Answer to Dik T. Winter
WM
> In article <15af9b47-eac6-4d22-8720-0d6e7ebba...@e21g2000yqb.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes:
> > The problem boils down to the following:
> >
> > En Am: m =< n <==> Am En m =< n [*]
> > En Am: m =< n ==> Am En m =< n [**]
> >
> > You know: Classical logic was obtained from finite sets ...
> > Show me a finite set that obeys [**] but not [*].
>
> The above is not contested: [**] implies
> [*].
I said: For complete linear sets [*] is true.
You said [*] is not true, but [**] is true.
Weyl said: Classical logic was obtained from finite sets.
Therefore I asked you: Show me a linear complete finite set, that
makes your claim [**] right and my claim [*] wrong.
This is so simple that it should be understandable even for someone
who is not "very deep in logic".
> What is contested is that:
> En Am: m =< n <== Am En m =< n [***]
> implies [*]. And *that* is the form you do use.
No. I do not use the implication only, I use the full equivalence. Of
course the equivalence includes the implication [***] as well as the
implication [**]
There is a trivial finite
> counter-example. Take three dice where on each of the sides one of the
> numbers one to nine is printed (some of them repeated). Say the set is {d1,
> d2, d3}. Define di < dj when the probability to throw a higher number with
> dj than with di is larger than to 1/2. (I would submit that all this is
> quite physical.) There is a set of three dice such that d1 < d2, d2 < d3 and
> d3 < d1.
>
> And so we have:
> Am En d_m < d_n (d1 < d2 < d3 < d1)
> but not
> En Am d_m < d_n (there is no best die).
Of course there is no best die. Therefore this set is not linear.
Every finite set of natural numbers has a "best" number.
Why do you bother with such nonsense examples?
But the answer is easy: Because you have no other examples.
_____________________
> Doesn't it bother you that he gets letters from
> other mathematicians in Germany complaining
> about it, and that he is proud about that fact?
I have never got a letter from a mathematician complaining about that.
I would never publish a letter of my private correspondence without
consent of the correspondent. My university of applied sciences got a
letter from a greasy informer who may be whatever but certainly is not
a mathematician.
> Do you not think that it might lower the value of other
> degrees in Germany as well?
But you think it would increase this value if I taught, as you
propose, that the sum of all natural numbers can be zero? Or if I
taught, contrary to fact, Cantor's claim that a real number is the
limit of its finite initial segments but all real numbers are not the
limits of all their finite initial segments?
Regards, WM
WM
> > If we define:
>
> > 1 is a natural number
> > and
> > with n also n+1 is a natural number
> > and
> > N is the smallest set that satisfies both conditions
>
> > then N is uniquely specified.
> > Of course there can be different models for N and there can be
> > different names for the elements of N. But that does not matter. The
> > natural numbers do not enter mathematics because someone "defines"
> > them, names them, or makes models of them, but because the natural
> > numbers are simply existing and mathematics is built upon them.
> But according to WM, no such thing as N can exist.
> So WM wold throw out the naturals on which so much is built.
The question is not whether the complete set of all naturals exists.
That question alrady is nearly as ridiculous as any affirmative
answer.
The question is whether we could inform someone who does not yet know,
what we understand by the sequence of natural numbers.
In order to answer this question, we need not wait until SETI gets
contact. We can answer it in every first class of every elementary
school. Of course we can inform any child with average intelligence
what we understand by this sequence 1, 2, 3, ...
Only logicians seem to see problems where no problems are. (As some
kind of compensation they see no problems where problems are.)
Of course this sequence 1, 2, 3, ... is uniquely defined, because it
is possible to inform any intelligent being about it. There is no the
slightest difference between the idea of N and the idea of a sonata or
pi.
Regards, WM
On 26 Mai, 23:09, Virgil <virg...@nowhere.com> wrote:
> In article
> <b54a16b3-914d-47eb-bd89-967cc10d2...@u10g2000vbd.googlegroups.com>,
>
>
>
>
>
> WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 25 Mai, 17:40, WM <mueck...@rz.fh-augsburg.de> wrote:
> > > ... classical logic was abstracted from the mathematics of finite sets
> > > and their subsets .... Forgetful of this limited origin, one
> > > afterwards mistook that logic for something above and prior to all
> > > mathematics, and finally applied it, without justification, to the
> > > mathematics of infinite sets. [H. Weyl].
>
> > > One example is this. For infinite sets with a linear ordering ³=<² the
> > > implication
> > > En Am: m =< n ==> Am En m =< n (1)
> > > is held correct, but not the equivalence
> > > En Am: m =< n <==> Am En m =< n (2)
> > > For any finite set with linear ordering however, (2) is true. The
> > > reason is that all elements of a finite set are subject to
> > > investigation. Therefore, if in ³completed², i.e., ³actual² infinity,
> > > as used in set theory, all elements are available, then also (2)
> > > should be used. (2) can only be false, if potentially infinite sets
> > > are considered, i.e., non-static sets which are never complete but
> > > allow that elements can be added.
>
> But non-static sets do not exist in any mathematical set theory.
They do not exist in what is commonly called ste theory and what is
eternally false mathematics.
Regards, WM
Martin Musatov
WM: I offer this text as further proof of polynomial time
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Watch List What is a Watch List? Thread Subject: An exact
simplification challenge - 91 (eerie PolyGamma) Subject: An exact
simplification challenge - 91 (eerie PolyGamma) From: Vladimir
Bondarenko Date: 25 May, 2009 03:23:31 Message: 1 of 6 Reply to this
message Add author to My Watch List View original format Flag as
spam Hello, - PolyGamma[0, 1 + I] -
PolyGamma[0, 1 - I] - PolyGamma[0, 3/2 +
I] - PolyGamma[0, 3/2 - I] + 2 PolyGamma[0,
2 + 2 I] + 2 PolyGamma[0, 2 - 2 I] + (1/2 +
I) PolyGamma[1, 1 + I] + (1/2 - I) PolyGamma[1, 1 -
I] - (1/2 + I) PolyGamma[1, 3/2 + I] - (1/2
- I) PolyGamma[1, 3/2 - I] + (2 + 4 I) PolyGamma[1, 2 + 2
I] + (2 - 4 I) PolyGamma[1, 2 - 2 I] - 2 Re
[I PolyGamma[1, 1 + I]] + 2 Re[I PolyGamma[1, 1 -
I]] ?Folks, please give not just the
answer but the processing.Cheers,Vladimir BondarenkoCo-founder, CEO,
Mathematical Directorhttp://www.cybertester.com/ Cyber Tester
Ltd.----------------------------------------------------------"We must
understand that technologieslike these are the way of the
future."----------------------------------------------------------
Subject: An exact simplification challenge - 91 (eerie PolyGamma)
From: Martin Musatov Date: 25 May, 2009 03:36:14 Message: 2 of 6
Reply to this message Add author to My Watch List View original
format Flag as spam Vladimir Bondarenko wrote:> Hello,>> -
PolyGamma[0, 1 + I]> - PolyGamma[0, 1 - I]> - PolyGamma[0, 3/2 + I]> -
PolyGamma[0, 3/2 - I]> + 2 PolyGamma[0, 2 + 2 I]> + 2 PolyGamma[0, 2 -
2 I]> + (1/2 + I) PolyGamma[1, 1 + I]> + (1/2 - I) PolyGamma[1, 1 - I]
> - (1/2 + I) PolyGamma[1, 3/2 + I]> - (1/2 - I) PolyGamma[1, 3/2 - I]
> + (2 + 4 I) PolyGamma[1, 2 + 2 I]> + (2 - 4 I) PolyGamma[1, 2 - 2 I]
> - 2 Re[I PolyGamma[1, 1 + I]]> + 2 Re[I PolyGamma[1, 1 - I]]>> ?>>
Folks, please give not just the answer but the processing.>> Cheers,>>
Vladimir Bondarenko>> Co-founder, CEO, Mathematical Director>>
http://www.cybertester.com/ Cyber Tester Ltd.>>
---------------------------------------------------------->> "We must
understand that technologies> like these are the way of the future.">>
----------------------------------------------------------People, did
I or did I not call it: "eerie effectiveness". Googlesearch sci.math:
"Musatov" and "Eerie". Want to keep going? I enjoyit. It keeps getting
more gratifying each passing day, like puttingpennies into a piggy
bank. Subject: An exact simplification challenge - 91 (eerie
PolyGamma) From: Martin Musatov Date: 25 May, 2009 03:40:46 Message: 3
of 6 Reply to this message Add author to My Watch List View
original format Flag as spam (C)2009:Here you go folks, call me
the information prophet what youare witnessing is simple P=NP
mathematics over large data sets:"Eerie" isn't it?...The
"Effectiveness"...Martin Musatov wrote:> Vladimir Bondarenko wrote:> >
Hello,> >> > - PolyGamma[0, 1 + I]> > - PolyGamma[0, 1 - I]> > -
PolyGamma[0, 3/2 + I]> > - PolyGamma[0, 3/2 - I]> > + 2 PolyGamma[0, 2
+ 2 I]> > + 2 PolyGamma[0, 2 - 2 I]> > + (1/2 + I) PolyGamma[1, 1 + I]
> > + (1/2 - I) PolyGamma[1, 1 - I]> > - (1/2 + I) PolyGamma[1, 3/2 +
I]> > - (1/2 - I) PolyGamma[1, 3/2 - I]> > + (2 + 4 I) PolyGamma[1, 2
+ 2 I]> > + (2 - 4 I) PolyGamma[1, 2 - 2 I]> > - 2 Re[I PolyGamma[1, 1
+ I]]> > + 2 Re[I PolyGamma[1, 1 - I]]> >> > ?> >> > Folks, please
give not just the answer but the processing.> >> > Cheers,> >> >
Vladimir Bondarenko> >> > Co-founder, CEO, Mathematical Director> >> >
http://www.cybertester.com/ Cyber Tester Ltd.> >> >
----------------------------------------------------------> >> > "We
must understand that technologies> > like these are the way of the
future."> >> >
----------------------------------------------------------> People,
did I or did I not call it: "eerie effectiveness". Google> search
sci.math: "Musatov" and "Eerie". Want to keep going? I enjoy> it. It
keeps getting more gratifying each passing day, like putting> pennies
into a piggy bank.Martin MusatovFounderMeAmI.org Subject: An
exact simplification challenge - 91 (eerie PolyGamma) From:
clicl...@freenet.de Date: 25 May, 2009 18:32:02 Message: 4 of 6
Reply to this message Add author to My Watch List View original
format Flag as spam Vladimir Bondarenko schrieb:>> - PolyGamma[0,
1 + I]> - PolyGamma[0, 1 - I]> - PolyGamma[0, 3/2 + I]> - PolyGamma[0,
3/2 - I]> + 2 PolyGamma[0, 2 + 2 I]> + 2 PolyGamma[0, 2 - 2 I]> + (1/2
+ I) PolyGamma[1, 1 + I]> + (1/2 - I) PolyGamma[1, 1 - I]> - (1/2 + I)
PolyGamma[1, 3/2 + I]> - (1/2 - I) PolyGamma[1, 3/2 - I]> + (2 + 4 I)
PolyGamma[1, 2 + 2 I]> + (2 - 4 I) PolyGamma[1, 2 - 2 I]> - 2 Re[I
PolyGamma[1, 1 + I]]> + 2 Re[I PolyGamma[1, 1 - I]]>> ?>This simple
challenge seems to be specifically made for Derive 6.10:-POLYGAMMA
(0,1+#i)-POLYGAMMA(0,1-#i)-POLYGAMMA(0,3/2+#i)-POLYGAM~MA(0,3/2-#i)
+2*POLYGAMMA(0,2+2*#i)+2*POLYGAMMA(0,2-2*#i)+(1/2+#i~)*POLYGAMMA
(1,1+#i)+(1/2-#i)*POLYGAMMA(1,1-#i)-(1/2+#i)*POLYGAMM~A(1,3/2+#i)-(1/2-
#i)*POLYGAMMA(1,3/2-#i)+(2+4*#i)*POLYGAMMA(1,2+~2*#i)+(2-4*#i)
*POLYGAMMA(1,2-2*#i)-2*RE(#i*POLYGAMMA(1,1+#i))+2*~RE(#i*POLYGAMMA(1,1-
#i))" ... is automatically rewritten to ... "-DIGAMMA(3/2-#i)-DIGAMMA
(3/2+#i)+2*DIGAMMA(2-2*#i)+2*DIGAMMA(2+2~*#i)-DIGAMMA(1-#i)-DIGAMMA
(1+#i)-ZETA(2,1/2-#i)/2-ZETA(2,1/2+#i)~/2+2*ZETA(2,1-2*#i)+2*ZETA
(2,1+2*#i)+ZETA(2,1-#i)/2+ZETA(2,1+#i)~/2+#i*(-CONJ(ZETA(2,1-#i))+CONJ
(ZETA(2,1+#i))+ZETA(2,1/2-#i)-ZET~A(2,1/2+#i)-4*ZETA(2,1-2*#i)+4*ZETA
(2,1+2*#i))" judicious substitution of the helper functions ... "mpsi
(z,m):=1/m*SUM(DIGAMMA((z+k)/m),k,0,m-1)+LN(m)mzeta(s,z,m):=1/m^s*SUM
(ZETA(s,(z+k)/m),k,0,m-1)" ... along with manual help for CONJ
(ZETA) ... "-DIGAMMA(3/2-#i)-DIGAMMA(3/2+#i)+2*mpsi(2-2*#i,2)+2*mpsi
(2+2*#i,~2)-DIGAMMA(1-#i)-DIGAMMA(1+#i)-ZETA(2,1/2-#i)/2-ZETA
(2,1/2+#i)/2~+2*mzeta(2,1-2*#i,2)+2*mzeta(2,1+2*#i,2)+ZETA(2,1-#i)/
2+ZETA(2,1~+#i)/2+#i*(-ZETA(CONJ(2),CONJ(1-#i))+ZETA(CONJ(2),CONJ
(1+#i))+ZE~TA(2,1/2-#i)-ZETA(2,1/2+#i)-4*mzeta(2,1-2*#i,2)+4*mzeta
(2,1+2*#i~,2))ZETA(2,1-#i)+ZETA(2,1+#i)+4*LN(2)2*RE(ZETA(2,1+#i))+4*LN
(2)3.698588915Martin. Subject: An exact simplification challenge - 91
(eerie PolyGamma) From: Vladimir Bondarenko Date: 25 May, 2009
19:41:14 Message: 5 of 6 Reply to this message Add author to My
Watch List View original format Flag as spam On May 25, 9:32 pm,
cliclic...@freenet.de wrote:> Vladimir Bondarenko schrieb:>>>>>> >
- PolyGamma[0, 1 + I]> > - PolyGamma[0, 1 - I]
> > - PolyGamma[0, 3/2 + I]> > - PolyGamma
[0, 3/2 - I]> > + 2 PolyGamma[0, 2 + 2 I]> >
+ 2 PolyGamma[0, 2 - 2 I]> > + (1/2 + I) PolyGamma[1,
1 + I]> > + (1/2 - I) PolyGamma[1, 1 - I]> >
- (1/2 + I) PolyGamma[1, 3/2 + I]> > - (1/2 - I)
PolyGamma[1, 3/2 - I]> > + (2 + 4 I) PolyGamma[1, 2 + 2
I]> > + (2 - 4 I) PolyGamma[1, 2 - 2 I]> >
- 2 Re[I PolyGamma[1, 1 + I]]> > + 2 Re[I PolyGamma[1, 1
- I]]>> > ?>> This simple challenge
seems to be specifically made for Derive 6.10:>> -POLYGAMMA(0,1+#i)-
POLYGAMMA(0,1-#i)-POLYGAMMA(0,3/2+#i)-POLYGAM~> MA(0,3/2-#i)
+2*POLYGAMMA(0,2+2*#i)+2*POLYGAMMA(0,2-2*#i)+(1/2+#i~> )*POLYGAMMA
(1,1+#i)+(1/2-#i)*POLYGAMMA(1,1-#i)-(1/2+#i)*POLYGAMM~> A(1,3/2+#i)-
(1/2-#i)*POLYGAMMA(1,3/2-#i)+(2+4*#i)*POLYGAMMA(1,2+~> 2*#i)+(2-4*#i)
*POLYGAMMA(1,2-2*#i)-2*RE(#i*POLYGAMMA(1,1+#i))+2*~> RE(#i*POLYGAMMA
(1,1-#i))>> " ... is automatically rewritten to ... ">> -DIGAMMA(3/2-
#i)-DIGAMMA(3/2+#i)+2*DIGAMMA(2-2*#i)+2*DIGAMMA(2+2~> *#i)-DIGAMMA(1-
#i)-DIGAMMA(1+#i)-ZETA(2,1/2-#i)/2-ZETA(2,1/2+#i)~> /2+2*ZETA(2,1-2*#i)
+2*ZETA(2,1+2*#i)+ZETA(2,1-#i)/2+ZETA(2,1+#i)~> /2+#i*(-CONJ(ZETA(2,1-
#i))+CONJ(ZETA(2,1+#i))+ZETA(2,1/2-#i)-ZET~> A(2,1/2+#i)-4*ZETA
(2,1-2*#i)+4*ZETA(2,1+2*#i))>> " judicious substitution of the helper
functions ... ">> mpsi(z,m):=1/m*SUM(DIGAMMA((z+k)/m),k,0,m-1)+LN(m)>>
mzeta(s,z,m):=1/m^s*SUM(ZETA(s,(z+k)/m),k,0,m-1)>> " ... along with
manual help for CONJ(ZETA) ... ">> -DIGAMMA(3/2-#i)-DIGAMMA(3/2+#i)
+2*mpsi(2-2*#i,2)+2*mpsi(2+2*#i,~> 2)-DIGAMMA(1-#i)-DIGAMMA(1+#i)-ZETA
(2,1/2-#i)/2-ZETA(2,1/2+#i)/2~> +2*mzeta(2,1-2*#i,2)+2*mzeta(2,1+2*#i,
2)+ZETA(2,1-#i)/2+ZETA(2,1~> +#i)/2+#i*(-ZETA(CONJ(2),CONJ(1-#i))+ZETA
(CONJ(2),CONJ(1+#i))+ZE~> TA(2,1/2-#i)-ZETA(2,1/2+#i)-4*mzeta(2,1-2*#i,
2)+4*mzeta(2,1+2*#i~> ,2))>> ZETA(2,1-#i)+ZETA(2,1+#i)+4*LN(2)>> 2*RE
(ZETA(2,1+#i))+4*LN(2)>> 3.698588915>> Martin.Great!I'd only offer a
bit simpler answer1-pi^2*CSCH(pi)^2+4*LOG(2)3.698588915But there's
still something about this andmany other challenges untold :)
Cheers,Vladimir BondarenkoCo-founder, CEO, Mathematical
Directorhttp://www.cybertester.com/ Cyber Tester
Ltd.----------------------------------------------------------"We must
understand that technologieslike these are the way of the
future."----------------------------------------------------------
Subject: An exact simplification challenge - 91 (eerie PolyGamma)
From: clicl...@freenet.de Date: 25 May, 2009 21:34:03 Message: 6 of
6 Reply to this message Add author to My Watch List View original
format Flag as spam Vladimir Bondarenko schrieb:> On May 25, 9:32
pm, cliclic...@freenet.de wrote:> > Vladimir Bondarenko schrieb:> >> >
> - PolyGamma[0, 1 + I]> > > - PolyGamma[0, 1 - I]> > > - PolyGamma[0,
3/2 + I]> > > - PolyGamma[0, 3/2 - I]> > > + 2 PolyGamma[0, 2 + 2 I]>
> > + 2 PolyGamma[0, 2 - 2 I]> > > + (1/2 + I) PolyGamma[1, 1 + I]> >
> + (1/2 - I) PolyGamma[1, 1 - I]> > > - (1/2 + I) PolyGamma[1, 3/2 +
I]> > > - (1/2 - I) PolyGamma[1, 3/2 - I]> > > + (2 + 4 I) PolyGamma
[1, 2 + 2 I]> > > + (2 - 4 I) PolyGamma[1, 2 - 2 I]> > > - 2 Re[I
PolyGamma[1, 1 + I]]> > > + 2 Re[I PolyGamma[1, 1 - I]]> >> > > ?> >>
> This simple challenge seems to be specifically made for Derive
6.10:> >> > -POLYGAMMA(0,1+#i)-POLYGAMMA(0,1-#i)-POLYGAMMA(0,3/2+#i)-
POLYGAM~> > MA(0,3/2-#i)+2*POLYGAMMA(0,2+2*#i)+2*POLYGAMMA(0,2-2*#i)
+(1/2+#i~> > )*POLYGAMMA(1,1+#i)+(1/2-#i)*POLYGAMMA(1,1-#i)-(1/2+#i)
*POLYGAMM~> > A(1,3/2+#i)-(1/2-#i)*POLYGAMMA(1,3/2-#i)+(2+4*#i)
*POLYGAMMA(1,2+~> > 2*#i)+(2-4*#i)*POLYGAMMA(1,2-2*#i)-2*RE
(#i*POLYGAMMA(1,1+#i))+2*~> > RE(#i*POLYGAMMA(1,1-#i))> >> > " ... is
automatically rewritten to ... "> >> > -DIGAMMA(3/2-#i)-DIGAMMA(3/2+#i)
+2*DIGAMMA(2-2*#i)+2*DIGAMMA(2+2~> > *#i)-DIGAMMA(1-#i)-DIGAMMA(1+#i)-
ZETA(2,1/2-#i)/2-ZETA(2,1/2+#i)~> > /2+2*ZETA(2,1-2*#i)+2*ZETA
(2,1+2*#i)+ZETA(2,1-#i)/2+ZETA(2,1+#i)~> > /2+#i*(-CONJ(ZETA(2,1-#i))
+CONJ(ZETA(2,1+#i))+ZETA(2,1/2-#i)-ZET~> > A(2,1/2+#i)-4*ZETA(2,1-2*#i)
+4*ZETA(2,1+2*#i))> >> > " judicious substitution of the helper
functions ... "> >> > mpsi(z,m):=1/m*SUM(DIGAMMA((z+k)/m),k,0,m-1)+LN
(m)> >> > mzeta(s,z,m):=1/m^s*SUM(ZETA(s,(z+k)/m),k,0,m-1)> >> > " ...
