> >
> > False! One can establish the existence and properties of the real number
> > field without ever relying on a single digit.
>
> But one need not do so. At least many people use decimal numbers.

But if one can do without digits, then nothing of the reals is dependent
on them.
>
> > > Exactly the numerals, the bricks of the numbers are determined.
> > > However, there are none, according to set theory.
> >
> > Set theories such as ZFC say nothing about the existence of numerals as
> > such, neither pro nor con. But there is nothing about numerals that is
> > incompatible wih any settheory that I know of.
>
> There is no contradiction, there is no contradiction, there is no
> contradiction, ...


If there really were any such contradictions, would someone as
demonstrably incompetent as WM be able to find it?
Only by sheer luck, and WM isn't that lucky!

And even if WM were lucky enough to find one,
could he prove it?
No chance at all!
--

[Virgil]

In article
<54b322c4-2fd4-4ac6...@d9g2000vbe.googlegroups.com>,

What you would get, if you could, is NAN!

Now a process or sequence, once started, may proceed ndefnitely. but a
process or sequence which has not started and cannot by its nature ever
get started does not exist.

At least outside of Wolkenmuekenheim.
--

[Virgil]

In article
<196a9e2e-96c9-4814...@m13g2000vbd.googlegroups.com>,

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

> On 23 Nov., 20:54, William Hughes <wpihug...@gmail.com> wrote:
>
> > Well, looks like communication
> > has come to a halt.
> >
> > You are correct, if analysis requires
> > that oo be described by digits
> > then there is a contradiction in mathematics.
> > I see no way of convincing you that oo
> > cannot be expressed by digits, nor have
> > you given any evidence other
> > that vague handwaving that it can.
>
> Clear mathematical formulas are not handwaving.

I have yet to see any "clear mathematical formulas" which can explicitly
display a sequence that has no beginning.

>
> An analytical expression of infinity is this
> Limit[n-->oo] SUM[k=0 to n] a_k*10^k = oo

Not as it stands! You would have to first establish that at least
infinitely many of those a_k's are to be strictly positive.

>
> in any case the limit of my sequence
> > > 01.
> > > 0.1
> > > 010.1
> > > 01.01
> > > 0101.01
> > > 010.101
> > > 01010.101
> > > 0101.0101
> > > ...
> has infinitely many digits right to the point as well as left to the
> point.

If the limit of your sequence were a real number then it would NOT have
any such representation, and only real numbers do have such any basal
representations

>
> So it is incorrect to say that communication has come to a halt but it
> is correct to say that you have no further arguments to defend your
> position but do not want to admit that for reasons that I don't know.

Those reasons have been quite clearly and repeatedly represented.

That WM remains ignorant of them, or at least clams to, is only by his
own choice, rather than by any necessity.
--

[Virgil]

In article
<49b3cb68-22ab-423a...@w1g2000vbx.googlegroups.com>,

The sequence
.01
.0101
.010101
.01010101
...
has limit 1/99, which is finitely expressible both as a repeating
decimal and as a rational number, but that does not mean that either

01.
01.01
0101.01
0101.0101
...

or

01.
0.1
010.1
01.01
0101.01
010.101
01010.101
0101.0101
...

has any limit expressible in the positional notation of decimals, or any
other such base.
--

[Musatov]

On Nov 23, 2:44 pm, Virgil <vir...@ligriv.com> wrote:
> In article

> <49b3cb68-22ab-423a-ac87-048870090...@w1g2000vbx.googlegroups.com>,

anno

[WM]

On 23 Nov., 23:32, Virgil <vir...@ligriv.com> wrote:
> In article

> <196a9e2e-96c9-4814-a3e6-4c2fd795c...@m13g2000vbd.googlegroups.com>,

>
>  WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > Clear mathematical formulas are not handwaving.
>
> I have yet to see any "clear mathematical formulas" which can explicitly
> display a sequence that has no beginning.

The sequence has no ending. In decimal represenation, i.e.,
representing only the coefficients, we write from right to left.

>
>
> > An analytical expression of infinity is this
> > Limit[n-->oo] SUM[k=0 to n] a_k*10^k = oo
>
> Not as it stands! You would have to first establish that at least
> infinitely many of those a_k's are to be strictly positive.

I did that. I wrote: ..., a_k, ..., a_3, a_2, a_1, a_0

where for every k there exists a digit
a_(k+m) =/= 0, with m in |N.

