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Matheology § 154: Consistency Proof!
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More options Nov 18 2012, 3:58 am
Newsgroups: sci.logic, sci.math
From: WM <mueck...@rz.fh-augsburg.de>
Date: Sun, 18 Nov 2012 00:58:22 -0800 (PST)
Local: Sun, Nov 18 2012 3:58 am
Subject: Matheology § 154: Consistency Proof!
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

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More options Nov 18 2012, 7:04 pm
Newsgroups: sci.logic, sci.math
From: Vurgil <Vur...@arg.erg>
Date: Sun, 18 Nov 2012 17:04:48 -0700
Local: Sun, Nov 18 2012 7:04 pm
Subject: Re: Matheology 154: Consistency Proof!
In article

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.

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)+ "

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More options Nov 18 2012, 7:15 pm
Newsgroups: sci.logic, sci.math
From: William Hughes <wpihug...@gmail.com>
Date: Sun, 18 Nov 2012 16:15:44 -0800 (PST)
Local: Sun, Nov 18 2012 7:15 pm
Subject: Re: Matheology § 154: Consistency Proof!
On Nov 18, 8:04 pm, Vurgil <Vur...@arg.erg> wrote:

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.

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More options Nov 18 2012, 7:25 pm
Newsgroups: sci.logic, sci.math
From: Vurgil <Vur...@arg.erg>
Date: Sun, 18 Nov 2012 17:25:24 -0700
Local: Sun, Nov 18 2012 7:25 pm
Subject: Re: Matheology § 154: Consistency Proof!
In article
William Hughes <wpihug...@gmail.com> wrote:

I have long since given up resisting it.

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More options Nov 19 2012, 5:28 am
Newsgroups: sci.logic, sci.math
From: WM <mueck...@rz.fh-augsburg.de>
Date: Mon, 19 Nov 2012 02:28:32 -0800 (PST)
Local: Mon, Nov 19 2012 5:28 am
Subject: Re: Matheology § 154: Consistency Proof!
On 19 Nov., 01:15, William Hughes <wpihug...@gmail.com> wrote:

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

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More options Nov 19 2012, 6:55 am
Newsgroups: sci.logic, sci.math
From: Alan Smaill <sma...@SPAMinf.ed.ac.uk>
Date: Mon, 19 Nov 2012 11:54:51 +0000
Local: Mon, Nov 19 2012 6:54 am
Subject: Re: Matheology § 154: Consistency Proof!

William Hughes <wpihug...@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

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More options Nov 19 2012, 7:58 am
Newsgroups: sci.logic, sci.math
From: William Hughes <wpihug...@gmail.com>
Date: Mon, 19 Nov 2012 04:58:09 -0800 (PST)
Local: Mon, Nov 19 2012 7:58 am
Subject: Re: Matheology § 154: Consistency Proof!
On Nov 19, 6:28 am, WM <mueck...@rz.fh-augsburg.de> wrote:

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

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More options Nov 19 2012, 8:53 am
Newsgroups: sci.logic, sci.math
From: WM <mueck...@rz.fh-augsburg.de>
Date: Mon, 19 Nov 2012 05:53:25 -0800 (PST)
Local: Mon, Nov 19 2012 8:53 am
Subject: Re: Matheology § 154: Consistency Proof!
On 19 Nov., 13:58, William Hughes <wpihug...@gmail.com> wrote:

A very good answer. So you can avoid any mathematical arguing. In
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

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More options Nov 19 2012, 5:08 pm
Newsgroups: sci.logic, sci.math
From: Vurgil <Vur...@arg.erg>
Date: Mon, 19 Nov 2012 15:08:52 -0700
Local: Mon, Nov 19 2012 5:08 pm
Subject: Re: Matheology § 154: Consistency Proof!
In article

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

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More options Nov 19 2012, 5:17 pm
Newsgroups: sci.logic, sci.math
From: Vurgil <Vur...@arg.erg>
Date: Mon, 19 Nov 2012 15:17:02 -0700
Local: Mon, Nov 19 2012 5:17 pm
Subject: Re: Matheology § 154: Consistency Proof!
In article

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)+ = 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)

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.

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More options Nov 19 2012, 5:27 pm
Newsgroups: sci.logic, sci.math
From: Vurgil <Vur...@arg.erg>
Date: Mon, 19 Nov 2012 15:27:44 -0700
Local: Mon, Nov 19 2012 5:27 pm
Subject: Re: Matheology § 154: Consistency Proof!
In article

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!