along with manual help for CONJ(ZETA) ... "> >> > -DIGAMMA(3/2-#i)-
DIGAMMA(3/2+#i)+2*mpsi(2-2*#i,2)+2*mpsi(2+2*#i,~> > 2)-DIGAMMA(1-#i)-
DIGAMMA(1+#i)-ZETA(2,1/2-#i)/2-ZETA(2,1/2+#i)/2~> > +2*mzeta(2,1-2*#i,
2)+2*mzeta(2,1+2*#i,2)+ZETA(2,1-#i)/2+ZETA(2,1~> > +#i)/2+#i*(-ZETA
(CONJ(2),CONJ(1-#i))+ZETA(CONJ(2),CONJ(1+#i))+ZE~> > TA(2,1/2-#i)-ZETA
(2,1/2+#i)-4*mzeta(2,1-2*#i,2)+4*mzeta(2,1+2*#i~> > ,2))> >> > ZETA
(2,1-#i)+ZETA(2,1+#i)+4*LN(2)> >> > 2*RE(ZETA(2,1+#i))+4*LN(2)> >> >
3.698588915> >>> Great!>> I'd only offer a bit simpler answer>> 1-
pi^2*CSCH(pi)^2+4*LOG(2)>> 3.698588915>Sorry, I didn't try to
"elementarize" the ZETA's.ZETA(2, 1+#i) + ZETA(2, 1-#i) =ZETA(2, 1+#i)
+ ZETA(2, -#i) + 1 =-(2 pi)^2 LI(-1, #e^(-2 pi)) + 1which is
equivalent to your expression since LI(-1,z) = z/(1-z)^2.Can Maple or
Mathematica perform the conversion automatically?Martin.Tags for this
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Applied By Date/Time spam per isakson 25 May, 2009 00:25:24 spam
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__
The unpopular opinion is not always wrong. Sometimes it is simply
perfect.
David C. Ullrich
<muec...@rz.fh-augsburg.de> wrote:
>On 26 Mai, 21:49, Virgil <virg...@nowhere.com> wrote:
>
>> > If we define:
>>
>> > 1 is a natural number
>> > and
>> > with n also n+1 is a natural number
>> > and
>> > N is the smallest set that satisfies both conditions
>>
>> > then N is uniquely specified.
>> > Of course there can be different models for N and there can be
>> > different names for the elements of N. But that does not matter. The
>> > natural numbers do not enter mathematics because someone "defines"
>> > them, names them, or makes models of them, but because the natural
>> > numbers are simply existing and mathematics is built upon them.
>
>> But according to WM, no such thing as N can exist.
>> So WM wold throw out the naturals on which so much is built.
>
>The question is not whether the complete set of all naturals exists.
>That question alrady is nearly as ridiculous as any affirmative
>answer.
>
>The question is whether we could inform someone who does not yet know,
>what we understand by the sequence of natural numbers.
Sorry, I can't follow any of this, because I don't already know what
you mean by the words "question", "is", "whether", "we".
"could", "inform", "someone", "who", "does", "not", "yet",
and "understand".
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
Dik T. Winter
> On 26 Mai, 04:07, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > In article <15af9b47-eac6-4d22-8720-0d6e7ebba...@e21g2000yqb.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes:
> > > The problem boils down to the following:
> > >
> > > En Am: m =< n <==> Am En m =< n [*]
> > > En Am: m =< n ==> Am En m =< n [**]
> > >
> > > You know: Classical logic was obtained from finite sets ...
> > > Show me a finite set that obeys [**] but not [*].
> >
> > The above is not contested: [**] implies
> > [*].
>
> I said: For complete linear sets [*] is true.
Not in the article to which I responded.
> You said [*] is not true, but [**] is true.
That is not what I said. I said that for the case involved you have to
*prove* that it is true, because it is not generally true.
> Weyl said: Classical logic was obtained from finite sets.
Right.
> Therefore I asked you: Show me a linear complete finite set, that
> makes your claim [**] right and my claim [*] wrong.
But now you include the word "linear". Where did "Weyl" include the
word "linear"?
> This is so simple that it should be understandable even for someone
> who is not "very deep in logic".
>
> > What is contested is that:
> > En Am: m =< n <== Am En m =< n [***]
> > implies [*]. And *that* is the form you do use.
>
> No. I do not use the implication only, I use the full equivalence. Of
> course the equivalence includes the implication [***] as well as the
> implication [**]
Because the implication [**] is always true, the only part of the equivalence
that is new is the implication [***].
> > There is a trivial finite
> > counter-example. Take three dice where on each of the sides one of the
> > numbers one to nine is printed (some of them repeated). Say the set is
> > {d1, d2, d3}. Define di < dj when the probability to throw a higher
> > number with dj than with di is larger than to 1/2. (I would submit that
> > all this is quite physical.) There is a set of three dice such that
> > d1 < d2, d2 < d3 and d3 < d1.
> >
> > And so we have:
> > Am En d_m < d_n (d1 < d2 < d3 < d1)
> > but not
> > En Am d_m < d_n (there is no best die).
>
> Of course there is no best die. Therefore this set is not linear.
> Every finite set of natural numbers has a "best" number.
>
> Why do you bother with such nonsense examples?
> But the answer is easy: Because you have no other examples.
Because they are answers to what you actually ask. It proves, in general,
that [***] does *not* imply [*], even not in finite cases. So in order
to use such implication you have to *prove* it.
Now you state that classical logic is derived from the finite case. But in
the finite case that implication is not generally available, so it is not
part of classical logic, I would say.
--
dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
WM
1. Then let us start with the meaning of your sentence
"because I don't already know what you mean by the words"
Obviously you can understand that sentence, because you even corrected
a typo (alrady) of mine.
"does" is closely related to "do", "don't" is the negation of "do",
apending a truncated version of "not".
So much for the first lesson. "First", abbreviated by "1." or "1^st"
points to the last member of an ordered set of cardinality 1. Any open
questions?
Regards, WM
WM
> In article <351d0bd7-fe7d-4005-a10a-42b63a903...@q16g2000yqg.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes:
> > On 26 Mai, 04:07, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > > In article <15af9b47-eac6-4d22-8720-0d6e7ebba...@e21g2000yqb.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes:
> > > > The problem boils down to the following:
> > > >
> > > > En Am: m =< n <==> Am En m =< n [*]
> > > > En Am: m =< n ==> Am En m =< n [**]
> > > >
> > > > You know: Classical logic was obtained from finite sets ...
> > > > Show me a finite set that obeys [**] but not [*].
> > >
> > > The above is not contested: [**] implies
> > > [*].
> >
> > I said: For complete linear sets [*] is true.
>
> Not in the article to which I responded.
But frequently I made use of what you call quatifier exchange and what
is allowed in case of complete linear sets.
>
> > You said [*] is not true, but [**] is true.
>
> That is not what I said.
You said:
The only thing that can be stated is (symbolically):
E n A m P(m, n) -> A m E n P(m, n)
not the reverse, this is just basic logic.
> I said that for the case involved you have to
> *prove* that it is true, because it is not generally true.
It is generally true for complete linear sets. You have to prove that
it is not.
>
> > Weyl said: Classical logic was obtained from finite sets.
>
> Right.
>
> > Therefore I asked you: Show me a linear complete finite set, that
> > makes your claim [**] right and my claim [*] wrong.
>
> But now you include the word "linear". Where did "Weyl" include the
> word "linear"?
I did never claim that quantifier exchange is allowed in case of non-
linear sets, like cyclic sets as, for instance, your dice. That would
be nonsense. A simple example: Every country has a country that lies
west of it. But there is no country that lies west of all countries.
>
> > This is so simple that it should be understandable even for someone
> > who is not "very deep in logic".
> >
> > > What is contested is that:
> > > En Am: m =< n <== Am En m =< n [***]
> > > implies [*]. And *that* is the form you do use.
> >
> > No. I do not use the implication only, I use the full equivalence. Of
> > course the equivalence includes the implication [***] as well as the
> > implication [**]
>
> Because the implication [**] is always true, the only part of the equivalence
> that is new is the implication [***].
For complete linear sets both are true, therefore [*] holds. And you
should recognize that actually complete linear sets obey [*]. Only
potentially infinite sets do not. But you mix up things. You claim the
existence of a complete linear set but disregard the necessary
consequence of completeness or linearity, namely the validity of [*].
Regards, WM
WM
want to get involved in too many threads, I answer here:
-possums.net> wrote:
> On 2009-05-27, Brian Chandler <imaginator...@despammed.com> wrote:
>
> > The "is" of "there is" is not the exists() predicate, so (I suppose!)
> > if you could formalise this it would not be a contradiction "as such".
That is correct!
>
> As I recall, he actually used the word "exist" for both aspects: both
> that the number exists and that it cannot exist. He may well have
> intended different meanings for each use without clarification or
> distinction.
Due to the lack of tools for representing numbers with large
information contents (larger than 2^80 bits, or 2^365 bits, or 2^X
bits, where X is a number that may depend on the progress of physics
but in any case is finite) we must accept that numbers with larger
information contents (that canot be reduced) do not exist.
That means, there exist numbers (namely according to current
mathematics, which calls itself realism but is simply a form of
idealism) that do not exist according to fact.
"to exist" is used here in two different meanings such that "there
exist(1) numbers that do not exist(2)" is not a self-contradiction.
Regards, WM
Virgil
<5f791699-355d-41db...@s20g2000vbp.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> I have seen MatheRealism being discussed in other threads. As I don't
> want to get involved in too many threads, I answer here:
>
> -possums.net> wrote:
> > On 2009-05-27, Brian Chandler <imaginator...@despammed.com> wrote:
> >
> > > The "is" of "there is" is not the exists() predicate, so (I suppose!)
> > > if you could formalise this it would not be a contradiction "as such".
>
> That is correct!
> >
> > As I recall, he actually used the word "exist" for both aspects: both
> > that the number exists and that it cannot exist. He may well have
> > intended different meanings for each use without clarification or
> > distinction.
>
> Due to the lack of tools for representing numbers with large
> information contents (larger than 2^80 bits, or 2^365 bits, or 2^X
> bits, where X is a number that may depend on the progress of physics
> but in any case is finite) we must accept that numbers with larger
> information contents (that canot be reduced) do not exist.
WM may have to accept if, but o one else need do so.
WM conflates our ability to name a number with its existence.
There is nothing inconsistent or illogical in the existence of things
for which we do not have names.
>
> That means, there exist numbers (namely according to current
> mathematics, which calls itself realism but is simply a form of
> idealism) that do not exist according to fact.
It only means that there are numbers for which specific forms of names
do not exist, but that is only a fault in those specific naming
mechanisms.
>
> "to exist" is used here in two different meanings such that "there
> exist(1) numbers that do not exist(2)" is not a self-contradiction.
It is a contradiction in English. It probably is a contradiction in the
German of scholars. But WM would not know about that.
--
Virgil
WM
>
>
>
>
>
> WM <mueck...@rz.fh-augsburg.de> wrote:
> > I have seen MatheRealism being discussed in other threads. As I don't
> > want to get involved in too many threads, I answer here:
>
> > -possums.net> wrote:
> > > On 2009-05-27, Brian Chandler <imaginator...@despammed.com> wrote:
>
> > > > The "is" of "there is" is not the exists() predicate, so (I suppose!)
> > > > if you could formalise this it would not be a contradiction "as such".
>
> > That is correct!
>
> > > As I recall, he actually used the word "exist" for both aspects: both
> > > that the number exists and that it cannot exist. He may well have
> > > intended different meanings for each use without clarification or
> > > distinction.
>
> > Due to the lack of tools for representing numbers with large
> > information contents (larger than 2^80 bits, or 2^365 bits, or 2^X
> > bits, where X is a number that may depend on the progress of physics
> > but in any case is finite) we must accept that numbers with larger
> > information contents (that canot be reduced) do not exist.
>
> WM may have to accept if, but o one else need do so.
> WM conflates our ability to name a number with its existence.
> There is nothing inconsistent or illogical in the existence of things
> for which we do not have names.
Unless these things are names only.
>
>
>
> > That means, there exist numbers (namely according to current
> > mathematics, which calls itself realism but is simply a form of
> > idealism) that do not exist according to fact.
>
> It only means that there are numbers for which specific forms of names
> do not exist, but that is only a fault in those specific naming
> mechanisms.
Replace "do not exist" by "cannot exist" and you see that such numbers
cannot be used for enumerating anything, hence are not numbers.
>
>
>
> > "to exist" is used here in two different meanings such that "there
> > exist(1) numbers that do not exist(2)" is not a self-contradiction.
>
> It is a contradiction in English. It probably is a contradiction in the
> German of scholars. But WM would not know about that.
In philosophical texts you can find frequently words with different
meanings that are distinguished by numbers as I did. To be(1) or to be
(2) or to be(14) that is the question. But you seem not to be(0) aware
of such scholarly texts.
Regards, WM
Virgil
<1d5fcfb8-e4f6-496f...@k8g2000yqn.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 26 Mai, 21:49, Virgil <virg...@nowhere.com> wrote:
>
> > > If we define:
> >
> > > 1 is a natural number
> > > and
> > > with n also n+1 is a natural number
> > > and
> > > N is the smallest set that satisfies both conditions
> >
> > > then N is uniquely specified.
> > > Of course there can be different models for N and there can be
> > > different names for the elements of N. But that does not matter. The
> > > natural numbers do not enter mathematics because someone "defines"
> > > them, names them, or makes models of them, but because the natural
> > > numbers are simply existing and mathematics is built upon them.
>
> > But according to WM, no such thing as N can exist.
> > So WM wold throw out the naturals on which so much is built.
>
> The question is not whether the complete set of all naturals exists.
That is a question on which WM differs from the mainstream.
But WM cannot grant its existence in a given argument and then deny it
in the same argument and expect anyone to accept that sort of argument.
> That question alrady is nearly as ridiculous as any affirmative
> answer.
Wrong!
>
> The question is whether we could inform someone who does not yet know,
> what we understand by the sequence of natural numbers.
Children know from a very early age. Whom does WM think is left to
inform?
>
But non-static sets do not exist in any mathematical set theory.
>
> They do not exist in what is commonly called ste theory and what is
> eternally false mathematics.
Only in WM's MathUnrealism is it false. Everywhere else it is the true
math.
>
> Regards, WM
--
Virgil
Virgil
<6590202d-598a-41fd...@x6g2000vbg.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
Then different names, being names only, would necessarily be different
things and then the name 1/2 and the name 2/4, being different things,
could not be equal.
> >
> >
> >
> > > That means, there exist numbers (namely according to current
> > > mathematics, which calls itself realism but is simply a form of
> > > idealism) that do not exist according to fact.
> >
> > It only means that there are numbers for which specific forms of names
> > do not exist, but that is only a fault in those specific naming
> > mechanisms.
>
> Replace "do not exist" by "cannot exist" and you see that such numbers
> cannot be used for enumerating anything, hence are not numbers.
Only cardinals can ennumerate anything, so WM would have us eliminate
all rationals, ordinals, algebraics, etc.
> >
> >
> >
> > > "to exist" is used here in two different meanings such that "there
> > > exist(1) numbers that do not exist(2)" is not a self-contradiction.
> >
> > It is a contradiction in English. It probably is a contradiction in the
> > German of scholars. But WM would not know about that.
>
> In philosophical texts you can find frequently words with different
> meanings that are distinguished by numbers as I did. To be(1) or to be
> (2) or to be(14) that is the question. But you seem not to be(0) aware
> of such scholarly texts.
If WM had indexed those usages of "exist" in his original claim, he
might be able to justify his present claim, but he didn't and thus can't.
--
Virgil
Virgil
<351d0bd7-fe7d-4005...@q16g2000yqg.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 26 Mai, 04:07, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > In article
> > <15af9b47-eac6-4d22-8720-0d6e7ebba...@e21g2000yqb.googlegroups.com> WM
> > <mueck...@rz.fh-augsburg.de> writes:
> > > The problem boils down to the following:
> > >
> > > En Am: m =< n <==> Am En m =< n [*]
> > > En Am: m =< n ==> Am En m =< n [**]
> > >
> > > You know: Classical logic was obtained from finite sets ...
> > > Show me a finite set that obeys [**] but not [*].
> >
> > The above is not contested: [**] implies
> > [*].
>
> I said: For complete linear sets [*] is true.
> You said [*] is not true, but [**] is true.
> Weyl said: Classical logic was obtained from finite sets.
> Therefore I asked you: Show me a linear complete finite set, that
> makes your claim [**] right and my claim [*] wrong.
What Weyl may have said in no way limits logic, classical or otherwise,
to finite sets. Or does WM claim that Weyl was speaking ex cathedra?
>
> This is so simple that it should be understandable even for someone
> who is not "very deep in logic".
>
> > What is contested is that:
> > En Am: m =< n <== Am En m =< n [***]
> > implies [*]. And *that* is the form you do use.
>
> No. I do not use the implication only, I use the full equivalence. Of
> course the equivalence includes the implication [***] as well as the
> implication [**]
Then WM is not operating in the same world as everyone else.
>
> > There is a trivial finite
> > counter-example. Take three dice where on each of the sides one of the
> > numbers one to nine is printed (some of them repeated). Say the set is
> > {d1,
> > d2, d3}. Define di < dj when the probability to throw a higher number with
> > dj than with di is larger than to 1/2. (I would submit that all this is
> > quite physical.) There is a set of three dice such that d1 < d2, d2 < d3
> > and
> > d3 < d1.
> >
> > And so we have:
> > Am En d_m < d_n (d1 < d2 < d3 < d1)
> > but not
> > En Am d_m < d_n (there is no best die).
>
> Of course there is no best die. Therefore this set is not linear.
> Every finite set of natural numbers has a "best" number.