>
>
>

> > in any case the limit of my sequence
> > > > 01.
> > > > 0.1
> > > > 010.1
> > > > 01.01
> > > > 0101.01
> > > > 010.101
> > > > 01010.101
> > > > 0101.0101
> > > > ...
> > has infinitely many digits right to the point as well as left to the
> > point.
>
> If the limit of your sequence were a  real number then it would NOT have
> any such representation, and only real numbers do have such any basal
> representations

The limit is not a real number, but an element of the extended reals.
Obviously then also the representation has to be extended from a
finite number of digits to an infinite number.

Regards, WM

[WM]

On 23 Nov., 22:37, Virgil <vir...@ligriv.com> wrote:

> Analysis can show that the limit VALUE is oo in the extended reals, but
> does not presume to claim that there is a decimal, or any other place
> value based numeral, representing that limit value.

If the limit value

Limit[n-->oo] SUM[k=0 to n] a_k*10^k = oo

is accepted in the extended reals, then it is simply ridiculous to
claim that the abbreviation

..., a_k, ..., a_3, a_2, a_1, a_0

is not in the abbreviations of the extended reals.

But William had agreed: "On the contrary, the fact that the analytic

*limit* cannot be described in terms of digits is the point."

And he stated proudly:

Analysis:
limit in real numbers: unbounded
(oo in extended reals)

limit of set of 1's: not estimated

Set Theory
limit in real numbers: not estimated
limit of set of 1's: {}

Therefore he would have to confess now that there is a contradiction
between set theory and analysis. On the other hand we know the first
commandment of matheology:

There's no con-
tra-dic-tion!
There's no con-
tra-dic-tion!
There's no con-
tra-dic-tion!
...

Regards, WM

[Virgil]

In article
<0ee396a8-91e7-4221...@o30g2000vbu.googlegroups.com>,

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

> On 23 Nov., 22:37, Virgil <vir...@ligriv.com> wrote:
>
> > Analysis can show that the limit VALUE is oo in the extended reals, but
> > does not presume to claim that there is a decimal, or any other place
> > value based numeral, representing that limit value.
>
> If the limit value
> Limit[n-->oo] SUM[k=0 to n] a_k*10^k = oo
> is accepted in the extended reals, then it is simply ridiculous to
> claim that the abbreviation
> ..., a_k, ..., a_3, a_2, a_1, a_0
> is not in the abbreviations of the extended reals.

The only such "abbreviations" in standard use for "extended reals are
oo for the one point compactification or +oo and -oo for the two point
compactification. Anything else exists only in Wolkenmuekenheim.

>
> But William had agreed: "On the contrary, the fact that the analytic
> *limit* cannot be described in terms of digits is the point."
>
> And he stated proudly:
>
> Analysis:
> limit in real numbers: unbounded
> (oo in extended reals)
> limit of set of 1's: not estimated
>
> Set Theory
> limit in real numbers: not estimated
> limit of set of 1's: {}
>
> Therefore he would have to confess now that there is a contradiction
> between set theory and analysis.

To say that different definitions of "limit" can give different results
only confuses WM, not anyone else.

But confusion is the norm in WMytheology.

> On the other hand we know the first commandment of

WMytheology

>
> There's no con-
> tra-dic-tion!
> There's no con-
> tra-dic-tion!
> There's no con-
> tra-dic-tion!
> ...
>
> Regards, WM

--

[Virgil]

In article
<7b365118-1b3d-4d1a...@q5g2000vbp.googlegroups.com>,

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

> > > in any case the limit of my sequence
> > > > > 01.
> > > > > 0.1
> > > > > 010.1
> > > > > 01.01
> > > > > 0101.01
> > > > > 010.101
> > > > > 01010.101
> > > > > 0101.0101
> > > > > ...
> > > has infinitely many digits right to the point as well as left to the
> > > point.
> >
> > If the limit of your sequence were a  real number then it would NOT have
> > any such representation, and only real numbers do have such any basal
> > representations
>
> The limit is not a real number, but an element of the extended reals.

The only standard representations of any additional members of the
extended reals, over and above the members of the standard reals, at
least outside of Wolkenmuekenheim, are oo, +oo and -oo.

> Obviously then also the representation has to be extended from a
> finite number of digits to an infinite number.

