Re: Peer-reviewed arguments against Cantor Diagonalization

[INFINITY POWER]

On Oct 28, 1:03 pm, Tonic...@yahoo.com wrote:
> On Saturday, October 27, 2012 11:49:32 PM UTC+2, JRStern wrote:
> > Are there any such published?
>
> > I can see in the archives here it's a common topic, and I have my own
>
> > crackpot theories which certainly overlap a lot of the more popular
>
> > objections.
>
> > I don't want to prove or assert or reject any statement about the
>
> > countability of reals, I just want to consider the validity of the
>
> > diagonalization argument.
>
> > Has anybody put that out in a refereed journal or a respectable
>
> > publisher? Even if it's just a prettier rejection of crank theories,
>
> > it would seem worthwhile.
>
> > Thanks,
>
> > J.
>
> Ask for Herc (Cooper), WM and other glorious local cranks who think (just
> a figure of speech) they have debunked Cantor, his theorems, his proofs
> and his theories.
>
> Of course, the only peer reviewed papers "against Cantor" could exist, SO
> FAR, in a journal abiding by the rules of Sumo in Tokio, Japan, and not by
> the rules of mathematics.
>
> Tonio
>


Thanks Tonio,

I should reciprocate the favor of mentioning my theories with a cite of your
famous formula to make infinity even bigger!

[TONICO]
Then I choose the number 0.a_1a_2a_3...., where a_i = 0 if the i-th
number in your list had zero in its i-position, a_i = 1 otherwise.
TADAAAA!

That's what real number theory is based on! Believe it or not!

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

Here are 2 LIVE MACROS where you can WATCH MORE_THAN_1
Antidiagonal and Powerset being constructed!

http://freewebs.com/namesort/matheology/powersets.html
http://freewebs.com/namesort/matheology/getreal.html



Nobody has ever got that far, yet alone to Question 6!

http://tinyurl.com/blueprints-questions
http://tinyurl.com/blueprints-hyperreals

HYPERREAL DIAGONAL (non-computable)
+ 0.1111111111... ANTIDIAG(DIAGONAL) (finite function)
-----------------------
HYPERREAL MISSING FROM COMPUTABLE REALS

i.e. the computable reals list is not missing any computable real.

So EITHER WAY
ANTIDIAG() is FINITE --> no missing computable real
ANTIDIAG() is INFINITE --> no computable missing real


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

But really, if you want something peer reviewed just take an established
theory and add a clock or something and put some ZFC equations in it!
CONTINOUS <<OBJECT>> always get's rave reviews!

Herc

[Graham Cooper]

On Oct 29, 1:14 am, Shmuel (Seymour J.) Metz
<spamt...@library.lspace.org.invalid> wrote:
> In <t6lo8899odoqh858k485vgffodlnb2n...@4ax.com>, on 10/27/2012

>    at 02:49 PM, JRStern <JRSt...@foobar.invalid> said:
>
> >Are there any such published?
>

> As in reviewed by other kooks? Such articles fall into the category of
> claims trisections of an arbitrary angle using only compass and
> straightedge; no serious journal will publish them.
>


HA FAVORTISM!

[TONICO]
Then I choose the number 0.a_1a_2a_3...., where a_i = 0 if the i-th
number in your list had zero in its i-position, a_i = 1 otherwise.
TADAAAA!

I guess you guys won't be able to use my algorithm to generate
POWERSET(N) then!


http://tinyurl.com/blueprints-POWERSET

a Universal Turing Machine to generate POWERSET(N).

Every <algorithm, input> pair that terminates populates a value into
P(N).

INPUT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
===========================================
TM1 H L H H H L L L L L H H H H ...
TM2 H H H H H H H H H H H H H H ...
TM3 H L L L L L L L L L L L L L ...
TM4 H H H H H H H H H H H H H H ...
...

If TM1(1) Halts then 1 e POWERSET_1
If TM1(2) Loops then 2 !e POWERSET_1
If TM2(1) Halts then 1 e POWERSET_2
If TM2(2) Halts then 2 e POWERSET_2
...

TAKEN FROM THE HALTING VALUES OF THIS EXAMPLE

INPUT 1 2 3 4 5 6 7 8 9 10 11 12 13 14...
TM1 H L
TM2 H H
...

POWERSET(N) = { {1,...} {1,2,...} ... }



Herc
--
www.microPROLOG.com

[INFINITY POWER]

On Oct 29, 8:25 am, Arturo Magidin <magi...@member.ams.org> wrote:

> On Saturday, October 27, 2012 4:49:32 PM UTC-5, JRStern wrote:
> > Are there any such published?
>