It may have a largest, but without a definition of being "better" that
is transitive, it need not have a "best".
>
> Why do you bother with such nonsense examples?
Why does WM bother with his much more nonsensical examples?
I doubt that even WM knows.
> But the answer is easy: Because you have no other examples.
We have them, but WM has steadfastly refused to understand them so we
are trying to simplify things to a point which WM can understand them.
So far we have not been able to get down to his level.
>
> > Doesn't it bother you that he gets letters from
> > other mathematicians in Germany complaining
> > about it, and that he is proud about that fact?
>
> I have never got a letter from a mathematician complaining about that.
> I would never publish a letter of my private correspondence without
> consent of the correspondent. My university of applied sciences got a
> letter from a greasy informer who may be whatever but certainly is not
> a mathematician.
>
> > Do you not think that it might lower the value of other
> > degrees in Germany as well?
>
> But you think it would increase this value if I taught, as you
> propose, that the sum of all natural numbers can be zero?
Considering some of the things you do claim to teach, it would hardly
change things.
> Or if I
> taught, contrary to fact, Cantor's claim that a real number is the
> limit of its finite initial segments but all real numbers are not the
> limits of all their finite initial segments?
That may be WM's claim, but it is certainly not Cantor's.
Cantor may have viewed reals as equivalence classes of Cauchy seequnces,
or, more likely, as Dedekind cuts, but where did WM get his delusion
that Cantor ever thought simultaneously that a real was something and at
the same time was not that thing?
--
Virgil
Virgil
<37a98f77-6bc9-4cae...@q2g2000vbr.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> >
> > Not in the article to which I responded.
>
> But frequently I made use of what you call quatifier exchange and what
> is allowed in case of complete linear sets.
You must prove it is allowed for the special case of what you call
"complete linear sets" as it is invalid in general.
> >
> > �> You �said [*] is not true, but [**] is true.
> >
> > That is not what I said.
>
> You said:
> The only thing that can be stated is (symbolically):
> E n A m P(m, n) -> A m E n P(m, n)
> not the reverse, this is just basic logic.
>
> > �I said that for the case involved you have to
> > *prove* that it is true, because it is not generally true.
>
> It is generally true for complete linear sets.
Not until you prove it to be so, which you have not done.
You have not even given an adequate definition of what you mean by
"complete linear sets", unless you mean only sets for which what you
claim is true.
> >
> > �> Weyl said: Classical logic was obtained from finite sets.
> >
> > Right.
> >
> > �> Therefore I asked you: Show me a linear complete finite set, that
> > �> makes your claim [**] right and my claim [*] wrong.
> >
> > But now you include the word "linear". �Where did "Weyl" include the
> > word "linear"?
>
> I did never claim that quantifier exchange is allowed in case of non-
> linear sets, like cyclic sets as, for instance, your dice. That would
> be nonsense. A simple example: Every country has a country that lies
> west of it. But there is no country that lies west of all countries.
If you concede that something does not hold for arbitrary cases and you
claim it does holds for certain special cases, then it is your
obligation to prove it for those special cases.
Which anyone competent in pure mathematics would have known, but which
such professors of impure math as WM does not know.
>
> For complete linear sets both are true
Often claimed but never proven (at least by WM).
--
Virgil
WM
> > > > Of course there can be different models for N and there can be
> > > > different names for the elements of N. But that does not matter. The
> > > > natural numbers do not enter mathematics because someone "defines"
> > > > them, names them, or makes models of them, but because the natural
> > > > numbers are simply existing and mathematics is built upon them.
>
> > > But according to WM, no such thing as N can exist.
> > > So WM wold throw out the naturals on which so much is built.
>
> > The question is not whether the complete set of all naturals exists.
>
> That is a question on which WM differs from the mainstream.
> But WM cannot grant its existence in a given argument and then deny it
> in the same argument and expect anyone to accept that sort of argument.
Grant the existence of two natural numbers m and n such that m/n = sqrt
(2). Then falsify it.
Grant the existence of a largest natural. Then falsify it.
Grant the existence of a largest prime number. Then falsify it.
Grant the existence of all natural numbers. Then falsify it.
All these proofs are proofs by contradiction.
>
> > That question alrady is nearly as ridiculous as any affirmative
> > answer.
>
> Wrong!
I said "nearly"! This is a fuzzy quantification, not a sharp one. It
cannot be wrong. One might qualify it as nearly wrong or nearly right,
at most.
>
>
>
> > The question is whether we could inform someone who does not yet know,
> > what we understand by the sequence of natural numbers.
>
> Children know from a very early age. Whom does WM think is left to
> inform?
How do children get to know what the real numbers are? Do they have
Peano's axioms in their genes or in their strollers?
>
> But non-static sets do not exist in any mathematical set theory.
That is orthodox nonsense. The natural numbers as a potetntially
infinite set exist in every mathematical theory before BC (before
Cantor) and in many mathematical theories AC, for instance in every
theory that is taught in elementary schools as well as in universities
outside the departments of "logics" and mathematics.
Regards, WM
WM
> > > There is nothing inconsistent or illogical in the existence of things
> > > for which we do not have names.
>
> > Unless these things are names only.
>
> Then different names, being names only, would necessarily be different
> things and then the name 1/2 and the name 2/4, being different things,
> could not be equal.
Why not? There are laws, i.e., other abstract things, that allow to
find out whether two names are the same or not. There are many names
identical to the name pi.
> > > > "to exist" is used here in two different meanings such that "there
> > > > exist(1) numbers that do not exist(2)" is not a self-contradiction.
>
> > > It is a contradiction in English. It probably is a contradiction in the
> > > German of scholars. But WM would not know about that.
>
> > In philosophical texts you can find frequently words with different
> > meanings that are distinguished by numbers as I did. To be(1) or to be
> > (2) or to be(14) that is the question. But you seem not to be(0) aware
> > of such scholarly texts.
>
> If WM had indexed those usages of "exist" in his original claim, he
> might be able to justify his present claim, but he didn't and thus can't.
If I had written a philosophical text, I would have explained this,
and in fact I did it in many of my contributions here, but nobody can
expect that I do so in every contribution. A minimum of brainwork is
required to understand my texts, but really, it is a very small
amount.
Regards, WM
WM
> > > > The problem boils down to the following:
>
> > > > En Am: m =< n <==> Am En m =< n [*]
> > > > En Am: m =< n ==> Am En m =< n [**]
>
> > > > You know: Classical logic was obtained from finite sets ...
> > > > Show me a finite set that obeys [**] but not [*].
> > I said: For complete linear sets [*] is true.
> > You said [*] is not true, but [**] is true.
> > Weyl said: Classical logic was obtained from finite sets.
> > Therefore I asked you: Show me a linear complete finite set, that
> > makes your claim [**] right and my claim [*] wrong.
>
> What Weyl may have said in no way limits logic, classical or otherwise,
> to finite sets. Or does WM claim that Weyl was speaking ex cathedra?
Weyl has recognized, by the end of his life (when obtaining his
doctorate under Hilbert he was preoccupied with Hilbert's opinion),
that logic is not included into the 10 commandments, but has to be
constructed or obtained as an abstraction from the reality. Therefore
we have no guarantee that it is suitable for sets that cannot belong
to reality. And I added: We cannot use logical theorems that are
counterfactual, i.e., contrary to what we obtained from reality. There
is no model of a complete linear set in reality that makes [*] false
and [**] true. But there are many models showing that [*] is true
whenever [**] is true.
>
> > No. I do not use the implication only, I use the full equivalence. Of
> > course the equivalence includes the implication [***] as well as the
> > implication [**]
>
> Then WM is not operating in the same world as everyone else.
Most of your ilk intermingle the worlds in which they work. They work
in potential infinity if the question is put whether there is a
largest natural number. But they work in actual infinity if the Cantor-
diagonal or any other irrational number is concerned. Only the
possibility to complete it by a limit process, leads to the erroneous
assumption of uncountability.
This had already been recognized by the late Alexander Zenkin, one of
the brave scientists who dared to condemn this hypocritical behaviour:
Cantor's 'paradise' as well as all modern axiomatic set theory is
based on the (self-contradictory) concept of actual infinity. Cantor
emphasized plainly and constantly that all transfinite objects of his
set theory are based on the actual infinity. Modern AST-people try to
persuade us to believe that the AST does not use actual infinity. It
is an intentional and blatant lie, since if infinite sets, X and N,
are potential, then the uncountability of the continuum becomes
unprovable, but without the notorious uncountablity of continuum the
modern AST as a whole transforms into a long twaddle about nothing.
>
> > Or if I
> > taught, contrary to fact, Cantor's claim that a real number is the
> > limit of its finite initial segments but all real numbers are not the
> > limits of all their finite initial segments?
>
> That may be WM's claim, but it is certainly not Cantor's.
> Cantor may have viewed reals as equivalence classes of Cauchy sequences,
> or, more likely, as Dedekind cuts, but where did WM get his delusion.
Cantor did not hold Dedekind's work in high esteem. Here is one
example:
Die Schrift von Dedekind „Was sind und was sollen die Zahlen" ist,
wenn auch ihrer Tendenz nach, die Arithmetik rein logisch zu
begründen, lobenswerth, nicht nach meinem Geschmack. ...
Das künstliche System der 172 sich nur um das Elementarste und zum
Theil Trivialste drehenden Dedekindschen Sätze scheint mir mehr
geeignet, die Natur der Zahlen zu verdunkeln als sie aufzuhellen.
[Letter from Cantor to Vivanti, 1888, April 2]
Cantor used his Fundamentalreihen:
Bemerkungen mit Bezug auf den Aufsatz: Zur Weierstraß-Cantorschen
Theorie der Irrationalzahlen.
[Math. Annalen Bd. 33, S. 476 (1889).]
Es möge mir gestattet sein, nur ganz kurz auf die Bedenken zu
antworten, welche Herr Illigens in bezug auf meine Theorie der
Irrationalzahlen ausgesprochen hat. ... sqrt(3) ist also nur ein
Zeichen für eine Zahl, welche erst noch gefunden werden soll, nicht
aber deren Definition. Letztere wird jedoch in meiner Weise etwa durch
(1,7, 1,73, 1,732, ...)
befriedigend gegeben.
Cantor talks about "his" theory of irrational numbers, neither
Cauchy's nor Dedekind's.
Regards, WM
Virgil
<2056d314-cabe-420c...@i6g2000yqj.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 28 Mai, 05:21, Virgil <virg...@nowhere.com> wrote:
> > > > > The problem boils down to the following:
> >
> > > > > En Am: m =< n <==> Am En m =< n [*]
> > > > > En Am: m =< n ==> Am En m =< n [**]
> >
> > > > > You know: Classical logic was obtained from finite sets ...
> > > > > Show me a finite set that obeys [**] but not [*].
>
> > > I said: For complete linear sets [*] is true.
> > > You said [*] is not true, but [**] is true.
> > > Weyl said: Classical logic was obtained from finite sets.
> > > Therefore I asked you: Show me a linear complete finite set, that
> > > makes your claim [**] right and my claim [*] wrong.
> >
> > What Weyl may have said in no way limits logic, classical or otherwise,
> > to finite sets. Or does WM claim that Weyl was speaking ex cathedra?
>
> Weyl has recognized, by the end of his life (when obtaining his
> doctorate under Hilbert he was preoccupied with Hilbert's opinion),
> that logic is not included into the 10 commandments, but has to be
> constructed or obtained as an abstraction from the reality. Therefore
> we have no guarantee that it is suitable for sets that cannot belong
> to reality. And I added: We cannot use logical theorems that are
> counterfactual, i.e., contrary to what we obtained from reality.
WM presumes that everything true can be, and must be, obtained only from
reality, but the laws of logic are derived from thoughts of an ideal
world and such thoughts are not obtained solely reality, but from
largely from an unreal vision of the ideal.
There
> is no model of a complete linear set in reality that makes [*] false
> and [**] true. But there are many models showing that [*] is true
> whenever [**] is true.
And many models for which "ExAy P(x,y) ==> AyEx P(x,y)" is false.
> >
> > > No. I do not use the implication only, I use the full equivalence. Of
> > > course the equivalence includes the implication [***] as well as the
> > > implication [**]
> >
> > Then WM is not operating in the same world as everyone else.
>
> Most of your ilk intermingle the worlds in which they work. They work
> in potential infinity if the question is put whether there is a
> largest natural number. But they work in actual infinity if the Cantor-
> diagonal or any other irrational number is concerned. Only the
> possibility to complete it by a limit process, leads to the erroneous
> assumption of uncountability.
Then WM better confine his attentions to areas in which no limiting
processes are wanted and no infinite sets are wanted, which excludes him
from all calculus.
>
> This had already been recognized by the late Alexander Zenkin, one of
> the brave scientists who dared to condemn this hypocritical behaviour
Scientists may mess with physics to their hearts content, but as
scientists, have no business messing with pure mathematics.
So if anyone introduced any "false logic", it would be physicists rather
than mathematicians.
--
Virgil
Virgil
<f8cdb05d-9469-4247...@3g2000yqk.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 28 Mai, 04:37, Virgil <virg...@nowhere.com> wrote:
>
> > > > There is nothing inconsistent or illogical in the existence of things
> > > > for which we do not have names.
> >
> > > Unless these things are names only.
> >
> > Then different names, being names only, would necessarily be different
> > things and then the name 1/2 and the name 2/4, being different things,
> > could not be equal.
>
> Why not? There are laws, i.e., other abstract things, that allow to
> find out whether two names are the same or not. There are many names
> identical to the name pi.
False. There is only one *name* that is identical to the name "pi", and
that is "pi" itself, but there are lots of names which name the same
number that "pi" names.
WM still does not understand the differnce between a name and the thing
named. That is only one of his major errors in logical comprehension.
> A minimum of brainwork is
> required to understand my texts, but really, it is a very small
> amount.
Actually, there is a major absence of brain work that is required to
accept WM's theses.
--
Virgil
Virgil
<ee23e72e-206f-48c9...@c9g2000yqm.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 27 Mai, 23:15, Virgil <virg...@nowhere.com> wrote:
>
> > > > > Of course there can be different models for N and there can be
> > > > > different names for the elements of N. But that does not matter. The
> > > > > natural numbers do not enter mathematics because someone "defines"
> > > > > them, names them, or makes models of them, but because the natural
> > > > > numbers are simply existing and mathematics is built upon them.
> >
> > > > But according to WM, no such thing as N can exist.
> > > > So WM wold throw out the naturals on which so much is built.
> >
> > > The question is not whether the complete set of all naturals exists.
> >
> > That is a question on which WM differs from the mainstream.
> > But WM cannot grant its existence in a given argument and then deny it
> > in the same argument and expect anyone to accept that sort of �argument.
>
> Grant the existence of two natural numbers m and n such that m/n = sqrt
> (2). Then falsify it.
> Grant the existence of a largest natural. Then falsify it.
> Grant the existence of a largest prime number. Then falsify it.
> Grant the existence of all natural numbers. Then falsify it.
>
> All these proofs are proofs by contradiction.
I have never seen a logically correct or complete falsification of the
existence of all naturals.
Note that an argument against the existence of numerals (names) for all
naturals does not suffice.
> >
> > > That question alrady is nearly as ridiculous as any affirmative
> > > answer.
> >
> > Wrong!
>
> I said "nearly"! This is a fuzzy quantification, not a sharp one. It
> cannot be wrong. One might qualify it as nearly wrong or nearly right,
> at most.
All of WM's logic is fuzzy, much to fuzzy for serious mathematics.
> >
> >
> >
> > > The question is whether we could inform someone who does not yet know,
> > > what we understand by the sequence of natural numbers.
> >
> > Children know from a very early age. Whom does WM think is left to
> > inform?
>
> How do children get to know what the real numbers are? Do they have
> Peano's axioms in their genes or in their strollers?
Quite possible in their genes.
> >
> > But non-static sets do not exist in any mathematical set theory.
>
> That is orthodox nonsense.
Then produce an example of a mathematical set theory is which there are
non-static sets.
> The natural numbers as a potetntially
> infinite set exist in every mathematical theory before BC (before
> Cantor) and in many mathematical theories AC, for instance in every
> theory that is taught in elementary schools as well as in universities
> outside the departments of "logics" and mathematics.
Balls! Set theories as taught in elementary schools almost all require a
fixed universal set in which all other sets are fixed proper subsets, a
la Venn diagrams. And this approach even carries over into some
introductory math classes in universities.
Certainly I have never seen such variable sets anywhere in either
England or the United states at any level from kindergarten through
postgraduate.
>
> Regards, WM
--
Virgil
Dik T. Winter
> On 27 Mai, 15:50, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > > > > The problem boils down to the following:
> > > > >
> > > > > En Am: m =< n <==> Am En m =< n [*]
> > > > > En Am: m =< n ==> Am En m =< n [**]
> > > > >
> > > > > You know: Classical logic was obtained from finite sets ...
> > > > > Show me a finite set that obeys [**] but not [*].
> > > >
> > > > The above is not contested: [**] implies [*].
> > >
> > > I said: For complete linear sets [*] is true.
> >
> > Not in the article to which I responded.
>
> But frequently I made use of what you call quatifier exchange and what
> is allowed in case of complete linear sets.
You think so, but you have to prove that it is valid for infinite complete
linear sets. Note that "classical logic is obtained from finite sets".
Nowhere in that quote the word linear is mentioned.
> > > You said [*] is not true, but [**] is true.
> >
> > That is not what I said.
>
> You said:
> The only thing that can be stated is (symbolically):
> E n A m P(m, n) -> A m E n P(m, n)
> not the reverse, this is just basic logic.
The reverse of
E n A m P(m, n) -> A m E n P(m, n)
is
E n A m P(m, n) <- A m E n P(m, n)
which is [***], neither [*] nor [**].
> > I said that for the case involved you have to
> > *prove* that it is true, because it is not generally true.
>
> It is generally true for complete linear sets. You have to prove that
> it is not.
It is not true for the infinite set of naturals.
(1) define FISON(n) be the set of naturals from 1 to n, that is: {1, ..., n}.
(2) A{m in N} E(n in N} such that FISON(m) subset FISON(n), trivial, take
n = m + 1.
(3) E{n in N} A{m in N} such that FISON(m) subset FISON(n), trivially false,
take m = n + 1.
Which part of this proof is wrong?
It clearly shows that
E n A m P(m, n) <- A m En P(m, n)
is false. Here with:
E n meaning E{n in N}
A n meaning A{n in N}
P(m, n) meaning FISON(m) subset FISON(n).
The first part of the implication is false while the second part of the
implication is true, and so the implication is false (all by classical
logic).
> > But now you include the word "linear". Where did "Weyl" include the
> > word "linear"?
>
> I did never claim that quantifier exchange is allowed in case of non-
> linear sets, like cyclic sets as, for instance, your dice. That would
> be nonsense. A simple example: Every country has a country that lies
> west of it. But there is no country that lies west of all countries.
But as Weyl did not include "linear" in his words, how can that quote
support your claim?
> > > > What is contested is that:
> > > > En Am: m =< n <== Am En m =< n [***]
> > > > implies [*]. And *that* is the form you do use.
...
> > Because the implication [**] is always true, the only part of the
> > equivalence that is new is the implication [***].
>
> For complete linear sets both are true, therefore [*] holds.
You just state without proof. Where in my proof above that it is false
did I go wrong?
> And you
> should recognize that actually complete linear sets obey [*].
Strange, I give above a proof that it does not hold. I did not use
"actual infinity" nor "potentially infinity", only logic and the axioms
of ZF.
> Only
> potentially infinite sets do not. But you mix up things. You claim the
> existence of a complete linear set but disregard the necessary
> consequence of completeness or linearity, namely the validity of [*].
Why is that a necessary consequence? Because you want it to be so? Can
you give a *mathematical* reason?
Herbert Newman
> Why is that a necessary consequence? Because you want it to be so? Can
> you give a *mathematical* reason?
Oh, you REALLY can't stop "arguing" with WM, right? Is this some sort of
compulsive act?
Herb
Jesse F. Hughes
Why don't you mind your own business? What compels you to these
confrontational posts? Just a natural asshole?