What is "obvious" in Wolkenmuekenheim is often not even true elsewhere.
--

[WM]

On 24 Nov., 21:34, Virgil <vir...@ligriv.com> wrote:
> In article

> <0ee396a8-91e7-4221-82e6-bed81af08...@o30g2000vbu.googlegroups.com>,

>
>  WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 23 Nov., 22:37, Virgil <vir...@ligriv.com> wrote:
>
> > > Analysis can show that the limit VALUE is oo in the extended reals, but
> > > does not presume to claim that there is a decimal, or any other place
> > > value based numeral, representing that limit value.
>
> > If the limit value
> > Limit[n-->oo] SUM[k=0 to n] a_k*10^k = oo
> > is accepted in the extended reals, then it is simply ridiculous to
> > claim that the abbreviation
> > ..., a_k, ..., a_3, a_2, a_1, a_0
> > is not in the abbreviations of the extended reals.
>
> The only such "abbreviations" in standard use for "extended reals  are
> oo for the one point compactification or +oo and -oo for the two point
> compactification.

My proof does not need this abbreviation. My proof only needs the
facts
that the set { a_k | k in |N } of coefficients of the series is not
empty and the number |{ a_k | k in |N }| of coefficients of the
series is not zero. More is not required.

>
>
>
>
>
>
>
> > But William had agreed: "On the contrary, the fact that the analytic
> > *limit* cannot be described in terms of digits is the point."
>
> > And he stated proudly:
>
> > Analysis:
> >    limit in real numbers: unbounded
> >                          (oo in extended reals)
> >    limit of set of 1's:  not estimated
>
> > Set Theory
> >    limit in real numbers: not estimated
> >    limit of set of 1's: {}
>
> > Therefore he would have to confess now that there is a contradiction
> > between set theory and analysis.
>
> To say that different definitions of "limit" can give different results
> only confuses WM, not anyone else.

The limit of a sequence is defined solely by the finite terms of the
sequence. Mathematics is the art of finding this limit, not the art of
inventing arbitrary limits.

Regards, WM

[Virgil]

In article
<4cc6c866-5b62-4fe4...@bq2g2000vbb.googlegroups.com>,

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

> On 24 Nov., 21:34, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <0ee396a8-91e7-4221-82e6-bed81af08...@o30g2000vbu.googlegroups.com>,
> >
> >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 23 Nov., 22:37, Virgil <vir...@ligriv.com> wrote:
> >
> > > > Analysis can show that the limit VALUE is oo in the extended reals, but
> > > > does not presume to claim that there is a decimal, or any other place
> > > > value based numeral, representing that limit value.
> >
> > > If the limit value
> > > Limit[n-->oo] SUM[k=0 to n] a_k*10^k = oo
> > > is accepted in the extended reals, then it is simply ridiculous to
> > > claim that the abbreviation
> > > ..., a_k, ..., a_3, a_2, a_1, a_0
> > > is not in the abbreviations of the extended reals.
> >
> > The only such "abbreviations" in standard use for "extended reals  are
> > oo for the one point compactification or +oo and -oo for the two point
> > compactification.
>
> My proof does not need this abbreviation. My proof only needs the
> facts
> that the set { a_k | k in |N } of coefficients of the series is not
> empty and the number |{ a_k | k in |N }| of coefficients of the
> series is not zero. More is not required.

In standard mathematics, series and sequences do not have coefficients
at all, but only terms, though sometimes, as in power series, the terms
can have coefficients.

But WMatheology has very little in common with standard mathematics.

> > > But William had agreed: "On the contrary, the fact that the analytic
> > > *limit* cannot be described in terms of digits is the point."
> >
> > > And he stated proudly:
> >
> > > Analysis:
> > >    limit in real numbers: unbounded
> > >                          (oo in extended reals)
> > >    limit of set of 1's:  not estimated
> >
> > > Set Theory
> > >    limit in real numbers: not estimated
> > >    limit of set of 1's: {}
> >
> > > Therefore he would have to confess now that there is a contradiction
> > > between set theory and analysis.
> >
> > To say that different definitions of "limit" can give different results
> > only confuses WM, not anyone else.
>
> The limit of a sequence is defined solely by the finite terms of the
> sequence.

It is also defined by the context of the sequence, which, among other
things, defines the allowable interpretations of those "finite terms".

A sequence of sets differs in essential ways from a sequence of reals so
that they necessarily will have different limits.

Thus WM's attempts to find a contradiction in such a difference is a
violation of common sense, as well as of mathematics.

> Mathematics is the art of finding this limit, not the art of
> inventing arbitrary limits.

Mathematics, before it can find a limit, must dfine what is expected of
that limit, and in sensible mathematics the limit of a sequence of real
numbers, if it exists, and the limit of a sequence of sets, if it also
exists, will never be the same thing.

And furthermore, the sequence of real number numerals that WM is harping
about does not exist as a real number numeral at all.
--