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More options Nov 20 2012, 1:39 am
Newsgroups: sci.logic, sci.math
From: WM <mueck...@rz.fh-augsburg.de>
Date: Mon, 19 Nov 2012 22:39:45 -0800 (PST)
Local: Tues, Nov 20 2012 1:39 am
Subject: Re: Matheology § 154: Consistency Proof!
On 19 Nov., 23:17, Vurgil <Vur...@arg.erg> wrote:

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

Regards, WM

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More options Nov 20 2012, 1:41 am
Newsgroups: sci.logic, sci.math
From: WM <mueck...@rz.fh-augsburg.de>
Date: Mon, 19 Nov 2012 22:41:22 -0800 (PST)
Local: Tues, Nov 20 2012 1:41 am
Subject: Re: Matheology § 154: Consistency Proof!
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

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More options Nov 20 2012, 3:05 am
Newsgroups: sci.logic, sci.math
From: Virgil <vir...@ligriv.com>
Date: Tue, 20 Nov 2012 01:05:58 -0700
Local: Tues, Nov 20 2012 3:05 am
Subject: Re: Matheology § 154: Consistency Proof!
In article

"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.
--

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More options Nov 20 2012, 3:08 am
Newsgroups: sci.logic, sci.math
From: Virgil <vir...@ligriv.com>
Date: Tue, 20 Nov 2012 01:08:54 -0700
Local: Tues, Nov 20 2012 3:08 am
Subject: Re: Matheology § 154: Consistency Proof!
In article

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.
--

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More options Nov 21 2012, 8:10 am
Newsgroups: sci.logic, sci.math
From: William Hughes <wpihug...@gmail.com>
Date: Wed, 21 Nov 2012 05:10:31 -0800 (PST)
Local: Wed, Nov 21 2012 8:10 am
Subject: Re: Matheology § 154: Consistency Proof!
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)

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More options Nov 21 2012, 10:10 am
Newsgroups: sci.logic, sci.math
From: WM <mueck...@rz.fh-augsburg.de>
Date: Wed, 21 Nov 2012 07:10:45 -0800 (PST)
Local: Wed, Nov 21 2012 10:10 am
Subject: Re: Matheology § 154: Consistency Proof!
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

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More options Nov 21 2012, 10:24 am
Newsgroups: sci.logic, sci.math
From: William Hughes <wpihug...@gmail.com>
Date: Wed, 21 Nov 2012 07:24:22 -0800 (PST)
Local: Wed, Nov 21 2012 10:24 am
Subject: Re: Matheology § 154: Consistency Proof!
Your basic problem remains.   You continue to talk
about "the" limit as if there was only one.

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More options Nov 21 2012, 10:37 am
Newsgroups: sci.logic, sci.math
From: WM <mueck...@rz.fh-augsburg.de>
Date: Wed, 21 Nov 2012 07:37:16 -0800 (PST)
Local: Wed, Nov 21 2012 10:37 am
Subject: Re: Matheology § 154: Consistency Proof!
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

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More options Nov 21 2012, 10:54 am
Newsgroups: sci.logic, sci.math
From: William Hughes <wpihug...@gmail.com>
Date: Wed, 21 Nov 2012 07:54:30 -0800 (PST)
Local: Wed, Nov 21 2012 10:54 am
Subject: Re: Matheology § 154: Consistency Proof!
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.

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More options Nov 21 2012, 11:00 am
Newsgroups: sci.logic, sci.math
From: WM <mueck...@rz.fh-augsburg.de>
Date: Wed, 21 Nov 2012 08:00:40 -0800 (PST)
Local: Wed, Nov 21 2012 11:00 am
Subject: Re: Matheology § 154: Consistency Proof!
On 21 Nov., 16:54, William Hughes <wpihug...@gmail.com> wrote:

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

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

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More options Nov 21 2012, 11:41 am
Newsgroups: sci.logic, sci.math
From: William Hughes <wpihug...@gmail.com>
Date: Wed, 21 Nov 2012 08:41:30 -0800 (PST)
Local: Wed, Nov 21 2012 11:41 am
Subject: Re: Matheology § 154: Consistency Proof!
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?

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More options Nov 21 2012, 12:20 pm
Newsgroups: sci.logic, sci.math
From: WM <mueck...@rz.fh-augsburg.de>
Date: Wed, 21 Nov 2012 09:20:57 -0800 (PST)
Local: Wed, Nov 21 2012 12:20 pm
Subject: Re: Matheology § 154: Consistency Proof!
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.

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

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

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More options Nov 21 2012, 12:43 pm
Newsgroups: sci.logic, sci.math
From: William Hughes <wpihug...@gmail.com>
Date: Wed, 21 Nov 2012 09:43:24 -0800 (PST)
Local: Wed, Nov 21 2012 12:43 pm
Subject: Re: Matheology § 154: Consistency Proof!
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.
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.

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More options Nov 21 2012, 12:57 pm
Newsgroups: sci.logic, sci.math
From: WM <mueck...@rz.fh-augsburg.de>
Date: Wed, 21 Nov 2012 09:57:57 -0800 (PST)
Local: Wed, Nov 21 2012 12:57 pm
Subject: Re: Matheology § 154: Consistency Proof!
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