--
Jesse F. Hughes
"Wiles made somewhere around half a million dollars U.S. that I heard
about, and I know he didn't take major endorsements."
--JSH on the rewards of proving Fermat's last theorem.
William Hughes
>
> But frequently I made use of what you call quatifier exchange and what
> is allowed in case of complete linear sets.
Nope.
It is easy to show
quantifier exchange is allowed for linear sets
if and only if there is a largest element.
Since complete linear sets without largest element exist,
quantifier exchange is not valid.
- William Hughes
Peter Webb
"WM" <muec...@rz.fh-augsburg.de> wrote in message
news:ee23e72e-206f-48c9...@c9g2000yqm.googlegroups.com...
On 27 Mai, 23:15, Virgil <virg...@nowhere.com> wrote:
> > > > Of course there can be different models for N and there can be
> > > > different names for the elements of N. But that does not matter. The
> > > > natural numbers do not enter mathematics because someone "defines"
> > > > them, names them, or makes models of them, but because the natural
> > > > numbers are simply existing and mathematics is built upon them.
>
> > > But according to WM, no such thing as N can exist.
> > > So WM wold throw out the naturals on which so much is built.
>
> > The question is not whether the complete set of all naturals exists.
>
> That is a question on which WM differs from the mainstream.
> But WM cannot grant its existence in a given argument and then deny it
> in the same argument and expect anyone to accept that sort of argument.
Grant the existence of two natural numbers m and n such that m/n = sqrt
(2). Then falsify it.
Grant the existence of a largest natural. Then falsify it.
Grant the existence of a largest prime number. Then falsify it.
Grant the existence of all natural numbers. Then falsify it.
All these proofs are proofs by contradiction.
*************************
How do you falsify the existence of the set of all Natural numbers in ZF ?
ZF includes an axiom of infinity, which pretty much directly guarantees that
there is an infinite set of all finite ordinals. Assuming ZF is consistent,
this is also known to be independent of the other ZF axioms. So unless you
are arguing that ZF is inconsistent, I think you are going to have a lot of
problems showing that there cannot be a set of all natural numbers.
Virgil
"Peter Webb" <webbf...@DIESPAMDIEoptusnet.com.au> wrote:
> How do you falsify the existence of the set of all Natural numbers in ZF ?
> ZF includes an axiom of infinity, which pretty much directly guarantees that
> there is an infinite set of all finite ordinals. Assuming ZF is consistent,
> this is also known to be independent of the other ZF axioms. So unless you
> are arguing that ZF is inconsistent, I think you are going to have a lot of
> problems showing that there cannot be a set of all natural numbers.
WM keeps claiming that any system which allows a set of ALL natural
number to exist is necessarily inconsistent, though he has not produced
anything that qualifies as mathematically or logically valid proof.
WM must have skipped plane geometry as a child, and ever since, as he
has no notion of what constitutes a mathematically valid or logically
valid proof, but argues more like a politician.
--
Virgil
Dik T. Winter
I think it is justified if my name is in the Subject. But if you do not
like it, please kilfile me.
Herbert Newman
> I think it is justified if my name is in the Subject.
Yes, you are right. But I guess you see now the dependence between a crank
and its codependent partners. The crank (or at least this one) is craving
for attention. If you stop responding he is suffering (from this lack of
attention). [Imho one of the MAIN reasons why this loon stil resides in
this NG is the _constant_ attention he gets from _certain_ posters...]
> But if you do not like it, please killfile me.
Hell, no! You are a competent and intelligent participant in this NG. It
would be a shame to ignore your (more intelligent) contributions. [Though
this does in no way relativize the regrettable effects of your codependent
behavior concerning WM, if so.]
Herb
WM
> In article <37a98f77-6bc9-4cae-88e5-fcbadd1d0...@q2g2000vbr.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes:
> > On 27 Mai, 15:50, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> ...
> > > > > > The problem boils down to the following:
> > > > > >
> > > > > > En Am: m =< n <==> Am En m =< n [*]
> > > > > > En Am: m =< n ==> Am En m =< n [**]
> > > > > >
> > > > > > You know: Classical logic was obtained from finite sets ...
> > > > > > Show me a finite set that obeys [**] but not [*].
> > > > >
> > > > > The above is not contested: [**] implies [*].
> > > >
> > > > I said: For complete linear sets [*] is true.
> > >
> > > Not in the article to which I responded.
> >
> > But frequently I made use of what you call quatifier exchange and what
> > is allowed in case of complete linear sets.
>
> You think so, but you have to prove that it is valid for infinite complete
> linear sets. Note that "classical logic is obtained from finite sets".
> Nowhere in that quote the word linear is mentioned.
Nowhere in that quote the word union in mentioned. Nevertheless the
logical rules of unions are obtained from unions of finite sets. The
logical rules of linear sets are obtained from finite linear sets.
> > > > You said [*] is not true, but [**] is true.
> > >
> > > That is not what I said.
> >
> > You said:
> > The only thing that can be stated is (symbolically):
> > E n A m P(m, n) -> A m E n P(m, n)
> > not the reverse, this is just basic logic.
>
> The reverse of
> E n A m P(m, n) -> A m E n P(m, n)
> is
> E n A m P(m, n) <- A m E n P(m, n)
> which is [***], neither [*] nor [**].
I never said so. But [*] is [**] & [***]. Therefore [*] differs from
[**] only by the reverse.
>
> > > I said that for the case involved you have to
> > > *prove* that it is true, because it is not generally true.
> >
> > It is generally true for complete linear sets. You have to prove that
> > it is not.
>
> It is not true for the infinite set of naturals.
That is your claim. It is justified for potential infinity. It is
wrong for complete sets.
> (1) define FISON(n) be the set of naturals from 1 to n, that is: {1, ..., n}.
> (2) A{m in N} E(n in N} such that FISON(m) subset FISON(n), trivial, take
> n = m + 1.
> (3) E{n in N} A{m in N} such that FISON(m) subset FISON(n), trivially false,
> take m = n + 1.
> Which part of this proof is wrong?
The proof is correct for potential infinity. The proof is incorrect
for actual infinity. In that latter case you need not take an n that
is surpassed by m. Why don't you start with an n that has no greater
m?
>
> It clearly shows that
> E n A m P(m, n) <- A m En P(m, n)
> is false. Here with:
> E n meaning E{n in N}
> A n meaning A{n in N}
> P(m, n) meaning FISON(m) subset FISON(n).
> The first part of the implication is false while the second part of the
> implication is true, and so the implication is false (all by classical
> logic).
Not at all. By classical logic, a complete linear set has a last
element.
>
> > > But now you include the word "linear". Where did "Weyl" include the
> > > word "linear"?
> >
> > I did never claim that quantifier exchange is allowed in case of non-
> > linear sets, like cyclic sets as, for instance, your dice. That would
> > be nonsense. A simple example: Every country has a country that lies
> > west of it. But there is no country that lies west of all countries.
>
> But as Weyl did not include "linear" in his words, how can that quote
> support your claim?
There are many finite sets with many special properties that follow
from classical logic. One of them is that a complete linear set has a
lst element.
You drop the completeness condition in certain cases but you assume it
in case of Cantor's proof. That is cheating.
>
> > > > > What is contested is that:
> > > > > En Am: m =< n <== Am En m =< n [***]
> > > > > implies [*]. And *that* is the form you do use.
> ...
> > > Because the implication [**] is always true, the only part of the
> > > equivalence that is new is the implication [***].
> >
> > For complete linear sets both are true, therefore [*] holds.
>
> You just state without proof. Where in my proof above that it is false
> did I go wrong?
State before beginning whether the set that you assume is complete and
static, i.e., every element is actually existing, or potentially
infinite.
>
> > And you
> > should recognize that actually complete linear sets obey [*].
>
> Strange, I give above a proof that it does not hold. I did not use
> "actual infinity" nor "potentially infinity"
That is the point! You use the absence of element m when you choose n
= m - 1. But you use the non-absence of m when you execute Cantor's
proof. Then you do not admit that for every FISON(n) there is an m = n
+ 1 that is not in the proof.
>
> > Only
> > potentially infinite sets do not. But you mix up things. You claim the
> > existence of a complete linear set but disregard the necessary
> > consequence of completeness or linearity, namely the validity of [*].
>
> Why is that a necessary consequence? Because you want it to be so? Can
> you give a *mathematical* reason?
Every finite linear set obeys [*]. That is the mathematical reason.
Regards, WM
WM
> On May 27, 11:04 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
>
>
> > But frequently I made use of what you call quatifier exchange and what
> > is allowed in case of complete linear sets.
>
> Nope.
>
> It is easy to show
>
> quantifier exchange is allowed for linear sets
> if and only if there is a largest element.
>
linear sets were obtained from linear finite sets.
Everything else is matheology of Wolkenkuckucksheim.
Proof: A complete linear set without a last element is self
contradictory, as can be proven by the only logical rules that must be
accepted, namely those obtained from finite sets.
Regards, WM
WM
wrote:
> "WM" <mueck...@rz.fh-augsburg.de> wrote in message
>
> news:ee23e72e-206f-48c9...@c9g2000yqm.googlegroups.com...
> On 27 Mai, 23:15, Virgil <virg...@nowhere.com> wrote:
>
> > > > > Of course there can be different models for N and there can be
> > > > > different names for the elements of N. But that does not matter. The
> > > > > natural numbers do not enter mathematics because someone "defines"
> > > > > them, names them, or makes models of them, but because the natural
> > > > > numbers are simply existing and mathematics is built upon them.
>
> > > > But according to WM, no such thing as N can exist.
> > > > So WM wold throw out the naturals on which so much is built.
>
> > > The question is not whether the complete set of all naturals exists.
>
> > That is a question on which WM differs from the mainstream.
> > But WM cannot grant its existence in a given argument and then deny it
> > in the same argument and expect anyone to accept that sort of argument.
>
> Grant the existence of two natural numbers m and n such that m/n = sqrt
> (2). Then falsify it.
> Grant the existence of a largest natural. Then falsify it.
> Grant the existence of a largest prime number. Then falsify it.
> Grant the existence of all natural numbers. Then falsify it.
>
> All these proofs are proofs by contradiction.
>
> *************************
>
> How do you falsify the existence of the set of all Natural numbers in ZF ?
> ZF includes an axiom of infinity, which pretty much directly guarantees that
> there is an infinite set of all finite ordinals.
What about classical arithmetics with an axiom that sqrt(2) is a
rational number?
Regards, WM
Virgil
<ea1c1694-68e6-44a3...@a36g2000yqc.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
WM's suggested system is provably inconsistent so his suggestion is
about as useful as tits on a bull, whereas ZF has no known
inconsistencies that are logically derivable without imposing other
assumptions.
--
Virgil
Virgil
<5df917e3-517f-4a38...@t21g2000yqi.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 29 Mai, 03:58, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > In article
> > <37a98f77-6bc9-4cae-88e5-fcbadd1d0...@q2g2000vbr.googlegroups.com>
> > WM <mueck...@rz.fh-augsburg.de> writes:
> > > On 27 Mai, 15:50, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > ...
> > > > > > > The problem boils down to the following:
> > > > > > >
> > > > > > > En Am: m =< n <==> Am En m =< n [*] En Am: m =< n
> > > > > > > ==> Am En m =< n [**]
> > > > > > >
> > > > > > > You know: Classical logic was obtained from finite
> > > > > > > sets ... Show me a finite set that obeys [**] but not
> > > > > > > [*].
> > > > > >
> > > > > > The above is not contested: [**] implies [*].
> > > > >
> > > > > I said: For complete linear sets [*] is true.
> > > >
> > > > Not in the article to which I responded.
> > >
> > > But frequently I made use of what you call quatifier exchange
> > > and what is allowed in case of complete linear sets.
> >
> > You think so, but you have to prove that it is valid for infinite
> > complete linear sets. Note that "classical logic is obtained from
> > finite sets". Nowhere in that quote the word linear is mentioned.
>
> Nowhere in that quote the word union in mentioned. Nevertheless the
> logical rules of unions are obtained from unions of finite sets.
While those rules may be suggested by looking at finite sets, the rules
themselves are not limited by the limitations of finite sets.
When one says that the union of a set of sets is the set of all members
of those member sets, there is no actual or implied restriction on any
of those sets to be finite.
> The logical rules of linear sets are obtained from finite linear
> sets.
That the rules for general well ordered sets may be suggested by finite
sets of finite well ordered sets, does not mean that when not finite
they must behave as if they were finite.
That WM is incapable of dealing honestly with infinite sets does not
limit everyone to his incapacity.
> >
> > > > I said that for the case involved you have to
> > > > *prove* that it is true, because it is not generally true.
> > >
> > > It is generally true for complete linear sets. You have to prove
> > > that it is not.
Complete in this case meaning finite. Since WM cannot prove that his
claims hold except for finite "linear sets", which are merely finite
well-ordered sets, no one has to assume that they must hold for
arbitrary well-ordered sets, and there is good reason to suppose
otherwise.
> >
> > It is not true for the infinite set of naturals.
>
> That is your claim. It is justified for potential infinity.
But outside of the tiny world of WM's MathUnrealism, there are
"potential infiniteness" cannot be applied to sets.
> It is wrong for complete sets.
By which WM means finite ones, but no one has claimed otherwise.
>
> The proof is correct for potential infinity.
But outside of the tiny world of WM's MathUnrealism, there are
"potential infiniteness" cannot be applied to sets.
> The proof is incorrect
> for actual infinity. In that latter case you need not take an n that
> is surpassed by m. Why don't you start with an n that has no greater
> m?
If one double quantifies with "AxEy" or " EyAx"one is not allowed to
constrain x, but must consider all x, even those ones wold prefer not to
consider.
So that one needs to consider not only n < m but also n = m and n > m.
> Not at all. By classical logic, a complete linear set has a last
> element.
Only if "complete" requires "finite", which,outside of the tiny world of
WM's MathUnrealism, it does not.
> >
> > > > But now you include the word "linear". Where did "Weyl"
> > > > include the word "linear"?
> > >
> > > I did never claim that quantifier exchange is allowed in case of
> > > non- linear sets, like cyclic sets as, for instance, your dice.
> > > That would be nonsense. A simple example: Every country has a
> > > country that lies west of it. But there is no country that lies
> > > west of all countries.
> >
> > But as Weyl did not include "linear" in his words, how can that
> > quote support your claim?
>
> There are many finite sets with many special properties that follow
> from classical logic. One of them is that a complete linear set has a
> lst element.
What is incomplete or non-linear about the set of rational integers?
>
> You drop the completeness condition in certain cases but you assume
> it in case of Cantor's proof. That is cheating.
Cantor, in his diagonal proof of the incompleteness of any list of
binary sequences, does NOT assume that the given list is complete.
He merely proves that any given list is incomplete.
The Cantor challenge is: Provide me with a list of infinite binary
sequences and I will show you how to find one not in that list.
So until WM, or someone, can provide a list of infinite binary sequences
for which the Cantor rule fails to provide a non-member of the list,
Cantor wins.
And WM loses.
> >
> > > > > > What is contested is that:
> > > > > > En Am: m =< n <== Am En m =< n [***]
> > > > > > implies [*]. And *that* is the form you do use.
> > ...
> > > > Because the implication [**] is always true, the only part of
> > > > the equivalence that is new is the implication [***].
> > >
> > > For complete linear sets both are true, therefore [*] holds.
> >
> > You just state without proof. Where in my proof above that it is
> > false did I go wrong?
>
> State before beginning whether the set that you assume is complete
> and static, i.e., every element is actually existing, or potentially
> infinite.
The set of all naturals, like every set in any mathematically sane set
theory, is static. It is "complete" only in the sense that it is not
merely potential but is actual. Anything that will be a member of it
tomorrow was already a member yesterday. It is well ordered, which means
that every non-empty subset has a smallest (but need not have a largest)
member.
> > > And
> > > you
> > > should recognize that actually complete linear sets obey [*].
> >
> > Strange, I give above a proof that it does not hold. I did not use
> > "actual infinity" nor "potentially infinity"
>
> That is the point! You use the absence of element m when you choose
> n = m - 1. But you use the non-absence of m when you execute Cantor's
> proof. Then you do not admit that for every FISON(n) there is an m =
> n + 1 that is not in the proof.
Nonsense. N is a well ordered set, but has no maximal member.
Thus for every m in N there is an n in N such that n > m.
"AmeN EneN m > n" says, in effect, that there must exist some function,
say f, from N to N such that f(n) = M.
In this case, f(n) = n+1 is such a function.
So that if WM objects, he must find some member, say n, of N for which
(n+1) is not a member of N.
>
>
> >
> > >
> > > Only
> > > potentially infinite sets do not. But you mix up things. You
> > > claim the existence of a complete linear set but disregard the
> > > necessary consequence of completeness or linearity, namely the
> > > validity of [*].
> >
> > Why is that a necessary consequence? Because you want it to be so?
> > Can you give a *mathematical* reason?
>
> Every finite linear set obeys [*]. That is the mathematical reason.
But no infinite well-ordered set does, and such sets exist outside of
WM's wee wee world.
--
Virgil
Virgil
<82f69cbc-0f68-4c74...@h18g2000yqj.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 29 Mai, 05:51, William Hughes <wpihug...@hotmail.com> wrote:
> > On May 27, 11:04 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> >
> >
> > > But frequently I made use of what you call quatifier exchange and what
> > > is allowed in case of �complete linear sets.
> >
> > Nope.
> >
> > �It is easy to show
> >
> > � � �quantifier exchange is allowed for linear sets
> > � � �if and only if there is a largest element.
> >
> Rules of logic were obtained from finite sets. Rules of logic for
> linear sets were obtained from linear finite sets.
But we have linear (meaning well ordered) non-finite sets outside of
WM's wee wee word of MathUnrealism.
>
> Everything else is matheology of Wolkenkuckucksheim.
>
> Proof: A complete linear set without a last element is self
> contradictory, as can be proven by the only logical rules that must be
> accepted, namely those obtained from finite sets.
If only finite sets can exist then there are only finitely many of them
, and the union of all finitely many of them must itself be a finite
set: a finite universe outside of which there can be nothing to add to
it when the things inside are used up.
But if there are only finitely many elements in that universal set, then
WM's potentially infinite sets, being of necessity subsets of a finite
universal set, can not be potentially infinite, as they must eventually
exhaust their finite universal set.
So WM's own rules kill off his idiotic notion of potential infiniteness.
--
Virgil
Peter Webb
>> Grant the existence of two natural numbers m and n such that m/n = sqrt
>> (2). Then falsify it.
>> Grant the existence of a largest natural. Then falsify it.
>> Grant the existence of a largest prime number. Then falsify it.
>> Grant the existence of all natural numbers. Then falsify it.
>>
>> All these proofs are proofs by contradiction.
>>
>> *************************
>>
>> How do you falsify the existence of the set of all Natural numbers in ZF
>> ?
>> ZF includes an axiom of infinity, which pretty much directly guarantees
>> that
>> there is an infinite set of all finite ordinals.
>
> What about classical arithmetics with an axiom that sqrt(2) is a
> rational number?
This does not answer my question.
But I will answer yours for you. If you create a version of "classical" (=
"standard" ?) arithmetic with an axiom that sqrt(2) is rational, it would be
inconsistent, and hence useless.
Now how about answering my question. How do you falsify the existence of the
set of all Natural numbers in ZF, as you claimed you could?
>
> Regards, WM
Dik T. Winter
> On Fri, 29 May 2009 11:28:53 GMT Dik T. Winter wrote:
>
> > I think it is justified if my name is in the Subject.
>
> Yes, you are right. But I guess you see now the dependence between a crank
> and its codependent partners. The crank (or at least this one) is craving
> for attention. If you stop responding he is suffering (from this lack of
> attention). [Imho one of the MAIN reasons why this loon stil resides in
> this NG is the _constant_ attention he gets from _certain_ posters...]
I do not think so. WM has now two books about mathematics on his name. The
first one he got published some time ago by some vanity press (Shaker Verlag).
(It is vanity press because you have to pay for your book to get published,
for what you pay you get a number of copies, everything further is handled by
print-on-demand.) I did a review of it that is available through my
web-pages. You can find it at:
<http://homepages.cwi.nl/~dik/english/mathematics/mueck/index.html>
The second (recently published, 2008 I think) is however by a reputable
publisher. It is intended as textbook for students, and used by WM at his
university. You may expect an influx of German students that think what
WM writes is gospel. There are already some around here: Albrecht (if I
remember well he is connected to a major German technical institution) and
also Eckard Blumschein (emiritus professor of the Magdeburg University in
some technical subject). Han de Bruijn was also one of them, but recently
he seems to have changed opinion.
What i find is a growing clash between technical physists and mathematicians.
The technical physists do not understand what mathematicians are doing, and
think they are subordinate to their needs, but on the other hand do not
understand numerical mathematics either (which caters for their needs).
They just think that numbers are absolute things.
Herbert Newman
>> ... you are right. But I guess you see now the dependence between a crank
>> and its codependent partners. The crank (or at least this one) is craving
>> for attention. If you stop responding he is suffering (from this lack of
>> attention). [Imho one of the MAIN reasons why this loon stil resides in
>> this NG is the _constant_ attention he gets from _certain_ posters...]
>>
> I do not think so. [...]
Of course not, since you are suffering from co-dependence. *sigh*
Anyway, WM was succesfully driven out from de.sci.mathematic since no one
was willing to "argue" with him the way _certain_ poster in this NG do (for
such a long time).
EOD
Herb
Jesse F. Hughes
> The second (recently published, 2008 I think) is however by a reputable
> publisher. It is intended as textbook for students, and used by WM at his
> university.
That is horrible news indeed.
WM must be the most successful crank I've ever seen. He *teaches* and
*publishes* his delusions! How remarkable and how regrettable.
(Yes, Abian was well-respected as a mathematician, but did not teach
or publish his cranky "discoveries", as far as I know.)
So who the hell published a WM textbook?
--
Jesse F. Hughes
"Just goes to tell you. If you make a major discovery, and some stupid
interviewer asks you if you're the greatest mathematician of all time,
just say no." -- practical advice from James S. Harris
Jesse F. Hughes
Mueckenheim is also cited as an authority on infinity at this
educational site:
http://www.learner.org/courses/mathilluminated/units/3/resources/index.php
I didn't see any evidence that WM's horrid arguments actually
influenced the text of the site. Nonetheless, students who want to
learn more about Cantor are directed to WM's illogical blatherings.
Still, I blame most WM's employer for putting him in a position of
authority to educate students on exactly that material he has shown no
capacity to understand. I can't comprehend how that situation has
remained. I'm sure that WM is tenured, but that doesn't entail that
he can teach bad mathematical reasoning in the classroom, does it?
--
"It's one of the easiest tickets to true fame--not this silly stuff
where people cheer you for a few years and then forget about you--but
the kind of fame where school kids have to read your biography and do
reports on you." -- Another reason to support James S. Harris.
WM
wrote:
> >> Grant the existence of two natural numbers m and n such that m/n = sqrt
> >> (2). Then falsify it.
> >> Grant the existence of a largest natural. Then falsify it.
> >> Grant the existence of a largest prime number. Then falsify it.
> >> Grant the existence of all natural numbers. Then falsify it.
>
> >> All these proofs are proofs by contradiction.
>
> >> *************************
>
> >> How do you falsify the existence of the set of all Natural numbers in ZF
> >> ?
> >> ZF includes an axiom of infinity, which pretty much directly guarantees
> >> that
> >> there is an infinite set of all finite ordinals.
>
> > What about classical arithmetics with an axiom that sqrt(2) is a
> > rational number?
>
> This does not answer my question.
>
> But I will answer yours for you. If you create a version of "classical" (=
> "standard" ?) arithmetic with an axiom that sqrt(2) is rational, it would > be
> inconsistent, and hence useless.
Same with the axiom of infinity: “There exists a complete linear
infinite set” is a self-contradictory similar to “there exists a pair
of natural numbers, a and b, such that b^2 = 2a^2.
>
> Now how about answering my question. How do you falsify the existence of the
> set of all Natural numbers in ZF, as you claimed you could?
It is simple: ... classical logic was abstracted from the mathematics
of finite sets and their subsets .... Forgetful of this limited
origin, one afterwards mistook that logic for something above and
prior to all mathematics, and finally applied it, without
justification, to the mathematics of infinite sets. [Hermann Weyl,
"Mathematics and logic: A brief survey serving as a preface to a
review of The Philosophy of Bertrand Russell", American Mathematical
Monthly 53: 2–13]
Show me a complete finite linear set that does not allow for
quantifier reversal.
En Am: m =< n <==> Am En m =< n [*] .
Therefore: Either [*] holds or the set is not complete but allows for
extension.
Then we have only
En Am: m =< n ==> Am En m =< n [**]
because not all elements are readily available.
That is called a potentially infinite set. But in this case there is
no chance to prove uncountability.
This had already been recognized by the late Alexander Zenkin, one of
the brave scientists who dared to condemn this hypocritical
behaviour:
Cantor's 'paradise' as well as all modern axiomatic set theory is
based on the (self-contradictory) concept of actual infinity. Cantor
emphasized plainly and constantly that all transfinite objects of his
set theory are based on the actual infinity. Modern AST-people try to
persuade us to believe that the AST does not use actual infinity. It
is an intentional and blatant lie, since if infinite sets, X and N,
are potential, then the uncountability of the continuum becomes
unprovable, but without the notorious uncountablity of continuum the
modern AST as a whole transforms into a long twaddle about nothing.
Resume: The internal contradiction in set theory is veiled by mixing
up potential and actual infinity. That is the reason why set theorists
usually refuse to specify which infinity they apply. Most even pretend
(or profess) not to know the difference.
Regards, WM
WM
> > It is wrong for complete sets.
>
> By which WM means finite ones, but no one has claimed otherwise.
Complete means finished = finite.
> Cantor, in his diagonal proof of the incompleteness of any list of
> binary sequences, does NOT assume that the given list is complete.
>
> He merely proves that any given list is incomplete.
>
> The Cantor challenge is: Provide me with a list of infinite binary
> sequences and I will show you how to find one not in that list.
That is not difficult to satisfy : The complete list of all real
numbers starts just at that position n+1 where you will cease to seek
it.
Regards, WM
WM
> But if there are only finitely many elements in that universal set, then
> WM's potentially infinite sets, being of necessity subsets of a finite
> universal set, can not be potentially infinite, as they must eventually
> exhaust their finite universal set.
Your universe of numbers is all numbers that you can construct. If you
increase your capabilities, your universe grows. That's why it is
infinite.
Regards, WM
WM
> but the laws of logic are derived from thoughts of an ideal
> world and such thoughts are not obtained solely reality, but from
> largely from an unreal vision of the ideal.
The result is in due shape.
>
> There
>
> > is no model of a complete linear set in reality that makes [*] false
> > and [**] true. But there are many models showing that [*] is true
> > whenever [**] is true.
>
> And many models for which "ExAy P(x,y) ==> AyEx P(x,y)" is false.
Not linear models. And no others!
>
> Then WM better confine his attentions to areas in which no limiting
> processes are wanted and no infinite sets are wanted, which excludes him
> from all calculus.
Either Cantor’s diagonal proof shows that the limit of all omega
indices can be reached by defining b_n =/= a_n for every n.
Then my binary tree proof shows that the limit of all omega levels can
be reached by showing that the number of distinct lines is countable
at every level n. Then all reals are countable. (Only a very confused
mind could consider the possibility that paths of the infinite tree
and decimal expansions of real numbers might represent different
mathematical objects).
Or the limit of the paths in the binary tree does not yield all real
numbers.
Then Cantors diagonal proof does not establish complete real number
either, and the proof is void.
>
>
>
> > This had already been recognized by the late Alexander Zenkin, one of
> > the brave scientists who dared to condemn this hypocritical behaviour
>
> Scientists may mess with physics to their hearts content, but as
> scientists, have no business messing with pure mathematics.
Mathematics is science done by scientists. Matheology is what you may
have in mind. And in fact : That has as much to do with science as has
astrology to with astronomy.
>
> So if anyone introduced any "false logic", it would be physicists rather
> than mathematicians.
I should refrain from calling what you and your ilk do : matheology.
It could be understood by lurkers as if your hobby was based upon
logic.
Regards, WM
Virgil
<aa348889-9e3b-4299...@s12g2000yqi.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
Perhaps WM will cease to seek it at some finite natural, but
mathematicians are not so easily discouraged, and will persist beyond
the successor of any such finite natural.
And according to the terms of the Cantor challenge in that proof, it is
up to those who challenger Cantor to provide the list of seequnces, and
Cantor only needs show there is a sequence not in that list. Which
Cantor succeeds in showing can always be done.
It is a shame that WM never seems able to comprehend what the Cantor
"diagonal" proof actually says.
--
Virgil
Virgil
Ralf Bader
>
> Mueckenheim is also cited as an authority on infinity at this
> educational site:
>
>
http://www.learner.org/courses/mathilluminated/units/3/resources/index.php
>
> I didn't see any evidence that WM's horrid arguments actually
> influenced the text of the site. Nonetheless, students who want to
> learn more about Cantor are directed to WM's illogical blatherings.
Did you consider writing to a person on the advisory board
http://www.learner.org/courses/mathilluminated/about/advisors.php
about this? That might have more effect than another round of discussion
with Mueckenheim in this group (but there could be an exercise in the main
text asking to explain why those papers are scrap)
> Still, I blame most WM's employer for putting him in a position of
> authority to educate students on exactly that material he has shown no
> capacity to understand. I can't comprehend how that situation has
> remained. I'm sure that WM is tenured, but that doesn't entail that
> he can teach bad mathematical reasoning in the classroom, does it?
That is a school for engineers and the like, not for science or mathematics
majors. Mueckenheim obviously came there as a physicist, and what he does
is to teach physics and mathematics to engineering students. I recently had
a short look into his textbook, and while the preface is kind of crazy, the
body of the book makes the impression of being fairly standard, on a first
sight.
Moreover, the students at that school seem to be obliged to take some
courses of a general interst nature which they can select according to
personal preference, and Mueckenheim's lecture on the "history of the
infinite" is among these courses.
As it is a state school one could send a formal complaint
("Dienstaufsichtsbeschwerde") to that school or the ministry responsible
for the state school system, of course in German language, and they would
have to answer it. If Mueckenheim gets to know this then he will blather
around that he had been denounced like Zermelo who had been driven out of
Freiburg university because he didn't perform the "Hitler greeting" well
enough.
A couple of years ago I stumbled in a public library across this book
http://www.amazon.de/Götzen-Computer-Kritik-unreinen-Vernunft/dp/3879592942/ref=sr_1_7?ie=UTF8&s=books&qid=1243715629&sr=1-7
"Gödel, Götzen und Computer", by Max Woitschach. The author was primarily
working for the german branch of IBM, and as a secondary job he lectured at
a school like Mueckenheim's. And that book contained his collected
misunderstandings (different from Mueckenheim's) about mathematics. So
Mueckenheim has predecessors. Woitschach deceased in 1993, and now there is
an endowment for "ideology-free science" with an annual award, see
http://www.woitschach-stiftung.de/
Ralf
Virgil
<503d7653-f1a6-4d24...@i6g2000yqj.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 28 Mai, 20:39, Virgil <virg...@nowhere.com> wrote:
> > but the laws of logic are derived from thoughts of an ideal
> > world and such thoughts are not obtained solely reality, but from
> > largely from an unreal vision of the ideal.
>
> The result is in due shape.
> >
> > �There
> >
> > > is no model of a complete linear set in reality that makes [*] false
> > > and [**] true. But there are many models showing that [*] is true
> > > whenever [**] is true.
> >
> > And many models �for which "ExAy P(x,y) ==> AyEx P(x,y)" is false.
>
> Not linear models.
The naturals and the integers, and the set of unit fractions, and a lot
of other models.
>
> >
> > Then WM better confine his attentions to areas in which no limiting
> > processes are wanted and no infinite sets are wanted, which excludes him
> > from all calculus.
>
> Either Cantor�s diagonal proof shows that the limit of all omega
> indices can be reached by defining b_n =/= a_n for every n.
Or what?
> Then my binary tree proof shows that the limit of all omega levels can
> be reached by showing that the number of distinct lines is countable
> at every level n.
Such a "proof" is no proof, as it requires assumptions contrary to fact.
Then all reals are countable.
In WM's personal world of MathUnrealism, he may, if he chooses , declare
that 2 = 1, but he has no power to declare anything new outside that
world without better "proofs" than he has yet been able to create.
> (Only a very confused
> mind could consider the possibility that paths of the infinite tree
> and decimal expansions of real numbers might represent different
> mathematical objects).
The WM must be saying that every path in a complete infinite binary tree
"is identical to" the decimal expansion of a real number AND every
decimal expansion of a real number "is identical to" a path in such a
binary tree.
Nonsense. For one thing, 0.1(0) and 0.0(1), where (x) indicates an
infinite sequence of repetitions of x, represent the same real but
different paths.
>
> Or the limit of the paths in the binary tree does not yield all real
> numbers.
They can be made to produce reals, but to get a bijection between
strings and reals is not anywhere as trivial as WM thinks it is, and
there is no single way to do it that is more natural and obvious that
anall others
> Then Cantors diagonal proof does not establish complete real number
> either, and the proof is void.
Cantor's diagonal proof does not refer to real numbers at all.
Suitable modifications of it done later by others also shows that no
list of reals can exhaust all reals. But Cantor had already proved that
separately before coming up with his diagonal argument.
>
> >
> >
> >
> > > This had already been recognized by the late Alexander Zenkin, one of
> > > the brave scientists who dared to condemn this hypocritical behaviour
> >
> > Scientists may mess with physics to their hearts content, but as
> > scientists, have no business messing with pure mathematics.
>
> Mathematics is science done by scientists.
That may be the egotist scientist's pint of view, but it is a false view.
> Matheology is what you may
> have in mind. And in fact : That has as much to do with science as has
> astrology to with astronomy.
What goes on in WM's wee weird world of MathUnrealism is not
mathematics, so much as bad engineering.
>
> >
> > So if anyone introduced any "false logic", it would be physicists rather
> > than mathematicians.
>
> I should refrain from calling what you and your ilk do : matheology.
> It could be understood by lurkers as if your hobby was based upon
> logic.
I have all sorts of hobbies. One of them is revealing the foolishness of
egotistists like WM.
--
Virgil
Ralf Bader
> In article
> <aa348889-9e3b-4299...@s12g2000yqi.googlegroups.com>,
> WM <muec...@rz.fh-augsburg.de> wrote:
>
>> On 30 Mai, 00:18, Virgil <virg...@nowhere.com> wrote:
>>
>> > > It is wrong for complete sets.
>> >
>> > By which WM means finite ones, but no one has claimed otherwise.
>>
>> Complete means finished = finite.
>> > Cantor, in his diagonal proof of the incompleteness of any list of
>> > binary sequences, does NOT assume that the given list is complete.
>> >
>> > He merely proves that any given list is incomplete.
>> >
>> > The Cantor challenge is: Provide me with a list of infinite binary
>> > sequences and I will show you how to find one not in that list.
>>
>> That is not difficult to satisfy : The complete list of all real
>> numbers starts just at that position n+1 where you will cease to seek
>> it.
>
> Perhaps WM will cease to seek it at some finite natural, but
> mathematicians are not so easily discouraged, and will persist beyond
> the successor of any such finite natural.
WHAT will they do? Persist to seek "the complete list of all real numbers"?
Certainly not.
Ralf
Virgil
<aa348889-9e3b-4299...@s12g2000yqi.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
If WM cannot even produce a list of binaries for Cantor's technique to
work on, or a list of reals for the modified diagonal proof to work on,
then WM loses by default.
And whenever WM, or anyone else, does produce one of those lists, it
will easily be shown to be incomplete.
--
Virgil
Virgil
<eebb8310-e58d-4f85...@c19g2000yqc.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
If all sets are finite, but ever changing as WM insists, then one can
never have a set theory at all, as sets, including universal sets if
any, do not change. Whatever WM is going on about, it is not sets.
So in WM's world there can never be any sets.
--
Virgil
Virgil
<4f20ac0a-a6b7-4a20...@c19g2000yqc.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> >
> > But I will answer yours for you. If you create a version of "classical" (=
> > "standard" ?) arithmetic with an axiom that sqrt(2) is rational, it would >
> > be
> > inconsistent, and hence useless.
>
> Same with the axiom of infinity: �There exists a complete linear
> infinite set� is a self-contradictory similar to �there exists a pair
> of natural numbers, a and b, such that b^2 = 2a^2.
WM keeps claiming this but never manages to prove it. All of his
attempts at proofs require assuming a priori and without proof that no
infinite set can exist.
> >
> > Now how about answering my question. How do you falsify the existence of
> > the
> > set of all Natural numbers in ZF, as you claimed you could?
>
> It is simple: ... classical logic was abstracted from the mathematics
> of finite sets and their subsets .... Forgetful of this limited
> origin, one afterwards mistook that logic for something above and
> prior to all mathematics, and finally applied it, without
> justification, to the mathematics of infinite sets. [Hermann Weyl,
> "Mathematics and logic: A brief survey serving as a preface to a
> review of The Philosophy of Bertrand Russell", American Mathematical
> Monthly 53: 2�13]
So from what cathedra was Weyl speaking?
Arguing from citing authorities may work in law courts, but carries
little weight mathematics.
>
> Show me a complete finite linear set that does not allow for
> quantifier reversal.
> En Am: m =< n <==> Am En m =< n [*] .
Show me that every infinite well ordered set does allow it.
>
> Therefore: Either [*] holds or the set is not complete but allows for
> extension.
Or the set is infintie, well-ordered, and has no maximal member.
> Then we have only
> En Am: m =< n ==> Am En m =< n [**]
> because not all elements are readily available.
Any set in a sane set theory has all members equally "available".
> That is called a potentially infinite set.
That is called nonsense in set theories, as is much of WM's
MathUnrealism.
--
Virgil
Peter Webb
"WM" <muec...@rz.fh-augsburg.de> wrote in message
news:4f20ac0a-a6b7-4a20...@c19g2000yqc.googlegroups.com...
On 30 Mai, 02:56, "Peter Webb" <webbfam...@DIESPAMDIEoptusnet.com.au>
wrote:
> >> Grant the existence of two natural numbers m and n such that m/n = sqrt
> >> (2). Then falsify it.
> >> Grant the existence of a largest natural. Then falsify it.
> >> Grant the existence of a largest prime number. Then falsify it.
> >> Grant the existence of all natural numbers. Then falsify it.
>
> >> All these proofs are proofs by contradiction.
>
> >> *************************
>
> >> How do you falsify the existence of the set of all Natural numbers in
> >> ZF
> >> ?
> >> ZF includes an axiom of infinity, which pretty much directly guarantees
> >> that
> >> there is an infinite set of all finite ordinals.
>
> > What about classical arithmetics with an axiom that sqrt(2) is a
> > rational number?
>
> This does not answer my question.
>
> But I will answer yours for you. If you create a version of "classical" (=
> "standard" ?) arithmetic with an axiom that sqrt(2) is rational, it would
> > be
> inconsistent, and hence useless.
Same with the axiom of infinity: �There exists a complete linear
infinite set� is a self-contradictory similar to �there exists a pair
of natural numbers, a and b, such that b^2 = 2a^2.
*****************************
I understand your claim. I was asking fir a proof of it, not you restating
it.
>
> Now how about answering my question. How do you falsify the existence of
> the
> set of all Natural numbers in ZF, as you claimed you could?
It is simple: ... classical logic was abstracted from the mathematics
of finite sets and their subsets .... Forgetful of this limited
origin, one afterwards mistook that logic for something above and
prior to all mathematics, and finally applied it, without
justification, to the mathematics of infinite sets. [Hermann Weyl,
"Mathematics and logic: A brief survey serving as a preface to a
review of The Philosophy of Bertrand Russell", American Mathematical
Monthly 53: 2�13]
*******************************
Well, Weyl never claimed or proved your assertion, so starting off with a
quote from him is a piss-poor starting point. I assume you have no proof,
which is why you are just in random quote mode.
Show me a complete finite linear set that does not allow for
quantifier reversal.
En Am: m =< n <==> Am En m =< n [*] .
************************
Its your proof. Either provide such a set yourself, or prove that none
exists.
Therefore: Either [*] holds or the set is not complete but allows for
extension.
*****************************
You better define "extension", and show its relationship to "complete". Just
in case, you better define "complete" as well, in case you are using it
differently to everybody else.
Then we have only
En Am: m =< n ==> Am En m =< n [**]
because not all elements are readily available.
**************************
You better define "readily available" while you are at it.
That is called a potentially infinite set. But in this case there is
no chance to prove uncountability.
*************************
Define "potentially infinite" as opposed to "infinite". Show that this does
not allow uncountability to be proved.
This had already been recognized by the late Alexander Zenkin, one of
the brave scientists who dared to condemn this hypocritical
behaviour:
Cantor's 'paradise' as well as all modern axiomatic set theory is
based on the (self-contradictory) concept of actual infinity. Cantor
emphasized plainly and constantly that all transfinite objects of his
set theory are based on the actual infinity. Modern AST-people try to
persuade us to believe that the AST does not use actual infinity. It
is an intentional and blatant lie, since if infinite sets, X and N,
are potential, then the uncountability of the continuum becomes
unprovable, but without the notorious uncountablity of continuum the
modern AST as a whole transforms into a long twaddle about nothing.
Resume: The internal contradiction in set theory is veiled by mixing
up potential and actual infinity. That is the reason why set theorists
usually refuse to specify which infinity they apply. Most even pretend
(or profess) not to know the difference.
*************************
What *is* the difference?
Regards, WM
**************************
Gee, such a big claim you make. Yet no proof. Not even a lame attempt. Even
cranks generally try and produce bad proofs, you don't even bother trying to
do that. You are actually sub-crank. To become merely crank, define some
terms, and try and make your rant at least look like a proof.
Virgil
<503d7653-f1a6-4d24...@i6g2000yqj.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 28 Mai, 20:39, Virgil <virg...@nowhere.com> wrote:
> > but the laws of logic are derived from thoughts of an ideal
> > world and such thoughts are not obtained solely reality, but from
> > largely from an unreal vision of the ideal.
>
> The result is in due shape.
> >
> > �There
> >
> > > is no model of a complete linear set in reality that makes [*] false
> > > and [**] true. But there are many models showing that [*] is true
> > > whenever [**] is true.
> >
> > And many models �for which "ExAy P(x,y) ==> AyEx P(x,y)" is false.
>
> Not linear models.
N is such a model, as are the set of all predecessors of any limit
ordinal.
> And no others!
Actually as many other as there are limit ordinals, which is more than
WM can count.
>
> >
> > Then WM better confine his attentions to areas in which no limiting
> > processes are wanted and no infinite sets are wanted, which excludes him
> > from all calculus.
>
> Either Cantor�s diagonal proof shows that the limit of all omega
> indices can be reached by defining b_n =/= a_n for every n.
Or WM is again dead wrong and an ass.
--
Virgil
WM
> And whenever WM, or anyone else, does produce one of those lists, it
> will easily be shown to be incomplete.
I told you alredy several times that logic fails in case of infinite
sets. Do not argue that Cantor's proof is correct. Argue why my proof
is incorrect. Argue facts, not opinions. See the thread concerning the
reactions to the binary tree. Choose one or more of those counter
arguments and try to convince people who are not trapped in
matheology. That would be more commendable than parroting your stuff.
Regards, WM
WM
> In article
> <eebb8310-e58d-4f85-9eda-37af9b74b...@c19g2000yqc.googlegroups.com>,
>
> WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 30 Mai, 00:31, Virgil <virg...@nowhere.com> wrote:
>
> > > But if there are only finitely many elements in that universal set, then
> > > WM's potentially infinite sets, being of necessity subsets of a finite
> > > universal set, can not be potentially infinite, as they must eventually
> > > exhaust their finite universal set.
>
> > Your universe of numbers is all numbers that you can construct. If you
> > increase your capabilities, your universe grows. That's why it is
> > infinite.
>
> If all sets are finite, but ever changing as WM insists, then one can
> never have a set theory at all, as sets, including universal sets if
> any, do not change.
This is stubborn nonsense. Of course sets are conceivable that can
change. The complete sets like N have only been introduced because
Cantor had a strong religious affinity.
Regards, WM
WM
> > Then we have only
> > En Am: m =< n ==> Am En m =< n [**]
> > because not all elements are readily available.
>
> Any set in a sane set theory has all members equally "available".
Then a complete set with linear order sould have a last element.
Regards, WM
WM
> > How do you falsify the existence of the
> > set of all Natural numbers in ZF, as you claimed you could?
>
> It is simple: ... classical logic was abstracted from the mathematics
> of finite sets and their subsets .... Forgetful of this limited
> origin, one afterwards mistook that logic for something above and
> prior to all mathematics, and finally applied it, without
> justification, to the mathematics of infinite sets. [Hermann Weyl,
> "Mathematics and logic: A brief survey serving as a preface to a
> review of The Philosophy of Bertrand Russell", American Mathematical
> Monthly 53: 2–13]
>
> *******************************
> Show me a complete finite linear set that does not allow for
> quantifier reversal.
> En Am: m =< n <==> Am En m =< n [*] .
>
> ************************
> Its your proof. Either provide such a set yourself, or prove that none
> exists.
[*] belongs to the basic logic of finite sets. As this logic has been
obtained by observing the behaviour of finite linear sets, there is no
further proof except that a finite linear set violating [*] has never
been observed.
>
> Therefore: Either [*] holds or the set is not complete but allows for
> extension.
>
> *****************************
> You better define "extension", and show its relationship to "complete". Just
> in case, you better define "complete" as well, in case you are using it
> differently to everybody else.
Complete means that every element of a set exists. In case of a linear
set, complete means that also the last element of the linear order
does exist.
>
> Then we have only
> En Am: m =< n ==> Am En m =< n [**]
> because not all elements are readily available.
>
> **************************
> You better define "readily available" while you are at it.
Every element of a complete set is readily available.
If not eery element is readily available, then the set is incomplete,
that mean not complete.
>
> That is called a potentially infinite set. But in this case there is
> no chance to prove uncountability.
>
> *************************
> Define "potentially infinite" as opposed to "infinite". Show that this does
> not allow uncountability to be proved.
Potentially infinite is not opposed to infinite but is one kind of
infinity. The other one is actually infinite.
>
> This had already been recognized by the late Alexander Zenkin, one of
> the brave scientists who dared to condemn this hypocritical
> behaviour:
> Cantor's 'paradise' as well as all modern axiomatic set theory is
> based on the (self-contradictory) concept of actual infinity. Cantor
> emphasized plainly and constantly that all transfinite objects of his
> set theory are based on the actual infinity. Modern AST-people try to
> persuade us to believe that the AST does not use actual infinity. It
> is an intentional and blatant lie, since if infinite sets, X and N,
> are potential, then the uncountability of the continuum becomes
> unprovable, but without the notorious uncountablity of continuum the
> modern AST as a whole transforms into a long twaddle about nothing.
>
> Resume: The internal contradiction in set theory is veiled by mixing
> up potential and actual infinity. That is the reason why set theorists
> usually refuse to specify which infinity they apply. Most even pretend
> (or profess) not to know the difference.
>
> *************************
> What *is* the difference?
>
Potential infinity is possible, actual infinity is not.
>
> **************************
> Gee, such a big claim you make. Yet no proof. Not even a lame attempt.
Sorry, you are only unable to understand it. Not everybody is able to
understand everthing. That is an odl wisdom.
> Even
> cranks generally try and produce bad proofs,
Yes, set theorists, for instance. But I have shown that they need to
use logic of potential infinity to "prove" their actual infinity.
Regards, WM
Peter Webb
So your theory really has two main components. One component is a sort of
psuedo-mystical word salad of undefined terms and fragments of philosophy,
and the other component is the occasional use of mathematical looking
equations and terms.
But is it art?
Virgil
<e705c332-c068-4e0f...@l12g2000yqo.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
Non sequitur, as usual.
There is no reason why having n+1 available whenever n is available
requires existence of an n with no n+1.
--
Virgil
Virgil
<eb95bd31-1206-411a...@g37g2000yqn.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 31 Mai, 03:12, "Peter Webb"
>
> > Show me a complete finite linear set that does not allow for
> > quantifier reversal.
> > En Am: m =< n <==> Am En m =< n � �[*] .
> >
> > ************************
> > Its your proof. Either provide such a set yourself, or prove that none
> > exists.
>
> [*] belongs to the basic logic of finite sets. As this logic has been
> obtained by observing the behaviour of finite linear sets, there is no
> further proof except that a finite linear set violating [*] has never
> been observed.
It still requires that which WM is incapable of providing, namely a
proof.
> >
> > Therefore: Either �[*] holds or the set is not complete but allows for
> > extension.
It still requires that which WM is incapable of providing, namely a
proof.
> >
> > *****************************
> > You better define "extension", and show its relationship to "complete". Just
> > in case, you better define "complete" as well, in case you are using it
> > differently to everybody else.
>
> Complete means that every element of a set exists.
Then N is complete.
> In case of a linear
> set, complete means that also the last element of the linear order
> does exist.
If that is a part of WM's definition, then WM need to prove that no
"incomplete" set can exist, something that he has been failing to do for
years, at least without assumes it a priori.
.
> > Then we have only
> > En Am: m =< n ==> Am En m =< n � �[**]
> > because not all elements are readily available.
> >
> > **************************
> > You better define "readily available" while you are at it.
>
> Every element of a complete set is readily available.
That does not define anything.
> If not eery element is readily available, then the set is incomplete,
> that mean not complete.
> >
> > That is called a potentially infinite set.
Except that there are no such things.
> >
> > *************************
> > Define "potentially infinite" as opposed to "infinite". Show that this does
> > not allow uncountability to be proved.
>
> Potentially infinite is not opposed to infinite but is one kind of
> infinity. The other one is actually infinite.
>
> >
> > Resume: The internal contradiction in set theory is veiled by mixing
> > up potential and actual infinity. That is the reason why set theorists
> > usually refuse to specify which infinity they apply. Most even pretend
> > (or profess) not to know the difference.
Sure they do. One major difference is that otentially infinite sets do
not ever exist but actually infinite sets exist in many set theories.
> >
> > *************************
> > What *is* the difference?
> >
> Potential infinity is possible, actual infinity is not.
Backwards as usual.
> >
> > **************************
> > Gee, such a big claim you make. Yet no proof. Not even a lame attempt.
>
> Sorry, you are only unable to understand it. Not everybody is able to
> understand everthing. That is an odl wisdom.
>
> > Even
> > cranks generally try and produce bad proofs,
>
> Yes, set theorists, for instance. But I have shown that they need to
> use logic of potential infinity to "prove" their actual infinity.
Wm keeps claiming to have "shown" all sorts of things, but until the
recipients of his "showings" acknowledge the validity of such
"showings", WM has shown nothing but his ignorance to anyone.
It was not Wiles "showing" a proof of FLT so much as the rest of the
world's acknowledging its validity that was important, since he was
nowhere near the first to claim a proof of FLT.
When WM can successfully square a circle, trisect an angle and duplicate
a cube using only the classical methods, only then will anyone here be
liable to accept WM's proofs that all sets are necessarily finite in any
and every set theory.
--
Virgil
Virgil
<27453436-4a7f-4501...@k38g2000yqh.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 30 Mai, 23:01, Virgil <virg...@nowhere.com> wrote:
>
> > And whenever WM, or anyone else, does produce one of those lists, it
> > will easily be shown to be incomplete.
>
> I told you alredy several times that logic fails in case of infinite
> sets.
Unlike the bellman, what WM tells me three times need not be , and
almost always isn't, true.
> Do not argue that Cantor's proof is correct.
Do not argue that it isn't until you can come up with a proof that it
isn't.
> Argue why my proof
> is incorrect.
Among other reasons, because it is in conflict with Two of Cantor's
theorems, and I have a great deal more trust in Cantor than I have in WM.
Argue facts, not opinions.
WM rekes not his own rede, but treads the primrose path.
> See the thread concerning the
> reactions to the binary tree. Choose one or more of those counter
> arguments and try to convince people who are not trapped in
> matheology. That would be more commendable than parroting your stuff.
Since WM has done nothing better that to parrot his own stuff, he is
hardly in a position to criticize others for doing it.
--
Virgil
Virgil
<c024871b-c949-4f54...@y7g2000yqa.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 30 Mai, 23:05, Virgil <virg...@nowhere.com> wrote:
> > In article
> > <eebb8310-e58d-4f85-9eda-37af9b74b...@c19g2000yqc.googlegroups.com>,
> >
> > �WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 30 Mai, 00:31, Virgil <virg...@nowhere.com> wrote:
> >
> > > > But if there are only finitely many elements in that universal set, then
> > > > WM's potentially infinite sets, being of necessity subsets of a finite
> > > > universal set, can not be potentially infinite, as they must eventually
> > > > exhaust their finite universal set.
> >
> > > Your universe of numbers is all numbers that you can construct. If you
> > > increase your capabilities, your universe grows. That's why it is
> > > infinite.
> >
> > If all sets are finite, but ever changing as WM insists, then one can
> > never have a set theory at all, as sets, including universal sets if
> > any, do not change.
>
> This is stubborn nonsense. Of course sets are conceivable that can
> change.
They have yet to be conceived in any mathematical set theory that I am
acquainted with. The only possible construction that might serve would
be set-valued functions of time, but even there, for any fixed time,
there would have to be a fixed set.
> The complete sets like N have only been introduced because
> Cantor had a strong religious affinity.
Wm must have a strong religious affinity of his own,believing God speaks
directly to him, to hold that N cannot be a set.
--
Virgil
Martin Musatov
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--
Virgil
WM
> Mueckenheim is also cited as an authority on infinity at this
> educational site:
>
>
> I didn't see any evidence that WM's horrid arguments actually
> influenced the text of the site.
You seem to have overlooked: “His polarizing results generated much
controversy that, to this day, is not completely resolved.”
> Nonetheless, students who want to
> learn more about Cantor are directed to WM's illogical blatherings.
Illogical blatherings you may find when reading Fools Of Mathematics
or something like that.
>
> Still, I blame most WM's employer for putting him in a position of
> authority to educate students on exactly that material he has shown no
> capacity to understand.
Wow, do you really belong to that small elite group of scholars who
understand cardinal and ordinal exponentiation? How did you manage
that task? Certainly you also understand the ordinary mistakes of set
theory. (Only the cardinal mistakes may have escaped you.)
How can someone believe that set theory is too difficult to understand
for an average intelligence? Are you so proud to have managed it that
you have lost all measure?
> I can't comprehend how that situation has
> remained. I'm sure that WM is tenured, but that doesn't entail that
> he can teach bad mathematical reasoning in the classroom, does it?
Therefore I don’t do so, but teach good mathematics, namely
mathematics that is free of confusing the different kinds of infinity
and free of the due silly results.
Regards, WM
WM
wrote:
> I see.
No, you don't. You intermingle mathematical theorems and logical
foundations. Mathematical theorems can be proven using axioms and
logic.
Logical foundations cannot be proven (how should they? By some pre-
logic laws?). Logical foundations can only be obtained from observing
the behaviour of sets --- necessarily finite sets, because there are
no infinite sets in reality that could be observed.
This observation results in a logical law that states: Every linear
complete set has a last element.
Think a while about that. Then you may come back and tell us the
result (not about the truth of what I just said, but about your level
of having understood it).
Regards, WM
WM
>
> WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 30 Mai, 23:16, Virgil <virg...@nowhere.com> wrote:
>
> > > > Then we have only
> > > > En Am: m =< n ==> Am En m =< n [**]
> > > > because not all elements are readily available.
>
> > > Any set in a sane set theory has all members equally "available".
>
> > Then a complete set with linear order sould have a last element.
>
> Non sequitur, as usual.
This is a logical truth obtained from observation of sets --- finite
sets of course, because actually infinite sets are not observable.
>
> There is no reason why having n+1 available whenever n is available
> requires existence of an n with no n+1.
Not for an infinite set that is not complete. But for every complete
set.
Regards, WM
Jesse F. Hughes
> How can someone believe that set theory is too difficult to understand
> for an average intelligence? Are you so proud to have managed it that
> you have lost all measure?
On the contrary, I *don't* believe that Cantor's theorem is too
difficult for the average person, given appropriate background
knowledge and skills. For this reason, I find you exceptional, not
average.
--
Jesse F. Hughes
"If you are a consumer that's taking advantage of the technologies
that exist ... then the spam problem for you is solved."
--MS spokesman verifying that the spam problem has been solved.
David Bernier
How do you define "Parallel lines" ?
Regards,
David Bernier
MoeBlee
> How do you define "Parallel lines" ?
A music album released in 1978 by the new wave rock band Blondie.
MoeBlee
Virgil
<70da2ace-75b1-46be...@y9g2000yqg.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 31 Mai, 16:56, "Peter Webb" <webbfam...@DIESPAMDIEoptusnet.com.au>
> wrote:
> > I see.
>
> No, you don't. You intermingle mathematical theorems and logical
> foundations. Mathematical theorems can be proven using axioms and
> logic.
>
> Logical foundations cannot be proven (how should they? By some pre-
> logic laws?). Logical foundations can only be obtained from observing
> the behaviour of sets
That could only be the case if one assumes a priori that nothing but
sets can exist,and not necessarily then.
>
> This observation results in a logical law that states: Every linear
> complete set has a last element.
Then that means there are non-complete linear complete sets which,
though ordered, do not have last elements.
>
> Think a while about that.
Did so, and, as usual, found that WM's arguments do not hold water.
--
Virgil
Virgil
<6ef6cae8-13cf-4983...@q16g2000yqg.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 31 Mai, 19:27, Virgil <virg...@nowhere.com> wrote:
> > In article
> > <e705c332-c068-4e0f-ade4-d3c25f33b...@l12g2000yqo.googlegroups.com>,
> >
> > �WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 30 Mai, 23:16, Virgil <virg...@nowhere.com> wrote:
> >
> > > > > Then we have only
> > > > > En Am: m =< n ==> Am En m =< n � �[**]
> > > > > because not all elements are readily available.
> >
> > > > Any set in a sane set theory has all members equally "available".
> >
> > > Then a complete set with linear order sould have a last element.
> >
> > Non sequitur, as usual.
>
> This is a logical truth obtained from observation of sets --- finite
> sets of course, because actually infinite sets are not observable.
At least not observable by WM, though they are as observable to others
as large finite sets.
> >
> > There is no reason why having n+1 available whenever n is available
> > requires existence of an n with no n+1.
>
> Not for an infinite set that is not complete. But for every complete
> set.
Then whatever WM's special notion of "completeness" may be in his world
of MathUnrealism, it is not universal among sets outside that world.
--
Virgil
Virgil
<6f02bd25-bb41-45f7...@s12g2000yqi.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 30 Mai, 14:59, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> > Mueckenheim is also cited as an authority on infinity at this
> > educational site:
> >
> > http://www.learner.org/courses/mathilluminated/units/3/resources/in
> > de...
> >
> > I didn't see any evidence that WM's horrid arguments actually
> > influenced the text of the site.
>
> You seem to have overlooked: �His polarizing results generated much
> controversy that, to this day, is not completely resolved.�
>
> > Nonetheless, students who want to
> > learn more about Cantor are directed to WM's illogical blatherings.
>
> Illogical blatherings you may find when reading Fools Of Mathematics
> or something like that.
> >
> > Still, I blame most WM's employer for putting him in a position of
> > authority to educate students on exactly that material he has shown
> > no capacity to understand.
>
> Wow, do you really belong to that small elite group of scholars who
> understand cardinal and ordinal exponentiation?
One does not have to understand all of it to undersatnd that it is both
valid mathematics and often quite interesting, whereas what WM is
trying to sell is neither.
> How did you manage
> that task?
By not taking WM's claims on faith.
At least one can spot many of the errors of WM's perverted version of
set theory.
>
> How can someone believe that set theory is too difficult to
> understand for an average intelligence?
WM is a true believer in his faith (see Eric Hoffer on true believers)
and is untouchable by any fact or logical argument not agreeing with
that faith.
But WM's claims require that names of numbers and the numbers they name
are identical, which as anyone can plainly see thy are not.
> Are you so proud to have managed it that you have lost all measure?
Mathematical measure theory requires infinite sets, so it is WM who has
lost all measure by denying its basis.
>
> > I can't comprehend how that situation has
> > remained. I'm sure that WM is tenured, but that doesn't entail
> > that he can teach bad mathematical reasoning in the classroom, does
> > it?
>
> Therefore I don�t do so, but teach good mathematics, namely
> mathematics that is free of confusing the different kinds of infinity
> and free of the due silly results.
>
> Regards, WM
--
Virgil
George Greene
> Therefore I don’t do so, but teach good mathematics, namely
> mathematics that is free of confusing the different kinds of infinity
> and free of the due silly results.
If you were posting this crap in German, you would've
been fired long ago.
George Greene
> Logical foundations cannot be proven (how should they? By some pre-
> logic laws?). Logical foundations can only be obtained from observing
> the behaviour of sets --- necessarily finite sets, because there are
> no infinite sets in reality that could be observed.
Most finite sets cannot be observed EITHER, dumbass.
They are WAY TOO BIG.
George Greene
> This is a logical truth obtained from observation
That is a contradiction.
Observation can INSPIRE you to adopt some AXIOMS that will thereAFTER
constitute "logical truth", but observation of the physical world
is simply irrelevant to logic in general. Logic is about DEFINITIONS,
NOT observations.
George Greene
> On 31 Mai, 19:27, Virgil <virg...@nowhere.com> wrote:
> > > > Any set in a sane set theory has all members equally "available".
On Jun 1, 6:54 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> > > Then a complete set with linear order should have a last element.
"should" is NOT a logical notion.
NEITHER is "complete", as YOU use it, SINCE YOU FAIL TO DEFINE IT.
It is, moreover, fairly comical that you say "set with linear order"
as though that were somehow ONE thing. ANY set can be put into ANY
order.
ANY ordering on ANY superset of ANY set can be RESTRICTED to a
smaller set. The point is that the set and the ordering SIMPLY HAVE
NOTHING TO DO WITH each other. If a set is infinite then YOU MAY
ORDER IT ANY WAY YOU LIKE. Insisting that the order be linear DOES
NOT AFFECT the question of whether there is a FIRST OR a last element!
Since the reverse/converse of any linear order IS ALSO A LINEAR order,
if it had to have a last element then it would also have to have a
first one.
The set of the integers HAS NEITHER.
George Greene
> On the contrary, I *don't* believe that Cantor's theorem is too
> difficult for the average person, given appropriate background
> knowledge and skills.
This should have been followed by
"You're just REALLY Stupid".
George Greene
YOU CAN'T *DEFINE* "complete", MORON!!!
If you want to define it AS having a last element,
then you will win BY CIRCULARITY! And lose thereby as well!
Denis Feldmann
>> mathematics that is free of confusing the different kinds of infinity
>> and free of the due silly results.
>
> If you were posting this crap in German, you would've
> been fired long ago.
>
they would have tranlated it and send it to Germans "authorities" long
ago. The logical conclusion is that true mathematicians are (relatively)
nice people. On the other hand, the damage done to some young Germans
should have been quite noticeable by now...
Jesse F. Hughes
I more or less said that in the part you snipped.
"For this reason, I find [WM] exceptional, not average."
--
Jesse F. Hughes
Baba: Spell checkers are bad.
Quincy (age 7): C-H-E-K-E-R-S A-R-E B-A-D.
Dik T. Winter
> On 29 Mai, 03:58, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
...
> > > > > > > The problem boils down to the following:
> > > > > > >
> > > > > > > En Am: m =< n <==> Am En m =< n [*]
> > > > > > > En Am: m =< n ==> Am En m =< n [**]
> > > > > I said: For complete linear sets [*] is true.
> > > >
> > > > Not in the article to which I responded.
> > >
> > > But frequently I made use of what you call quatifier exchange and what
> > > is allowed in case of complete linear sets.
> >
> > You think so, but you have to prove that it is valid for infinite complete
> > linear sets. Note that "classical logic is obtained from finite sets".
> > Nowhere in that quote the word linear is mentioned.
>
> Nowhere in that quote the word union in mentioned. Nevertheless the
> logical rules of unions are obtained from unions of finite sets.
What is the relevance of this? The logical rules of unions state that when
you have a union of sets that union does contain an element if it is in one
of the sets. There is no difference between finite unions and infinite
unions.
> The
> logical rules of linear sets are obtained from finite linear sets.
Aha, here we get to the heart of the matter. You do not believe in infinite
sets (or infinite unions or infinite sets of finite linear sets). That is
possible, of course, but does not rule out theories in which those things
do exist. Quoting philosophers of some time ago does not make that wrong.
> > > The only thing that can be stated is (symbolically):
> > > E n A m P(m, n) -> A m E n P(m, n)
> > > not the reverse, this is just basic logic.
> >
> > The reverse of
> > E n A m P(m, n) -> A m E n P(m, n)
> > is
> > E n A m P(m, n) <- A m E n P(m, n)
> > which is [***], neither [*] nor [**].
>
> I never said so. But [*] is [**] & [***]. Therefore [*] differs from
> [**] only by the reverse.
Being incomprehensible again. The reverse of *what*? The reverse of [**]
is (as I wrote) [***]. You may never have said so, but it is. To recap:
[*] En Am: m =< n <==> Am En m =< n
[**] En Am: m =< n ==> Am En m =< n
[***] En Am: m =< n <== Am En m =< n
But you ask as a counterexample something where [**] is true but [*] false.
But that is the wrong way around. Whenever [**] is true, [*] is also true.
What is contested is that there are case where [***] is true and [*] false.
And it is the latter implication that you do use.
> > > > I said that for the case involved you have to
> > > > *prove* that it is true, because it is not generally true.
> > >
> > > It is generally true for complete linear sets. You have to prove that
> > > it is not.
> >
> > It is not true for the infinite set of naturals.
>
> That is your claim. It is justified for potential infinity. It is
> wrong for complete sets.
Now you are using words that are again completely incomprehensible. You have
still failed to give a definition of "potential infinity" that is valid within
ZF. Moreover, you have not proven (within ZF) that the statement is wrong.
> > (1) define FISON(n) be the set of naturals from 1 to n, that is:
> > {1, ..., n}.
> > (2) A{m in N} E(n in N} such that FISON(m) subset FISON(n), trivial, take
> > n = m + 1.
> > (3) E{n in N} A{m in N} such that FISON(m) subset FISON(n), trivially
> > false, take m = n + 1.
> > Which part of this proof is wrong?
>
> The proof is correct for potential infinity. The proof is incorrect
> for actual infinity.
Can you provide me with definitions within ZF that shows the difference?
> In that latter case you need not take an n that
> is surpassed by m. Why don't you start with an n that has no greater
> m?
Because there is no such n. Remember: the set of natural numbers has no
largest element in ZF.
> > It clearly shows that
> > E n A m P(m, n) <- A m En P(m, n)
> > is false. Here with:
> > E n meaning E{n in N}
> > A n meaning A{n in N}
> > P(m, n) meaning FISON(m) subset FISON(n).
> > The first part of the implication is false while the second part of the
> > implication is true, and so the implication is false (all by classical
> > logic).
>
> Not at all. By classical logic, a complete linear set has a last
> element.
Oh. I think that the term "classical logic" has changed a bit since the
last time you looked at it. And, if you refer to Weyl's quote, he stated
that classical logic was *derived* from the logic on finite sets, not that
it was *identical* to logic on finite sets.
> > > I did never claim that quantifier exchange is allowed in case of non-
> > > linear sets, like cyclic sets as, for instance, your dice. That would
> > > be nonsense. A simple example: Every country has a country that lies
> > > west of it. But there is no country that lies west of all countries.
> >
> > But as Weyl did not include "linear" in his words, how can that quote
> > support your claim?
>
> There are many finite sets with many special properties that follow
> from classical logic. One of them is that a complete linear set has a
> lst element.
Not "a complete linear set". But "a complete finite linear set". Why do
you drop the word "finite" in the second sentence? To obfuscate?
> You drop the completeness condition in certain cases but you assume it
> in case of Cantor's proof. That is cheating.
You again misunderstand the proof completely. There is an assumption that
a complete list is provided and that is proven false.
> > > > > > What is contested is that:
> > > > > > En Am: m =< n <== Am En m =< n [***]
> > > > > > implies [*]. And *that* is the form you do use.
> > ...
> > > > Because the implication [**] is always true, the only part of the
> > > > equivalence that is new is the implication [***].
> > >
> > > For complete linear sets both are true, therefore [*] holds.
> >
> > You just state without proof. Where in my proof above that it is false
> > did I go wrong?
>
> State before beginning whether the set that you assume is complete and
> static, i.e., every element is actually existing, or potentially
> infinite.
What in the world is a "potentially infinite element"? And, in ZF all sets
are static, there is no place for non-static sets.
I will recap my proof, I assume the axiom of infinity. That is all.
> > (1) define FISON(n) be the set of naturals from 1 to n, that is:
> > {1, ..., n}.
> > (2) A{m in N} E(n in N} such that FISON(m) subset FISON(n), trivial, take
> > n = m + 1.
> > (3) E{n in N} A{m in N} such that FISON(m) subset FISON(n), trivially
> > false, take m = n + 1.
> > Strange, I give above a proof that it does not hold. I did not use
> > "actual infinity" nor "potentially infinity"
>
> That is the point! You use the absence of element m when you choose n
> = m - 1.
Where in the proof do I use an element m when I chose n = m - 1? In the proof
I chose only elements that *follow* given elements, as is assured by the
axiom of infinity. Quantifier dyslexia on (3)?
> But you use the non-absence of m when you execute Cantor's
> proof. Then you do not admit that for every FISON(n) there is an m = n
> + 1 that is not in the proof.
This is really incomprehensible.
> > > Only
> > > potentially infinite sets do not. But you mix up things. You claim the
> > > existence of a complete linear set but disregard the necessary
> > > consequence of completeness or linearity, namely the validity of [*].
> >
> > Why is that a necessary consequence? Because you want it to be so? Can
> > you give a *mathematical* reason?
>
> Every finite linear set obeys [*]. That is the mathematical reason.
Clearly, the only reason is that you want it to be so.
--
dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Dik T. Winter
> "Dik T. Winter" <Dik.W...@cwi.nl> writes:
> > The second (recently published, 2008 I think) is however by a reputable
> > publisher. It is intended as textbook for students, and used by WM at his
> > university.
>
> So who the hell published a WM textbook?
Not so difficult to find. Oldenbourg Wissenschaftsverlag. The title
is "Mathematik f�r die ersten Semester".
Dik T. Winter
> On Jun 1, 6:38=A0am, WM <mueck...@rz.fh-augsburg.de> wrote:
> > Therefore I don=92t do so, but teach good mathematics, namely
> > mathematics that is free of confusing the different kinds of infinity
> > and free of the due silly results.
>
> If you were posting this crap in German, you would've
> been fired long ago.
You apparently do not understand it at all. WM has posted this stuff for
years in the German newsgroup on mathematics, in German. Moreover, he
has two books about mathematics on his name, in German.
Virgil
wrote:
> In article
> <5df917e3-517f-4a38...@t21g2000yqi.googlegroups.com> WM
> <muec...@rz.fh-augsburg.de> writes:
> > You drop the completeness condition in certain cases but you assume it
> > in case of Cantor's proof. That is cheating.
>
> You again misunderstand the proof completely. There is an assumption that
> a complete list is provided and that is proven false.
As I understand the Cantor diagonal proof, the only assumption is that
whenever one is provided with a list then that list has to omit at least
one sequence. I do not think it was, in its original form, an indirect
proof as your statement seems to indicate.
--
Virgil
WM
> In article <5df917e3-517f-4a38-b28b-363843496...@t21g2000yqi.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes:
> > On 29 Mai, 03:58, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> ...
> > > > > > > > The problem boils down to the following:
> > > > > > > >
> > > > > > > > En Am: m =< n <==> Am En m =< n [*]
> > > > > > > > En Am: m =< n ==> Am En m =< n [**]
> ...
> > > > > > I said: For complete linear sets [*] is true.
> > > > >
> > > > > Not in the article to which I responded.
> > > >
> > > > But frequently I made use of what you call quatifier exchange and what
> > > > is allowed in case of complete linear sets.
> > >
> > > You think so, but you have to prove that it is valid for infinite complete
> > > linear sets. Note that "classical logic is obtained from finite sets".
> > > Nowhere in that quote the word linear is mentioned.
> >
> > Nowhere in that quote the word union in mentioned. Nevertheless the
> > logical rules of unions are obtained from unions of finite sets.
>
> What is the relevance of this?
The word union and the word linear are not mentioned in the quote.
Nevertheless the due logica rules were obtained from unions of finite
sets and linear finite sets.
>
> > The
> > logical rules of linear sets are obtained from finite linear sets.
>
> Aha, here we get to the heart of the matter. You do not believe in infinite
> sets (or infinite unions or infinite sets of finite linear sets). That is
> possible, of course, but does not rule out theories in which those things
> do exist.
But it does rule out theories which are contradicted by the
fundamental logical rules. And one of these rules is that a complete
linear set has a last element.
>
> > > > The only thing that can be stated is (symbolically):
> > > > E n A m P(m, n) -> A m E n P(m, n)
> > > > not the reverse, this is just basic logic.
> > >
> > > The reverse of
> > > E n A m P(m, n) -> A m E n P(m, n)
> > > is
> > > E n A m P(m, n) <- A m E n P(m, n)
> > > which is [***], neither [*] nor [**].
> >
> > I never said so. But [*] is [**] & [***]. Therefore [*] differs from
> > [**] only by the reverse.
>
> Being incomprehensible again. The reverse of *what*? The reverse of [**]
> is (as I wrote) [***]. You may never have said so, but it is. To recap:
> [*] En Am: m =< n <==> Am En m =< n
> [**] En Am: m =< n ==> Am En m =< n
> [***] En Am: m =< n <== Am En m =< n
>
> But you ask as a counterexample something where [**] is true but [*] false.
> But that is the wrong way around. Whenever [**] is true, [*] is also true.
> What is contested is that there are case where [***] is true and [*] false.
> And it is the latter implication that you do use.
No. I use the fact that for complete linear sets always both
implications are true :
[**] & [***]. This means that [*] is true.
If you disagree, then you should come up with a finite linear set for
which only one implication is true.
>
> > > > > I said that for the case involved you have to
> > > > > *prove* that it is true, because it is not generally true.
> > > >
> > > > It is generally true for complete linear sets. You have to prove that
> > > > it is not.
> > >
> > > It is not true for the infinite set of naturals.
> >
> > That is your claim. It is justified for potential infinity. It is
> > wrong for complete sets.
>
> Now you are using words that are again completely incomprehensible. You have
> still failed to give a definition of "potential infinity" that is valid within
> ZF. Moreover, you have not proven (within ZF) that the statement is wrong.
ZF uses potential infinity whenever the validity of
En Am: m =< n <== Am En m =< n [***]
for linear sets is denied.
ZF claims that this denial is correct for complete linear infinite
sets, but this is a wrong claim, as we can obtain from logic.
>
> > > (1) define FISON(n) be the set of naturals from 1 to n, that is:
> > > {1, ..., n}.
> > > (2) A{m in N} E(n in N} such that FISON(m) subset FISON(n), trivial, take
> > > n = m + 1.
> > > (3) E{n in N} A{m in N} such that FISON(m) subset FISON(n), trivially
> > > false, take m = n + 1.
> > > Which part of this proof is wrong?
> >
> > The proof is correct for potential infinity. The proof is incorrect
> > for actual infinity.
>
> Can you provide me with definitions within ZF that shows the difference?
>
> > In that latter case you need not take an n that
> > is surpassed by m. Why don't you start with an n that has no greater
> > m?
>
> Because there is no such n. Remember: the set of natural numbers has no
> largest element in ZF.
That is the logic of potential infinity, i.e., of incomplete sets.
In Cantor’s diagonal argument you can use the same logic : There is no
last line, therefore there is always a line beyond the checked lines.
But there you don’t.
>
> > > It clearly shows that
> > > E n A m P(m, n) <- A m En P(m, n)
> > > is false. Here with:
> > > E n meaning E{n in N}
> > > A n meaning A{n in N}
> > > P(m, n) meaning FISON(m) subset FISON(n).
> > > The first part of the implication is false while the second part of the
> > > implication is true, and so the implication is false (all by classical
> > > logic).
> >
> > Not at all. By classical logic, a complete linear set has a last
> > element.
>
> Oh. I think that the term "classical logic" has changed a bit since the
> last time you looked at it. And, if you refer to Weyl's quote, he stated
> that classical logic was *derived* from the logic on finite sets, not that
> it was *identical* to logic on finite sets.
When it is derived from logic of finite sets, then it is not the
reverse of the logic of finite sets. But that is claimed in ZF.
>
> > > > I did never claim that quantifier exchange is allowed in case of non-
> > > > linear sets, like cyclic sets as, for instance, your dice. That would
> > > > be nonsense. A simple example: Every country has a country that lies
> > > > west of it. But there is no country that lies west of all countries.
> > >
> > > But as Weyl did not include "linear" in his words, how can that quote
> > > support your claim?
> >
> > There are many finite sets with many special properties that follow
> > from classical logic. One of them is that a complete linear set has a
> > lst element.
>
> Not "a complete linear set". But "a complete finite linear set". Why do
> you drop the word "finite" in the second sentence? To obfuscate?
The logic is derived from finite complete linear sets. If logic is to
be applied to infinite complete linear sets, then we cannot change it
to the opposite. Either those sets obey that logic or they do not
exist in a science that is subject to the application of logic.
In fact thoses sets contradict their own existence under the
government of logic.
>
> > You drop the completeness condition in certain cases but you assume it
> > in case of Cantor's proof. That is cheating.
>
> You again misunderstand the proof completely. There is an assumption that
> a complete list is provided and that is proven false.
Small wonder. There cannot be a complete list, because the existence
of a complete infinite linear set like N contradicts logic.
>
> > > > > > > What is contested is that:
> > > > > > > En Am: m =< n <== Am En m =< n [***]
> > > > > > > implies [*]. And *that* is the form you do use.
> > > ...
> > > > > Because the implication [**] is always true, the only part of the
> > > > > equivalence that is new is the implication [***].
> > > >
> > > > For complete linear sets both are true, therefore [*] holds.
> > >
> > > You just state without proof. Where in my proof above that it is false
> > > did I go wrong?
> >
> > State before beginning whether the set that you assume is complete and
> > static, i.e., every element is actually existing, or potentially
> > infinite.
>
> What in the world is a "potentially infinite element"? And, in ZF all sets
> are static, there is no place for non-static sets.
That is a blatant lie. Every static linear set has a last element.
If you say you cannot choose the last elemement because there is none,
then you apply potentially infinite sets. Then for every element that
you choose there is a larger one. If it has been there all time, why
the hell did you not start with this one ?
>
> > > (1) define FISON(n) be the set of naturals from 1 to n, that is:
> > > {1, ..., n}.
> > > (2) A{m in N} E(n in N} such that FISON(m) subset FISON(n), trivial, take
> > > n = m + 1.
> > > (3) E{n in N} A{m in N} such that FISON(m) subset FISON(n), trivially
> > > false, take m = n + 1.
>
> > > Strange, I give above a proof that it does not hold. I did not use
> > > "actual infinity" nor "potentially infinity"
> >
> > That is the point! You use the absence of element m when you choose n
> > = m - 1.
>
> Where in the proof do I use an element m when I chose n = m - 1? In the proof
> I chose only elements that *follow* given elements, as is assured by the
> axiom of infinity.
That is assured by the axiom of potential infinity. If all elemeents
were there, why then do you have to select one that is diminishingly
small compared to most ?
>
> > But you use the non-absence of m when you execute Cantor's
> > proof. Then you do not admit that for every FISON(n) there is an m = n
> > + 1 that is not in the proof.
>
> This is really incomprehensible.
So it is, but you do it nevertheless.
>
> > > > Only
> > > > potentially infinite sets do not. But you mix up things. You claim the
> > > > existence of a complete linear set but disregard the necessary
> > > > consequence of completeness or linearity, namely the validity of [*].
> > >
> > > Why is that a necessary consequence? Because you want it to be so? Can
> > > you give a *mathematical* reason?
> >
> > Every finite linear set obeys [*]. That is the mathematical reason.
>
> Clearly, the only reason is that you want it to be so.
Show me a finite linear set that does not obey [*]. Or try to explain
why in your opinion this law must be reversed for infinite sets which
are actually existing, i.e., every elemenmt of them can be chosen for
comparison with others.
Regards, WM
WM
> In article <db512ef4-fbff-48ee-af30-ed25dc9b7...@s16g2000vbp.googlegroups.com> George Greene <gree...@email.unc.edu> writes:
> > On Jun 1, 6:38=A0am, WM <mueck...@rz.fh-augsburg.de> wrote:
> > > Therefore I don=92t do so, but teach good mathematics, namely
> > > mathematics that is free of confusing the different kinds of infinity
> > > and free of the due silly results.
> >
> > If you were posting this crap in German, you would've
> > been fired long ago.
>
> You apparently do not understand it at all. WM has posted this stuff for
> years in the German newsgroup on mathematics, in German. Moreover, he
> has two books about mathematics on his name, in German.
And both say, in effect, about the existence of actual infinity, what
Kant said about the proof of the existence of God: These assumptions
(proof of God, axiom of infinity) are as ridiculous as a merchant who
would try to improve his balance by adding some zeros behind his
result.
I would be glad if a court would have to decide about my position
based on the binary tree and my further arguments. Kant, at his time,
had to stop his lectures about religious topics by order of his
Prussian King. But I am sure that all this Cantor-nonsense would burst
like a bubble filled with foul gas once the public got to know what
crap has to be paid by the taxpayer.
Regards, WM
Regards, WM
WM
> In article <KKn5F7....@cwi.nl>, "Dik T. Winter" <Dik.Win...@cwi.nl>
> > > You drop the completeness condition in certain cases but you assume it
> > > in case of Cantor's proof. That is cheating.
>
> > You again misunderstand the proof completely. There is an assumption that
> > a complete list is provided and that is proven false.
>
> As I understand the Cantor diagonal proof, the only assumption is that
> whenever one is provided with a list then that list has to omit at least
> one sequence. I do not think it was, in its original form, an indirect
> proof as your statement seems to indicate.
Cantor understood it as a proof by contradiction. "da wir sonst vor
dem Widerspruch stehen würden, daß ein Ding E0 sowohl Element von M,
wie auch nicht Element von M wäre." But probably you know better.
The only thing that is interesting here is that the same holds for the
list of all natural numbers. Give me a list of natural numbers and I
will show you that it is incomplete.
Of course you are not allowed to say: All n in N! That would be a
contradiction, because there is no last n and consequently no chance
to check whether your list would contain all n in N. (And if you did
so, then one could also say: All r in R. The contracdiction would be
of same size.)
Regards, WM
WM
> There is no difference between finite unions and infinite
> unions.
There should not be, but there is --- at least if ZF is taken to be
true.
The union of FISONs (of natural numbers or of other finite linear
sets) is the last FISON.
Regards, WM
Virgil
<324b4dea-62b3-4a92...@j12g2000vbl.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
While they may be suggested by and be compatible with unions of finite
sets, definitions of union of a set of sets in any set theory
(excluding WM's) do not restricts sets, or their unions, to being finite
sets.
>
> >
> > > The
> > > logical rules of linear sets are obtained from finite linear sets.
> >
> > Aha, here we get to the heart of the matter. You do not believe in
> > infinite
> > sets (or infinite unions or infinite sets of finite linear sets). That is
> > possible, of course, but does not rule out theories in which those things
> > do exist.
>
> But it does rule out theories which are contradicted by the
> fundamental logical rules.
WM's notion of what contradicts "fundamental logical rules" conflicts
with logic's notion of what contradicts logical rules.
> And one of these rules is that a complete
> linear set has a last element.
A finite ordered set need not have a last element, so WM os wrong again.
> No. I use the fact that for complete linear sets always both
> implications are true :
But unless "completeness" requires "finiteness" it is false, and
nothing outside of WM's perverted vision of set theories requires
finiteness.
>
> If you disagree, then you should come up with a finite linear set for
> which only one implication is true.
(En in S) (Am in S) : m =< n <==> Am En m =< n
There is no logic of potentially infinite sets, as they are self
contradictory.
> In Cantor�s diagonal argument you can use the same logic : There is no
> last line, therefore there is always a line beyond the checked lines.
> But there you don�t.
Then WM does not understand that argument, since Cantor does not require
any constraints on a list other than that its members be binary
sequences, including whether or not there is a last "line".
> > Oh. I think that the term "classical logic" has changed a bit since the
> > last time you looked at it. And, if you refer to Weyl's quote, he stated
> > that classical logic was *derived* from the logic on finite sets, not that
> > it was *identical* to logic on finite sets.
>
> When it is derived from logic of finite sets, then it is not the
> reverse of the logic of finite sets. But that is claimed in ZF.
That may be how WM misinterpretes ZF, but no one else can be forced to
accept WM's misinterpretation, and no one else does.
> >
> > > > > I did never claim that quantifier exchange is allowed in case of
> > > > > non-
> > > > > linear sets, like cyclic sets as, for instance, your dice. That
> > > > > would
> > > > > be nonsense. A simple example: Every country has a country that
> > > > > lies
> > > > > west of it. But there is no country that lies west of all
> > > > > countries.
> > > >
> > > > But as Weyl did not include "linear" in his words, how can that quote
> > > > support your claim?
> > >
> > > There are many finite sets with many special properties that follow
> > > from classical logic. One of them is that a complete linear set has a
> > > lst element.
> >
> > Not "a complete linear set". But "a complete finite linear set". Why do
> > you drop the word "finite" in the second sentence? To obfuscate?
>
> The logic is derived from finite complete linear sets.
"Derived from" does not mean "identical to", and the set theory of ZF,
among others, is compatible with those set theories of finite sets which
do not impose finiteness.
> If logic is to
> be applied to infinite complete linear sets, then we cannot change it
> to the opposite. Either those sets obey that logic or they do not
> exist in a science that is subject to the application of logic.
>
> In fact thoses sets contradict their own existence under the
> government of logic.
That "government" does not rule anywhere except in WM's weird world of
MathuUnrealism.
If WM's arguments could be proved, he would long since have gathered
supporters, but he has not.
> >
> > > You drop the completeness condition in certain cases but you assume it
> > > in case of Cantor's proof. That is cheating.
> >
> > You again misunderstand the proof completely. There is an assumption that
> > a complete list is provided and that is proven false.
>
> Small wonder. There cannot be a complete list, because the existence
> of a complete infinite linear set like N contradicts logic.
Actually, the Cantor argument does not assume or require a complete
list, it allows ANY list, all of whose members are binary sequences and
then shows that that list must be incomplete in the sense of omitting at
least one potential member.
So WM does not even understand the very argument he is vainly trying to
oppose.
> >
> > What in the world is a "potentially infinite element"? And, in ZF all sets
> > are static, there is no place for non-static sets.
> "
> That is a blatant lie. Every static linear set has a last element.
Not necessarily in ZF. From what axiom of combination of axioms does WM
claim to derive his "Every static linear set has a last element"?
Unless he can do so, he lies.
> If you say you cannot choose the last elemement because there is none,
> then you apply potentially infinite sets.
Nonsense! There is no provision for potentially infinite sets in ZF.
If WM disputes this, he must show which definitions and/or axioms
produce this provision.
> Then for every element that
> you choose there is a larger one. If it has been there all time, why
> the hell did you not start with this one ?
For which element does WM argue there is no larger one?
Unless WM can tell us how to find "this one", we will continue to hold
that his his "this one", with no larger one, does not exist.
ZF says that for EVERY natural there is a successor natural.
> >
> > > > (1) define FISON(n) be the set of naturals from 1 to n, that is:
> > > > {1, ..., n}.
> > > > (2) A{m in N} E(n in N} such that FISON(m) subset FISON(n), trivial,
> > > > take
> > > > n = m + 1.
> > > > (3) E{n in N} A{m in N} such that FISON(m) subset FISON(n), trivially
> > > > false, take m = n + 1.
> >
> > > > Strange, I give above a proof that it does not hold. I did not use
> > > > "actual infinity" nor "potentially infinity"
> > >
> > > That is the point! You use the absence of element m when you choose n
> > > = m - 1.
> >
> > Where in the proof do I use an element m when I chose n = m - 1? In the
> > proof
> > I chose only elements that *follow* given elements, as is assured by the
> > axiom of infinity.
>
> That is assured by the axiom of potential infinity.
What does "the axiom of potentail infinity"
> say? If all elemeents
> were there, why then do you have to select one that is diminishingly
> small compared to most ?
Because every one of them is "diminishingly small compared to most".
WM keeps asking us to to select things which do not exist while
ignoring things which do.
> Show me a finite linear set that does not obey [*]. Or try to explain
> why in your opinion this law must be reversed for infinite sets which
> are actually existing, i.e., every elemenmt of them can be chosen for
> comparison with others.
No one can show anything to anyone as invincibly ignorant as WM.
--
Virgil
George Greene
> The question is whether we could inform someone who does not yet know,
> what we understand by the sequence of natural numbers.
Could we inform them COMPLETELY?
Could we ever FINISH informing them?
Wouldn't they just POTENTIALLY informed??
> > But non-static sets do not exist in any mathematical set theory.
Of COURSE not.
THE PARADIGM is static!
IF you want something variable as OPPOSED to static then
you CALL that something A VARIABLE or A PROCESS
AS OPPOSED to a static constant!
The theory is INHERENTLY ABOUT static, constant things!
AND some of these static constant things are statically constantly
LACKING first and last elements (like the set of all&only the
integers,
for example). Or at least they CAN be ordered that way -- order is
NOT an INHERENT property OF ANY set -- indeed, the fact that
ALL possible orders are EQUALLY legitimate -- STATICALLY -- is
what MAKES a thing a set!
> They do not exist in what is commonly called set theory and what is
> eternally false mathematics.
Because the paradigm IS STATIC, it does NOT do TIME, EITHER,
which makes your calling it "eternally" anything, well, even stupider
than usual.
The mathematics you are calling eternally false is in fact
PERFECTLY capable of dealing with variables and processes.
It's just that N *IS NOT ONE* of them.
N *is* static, PRECISELY AS your contention that we can finish
communicating it clearly SHOWS.
George Greene
> > This should have been followed by
> > "You're just REALLY Stupid".
On Jun 2, 5:22 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> I more or less said that in the part you snipped.
>
> "For this reason, I find [WM] exceptional, not average."
Well, more less than more, but sometimes, less is more.
George Greene
> In article <db512ef4-fbff-48ee-af30-ed25dc9b7...@s16g2000vbp.googlegroups.com> George Greene <gree...@email.unc.edu> writes:
> > been fired long ago.
On Jun 2, 10:37 pm, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> You apparently do not understand it at all.
Understand WHAT?? You owe us an antecedent for "it", there.
> WM has posted this stuff for
> years in the German newsgroup on mathematics, in German.
Shit. There is MORE THAN ONE mathematics newsgroup in the German
language. This one in America has become so degraded that what
few serious scholars still read it treat it as idle amusement, not
something
to try to get anybody fired about. WM has NOT been posting this stuff
in any forum that anybody CARES about (in German).
> Moreover, he has two books about mathematics on his name,
> in German.
Books ABOUT mathematics are NOT likely to be respected by
MATHEMATICIANS. He would NEED to write a book WITH SOME
MATH in it AS OPPOSED to a book "about" mathematics, in order
to be relevant.
George Greene
> On 3 Jun., 04:37, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > You apparently do not understand it at all. WM has posted this stuff for
> > years in the German newsgroup on mathematics, in German. Moreover, he
> > has two books about mathematics on his name, in German.
On Jun 3, 3:26 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> And both say, in effect, about the existence of actual infinity, what
> Kant said about the proof of the existence of God: These assumptions
> (proof of God, axiom of infinity) are as ridiculous as a merchant who
> would try to improve his balance by adding some zeros behind his
> result.
That is not something that can legitimately be said in any
mathematical
context. In the first place, WE DON'T prove that infinity exists.
WE JUST HAVE AN AXIOM of infinity.
YOU CAN USE THE DENIAL OF THAT AXIOM
and talk about A FINITE universe IF YOU LIKE!
NOTHING IS STOPPING you!
Except maybe for the fact that IF you allow EVERYthing to have
a successor that is distinct FROM ALL SMALLER things, then
THERE FACTUALLY ARE
an infinite number of things and that assuming otherwise
LEADS TO CONTRADICTIONS, EVEN IN YOUR impoverished
logic.
> I would be glad if a court would have to decide about my position
> based on the binary tree and my further arguments.
It has. You've been convicted.
Unanimously.
> Kant, at his time,
> had to stop his lectures about religious topics by order of his
> Prussian King. But I am sure that all this Cantor-nonsense would burst
> like a bubble filled with foul gas once the public got to know what
> crap has to be paid by the taxpayer.
The public doesn't know shit about math,
and if you are at a public university and taxpayers are paying
YOUR salary, well, be careful what you wish for.
Come the day.
George Greene
> What is the relevance of this? The logical rules of unions state that when
> you have a union of sets that union does contain an element if it is in one
> of the sets.
DAMN.
This is the only reason I am still here (because you and Virgil are).
YOU ARE TOO STUPID to be doing this!
THERE ARE NO SUCH THINGS as "the logical rules of unions"!
THERE IS AN *AXIOM* of union in ZFC!
This axiom IS NOT a logical truth, or it would NOT NEED to exist at
all!
To the extent that "rules of unions" exist at all, THEY ARE *NOT*
logical and MUST not be logical!
IN ADDITION to the axiom of union, there is also, USUALLY,
an axiom of pairing! THERE IS *NO LOGICAL* connection WHATEVER
between the two! Either could get along FINE WITHOUT the other!
But the axiom of pairing suffices to create ALL THE FINITE unions!
YOU ONLY NEED the axiom of union FOR INFINITE unions!
> There is no difference between finite unions and infinite
> unions.
THERE IS *SO*, TOO!!
Finite unions can be built up ONE ELEMENT AT A TIME,
THROUGH PAIRING! INFINITE UNIONS *CANNOT*!!
THAT IS PRECISELY WHY WM IS WRONG to be generalizing
from what happens with EVERY finite case TO the infinite case!
a
Virgil
<103a00da-91c5-47fc...@e24g2000vbe.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 3 Jun., 04:37, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > In article
> > <db512ef4-fbff-48ee-af30-ed25dc9b7...@s16g2000vbp.googlegroups.com> George
> > Greene <gree...@email.unc.edu> writes:
> > �> On Jun 1, 6:38=A0am, WM <mueck...@rz.fh-augsburg.de> wrote:
> > �> > Therefore I don=92t do so, but teach good mathematics, namely
> > �> > mathematics that is free of confusing the different kinds of infinity
> > �> > and free of the due silly results.
> > �>
> > �> If you were posting this crap in German, you would've
> > �> been fired long ago.
> >
> > You apparently do not understand it at all. �WM has posted this stuff for
> > years in the German newsgroup on mathematics, in German. �Moreover, he
> > has two books about mathematics on his name, in German.
>
> And both say, in effect, about the existence of actual infinity, what
> Kant said about the proof of the existence of God: These assumptions
> (proof of God, axiom of infinity) are as ridiculous as a merchant who
> would try to improve his balance by adding some zeros behind his
> result.
If I recall correctly, Kant also claimed that only Eulidean geometry
could be consistent.
So perhaps one ought not rely on Kant's opinions about mathematics.
>
> I would be glad if a court would have to decide about my position
> based on the binary tree and my further arguments.
Anyone who would submit the validity of a mathematical argument to a
court of law, knows very little about mathematics and very little about
the law.
> Kant, at his time,
> had to stop his lectures about religious topics by order of his
> Prussian King. But I am sure that all this Cantor-nonsense would burst
> like a bubble filled with foul gas once the public got to know what
> crap has to be paid by the taxpayer.
If the mathematicians of Germany knew what anti-mathematical nonsense WM
was pouring into his students, they might very well act to burst that
bubble filled with even fouler gas.
--
Virgil
Virgil
<d797f3cd-50a1-4403...@s21g2000vbb.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 3 Jun., 05:34, Virgil <virg...@nowhere.com> wrote:
> > In article <KKn5F7....@cwi.nl>, "Dik T. Winter" <Dik.Win...@cwi.nl>
> > wrote:
> >
> > > In article
> > > <5df917e3-517f-4a38-b28b-363843496...@t21g2000yqi.googlegroups.com> WM
> > > <mueck...@rz.fh-augsburg.de> writes:
> > > �> You drop the completeness condition in certain cases but you assume it
> > > �> in case of Cantor's proof. That is cheating.
> >
> > > You again misunderstand the proof completely. �There is an assumption that
> > > a complete list is provided and that is proven false.
> >
> > As I understand the Cantor diagonal proof, the only assumption is that
> > whenever one is provided with a list then that list has to omit at least
> > one sequence. I do not think it was, in its original form, an indirect
> > proof as your statement seems to indicate.
>
> Cantor understood it as a proof by contradiction. "da wir sonst vor
> dem Widerspruch stehen w�rden, da� ein Ding E0 sowohl Element von M,
> wie auch nicht Element von M w�re." But probably you know better.
A quote without a source cited is not evidence.
There is certainly no necessity for it to be cast as an indirect proof
when the direct proof is even simpler.
>
> The only thing that is interesting here is that the same holds for the
> list of all natural numbers. Give me a list of natural numbers and I
> will show you that it is incomplete.
A list in the context of the Cantor diagonal argument means a mapping
from the set of all naturals to the members of the listed set, so that
the function, f: N --> N: x |--> x is a list of natural numbers.
Now WM must shown how and why THAT listing is incomplete (in the sense
of missing some member of N either as argument or as value of the
function f).
>
> Of course you are not allowed to say: All n in N!
All n in N, without the factorial symbol, though, is quite legitimate.
> That would be a
> contradiction, because there is no last n and consequently no chance
> to check whether your list would contain all n in N.
Since my listing clearly includes every n in N for which n = n, WM must
show that there is some n for which n = n is false to falsify my
argument.
--
Virgil
Martin Musatov
Martin Musatov wrote:
my sight.
++
Martin Musatov
Marshall
>
> I would be glad if a court would have to decide about my position
> based on the binary tree and my further arguments. Kant, at his time,
> had to stop his lectures about religious topics by order of his
> Prussian King. But I am sure that all this Cantor-nonsense would burst
> like a bubble filled with foul gas once the public got to know what
> crap has to be paid by the taxpayer.
Do we have any evidence that JSH and WM are different
people? Have they ever been photographed together?
Marshall
j/k
MoeBlee
> Do we have any evidence that JSH and WM are different
> people? Have they ever been photographed together?
Yes they have been. See:
http://www.ltn.lv/~podnieks/slides/goedel/Goedel_Einstein.jpg
MoeBlee
Virgil
<5b9b7d5b-11d3-4125...@x5g2000yqk.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
If WM claims the existence of a last FISON, let him present it to us.
But outside of WM's U.N. Realistic world, every FISON is a proper subset
of another FISON, so that the union of all of them cannot itself be one
of them.
Or does WM's set theory require existence of sets which are proper
subsets of themselves?
--
Virgil