> You said elsewhere you are interested in "peer-reviewed" criticisms to
> Cantor's diagonal argument, but not from the point of view of intuitionism
> or some other logical framework, but strictly within the context of ZFC.
>
> There are no such things, because the argument is a valid argument within
> ZF. It is in fact pretty short and clear.
>
> Recall that given a set X, the Axiom of the Power Set states that there is
> a set Y such that z in Y if and only if z is a subset of X. We call this
> set the "power set of X", an denote it P(X).
>
> THEOREM (Cantor) Let X be any set, and let P(X) be the power set of X. If
> f:X->P(X) is any function, then f is not onto; that is, there exists B in
> P(X) that does not lie in the image of f.
>
> Proof. Let f:X->P(X) be a function. By the Axiom of Separation,
>
> B = {x in X | x is not an element of f(X)}
>
> is a subset of A, hence an element of P(X). We claim that f(y)=/=B for all
> y\in X.
>
> Indeed, let y in X. Either f(y)=/=B, or f(y)=B. If f(y)=B, then y in B ->
> y in f(y) -> y not in B; since (P->not(P))->not(P) is a tautology, we
> conclude that f(y)=/=B. So if y in X, then f(y)=/=B, proving that B is not
> in the image of f. QED
>
> LEMMA 1: There is a bijection g:(0,1)->P(N), where P(N) is the power set
> of the natural numbers.
>
> Proof: There is an injection (0,1) to P(N) as follows: given any number x
> in (0,1), we can express x in base 2; if there are two expressions for x
> in base 2, then select the one with finitely many 1s. Map the number x =
> 0.a_1a_2a_3... to the set S = {n in N | a_n=1}. The map is easily seen to
> be one-to-one (given that we have specified which expansion to use).
>
> And there is an injection from P(N) to (0,1): given a subset X of N, let x
> be the real number Sum (a_n/10^n), where a_n=5 if n is not in X, and a_n=6
> if n is in X. Again, this is easily seen to be an injection.
>
> Since we have an injection (0,1)->P(N), and an injection P(N)->(0,1), the
> Cantor-Bernstein-Schroeder Theorem guarantees (in ZF) the existence of a
> bijection g:(0,1)->P(N). QED
>
> COROLLARY: If f:N->R is any function, where N is the natural numbers and R
> is the real numbers, then f is not onto.
>
> Proof: First, there is a bijection h between R and (-pi/2,pi/2), given by
> the arctan function; and there is a bijection t between (-pi/2,pi/2) and
> (0,1), given by t(x) = (x+(pi/2))/pi. And from Lemma 1, we have a
> bijection g from (0,1) to P(N). Note that the compositum gth:R->P(N) is a
> bijection.
>
> Let f:N->R be any function. Then the compositum gthf is a function
> N->P(N). By Cantor's Theorem, this function is not onto. In particular,
> there exists a subset X of P(N) that is not in the image of gthf. Now let
> r = (gth)^{-1}(X) (the function (gth)^{-1} exists because gth is a
> bijection); then r is not in the image of f (for if r = f(n), then
> (gth)^{-1}(X) = f(n), so X = (gth)((gth)^{-1}(X)) = gth(f(n)) = gthf(n),
> which contradicts the choice of X).
>
> Thus, f is not onto, as claimed. QED
>
> --
> Arturo Magidin



In Peano Arithmetic, the relation e(#NUM,#NUM) can represent set membership.

e(1,1)
e(2,1)
e(3,1)
e(2,2)
e(4,2)
e(6,2)
e(7,3)
e(8,3)
e(9,3)

set1 = {1,2,3)
set2 = {2,4,6}
set3 = {7,8,9}


B = {x in X | x is not an element of f(X)}

SO
B = {x | ~xex}

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

THAT A PURE SET ARGUMENT

HERE IS AN INFINITE SET ARGUMENT

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

http://freewebs.com/namesort/matheology/powersets.html

For any 2 infinite sets f & g of subsets of N.

f(1) = { 3 4 7 8 9 11 }
f(2) = { 7 10 12 13 16 20 }
f(3) = { 1 6 8 12 18 19 }
f(4) = { 1 6 16 }
f(5) = { 2 4 5 6 8 13 18 }
f(6) = { 1 3 7 11 12 17 }
f(7) = { 1 4 5 7 10 11 13 17 }
f(8) = { 2 5 6 7 9 16 20 }
f(9) = { 4 9 10 }
f(10) = { 2 6 7 8 9 11 14 15 16 17 20 }
f(11) = { 5 11 12 20 }
f(12) = { 2 3 6 10 14 17 18 }
f(13) = { 3 4 6 9 18 }
f(14) = { 1 4 8 9 12 15 16 19 20 }
f(15) = { 1 2 7 11 14 16 19 }
f(16) = { 1 6 9 13 14 16 20 }
f(17) = { 2 4 5 10 11 13 16 17 }
f(18) = { 1 2 3 5 8 9 10 17 }
f(19) = { 6 7 11 13 }
f(20) = { 2 7 13 14 15 18 }
...

B = {x in X | x is not an element of f(X)}
= { 1 2 3 4 6 8 10 12 13 14 15 18 19 20 ...}

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

g(1) = { 1 2 9 11 12 14 17 19 }
g(2) = { 9 10 12 14 15 20 }
g(3) = { 1 3 8 10 12 20 }
g(4) = { 1 2 3 4 6 11 12 17 18 19 }
g(5) = { 3 4 5 7 8 9 11 13 15 }
g(6) = { 1 8 12 13 18 }
g(7) = { 7 10 11 20 }
g(8) = { 2 3 4 5 7 16 }
g(9) = { 1 5 11 13 18 }
g(10) = { 5 8 9 10 14 17 18 19 }
g(11) = { 3 }
g(12) = { 4 5 6 8 9 13 }
g(13) = { 1 2 4 5 7 9 11 12 18 }
g(14) = { 3 5 6 8 18 }
g(15) = { 1 2 3 6 9 13 15 17 19 }
g(16) = { 1 8 9 10 14 17 }
g(17) = { 1 3 4 5 10 11 14 17 }
g(18) = { 1 7 8 13 15 }
g(19) = { 2 3 4 6 9 12 14 15 19 }
g(20) = { 3 4 6 10 17 }
...

C = {x in Y | x is not an element of f(Y)}
{ 2 6 8 9 11 12 13 14 16 18 20 ...}

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

f'

f(1) = { 3 4 7 8 9 11 }
f(2) = { 7 10 12 13 16 20 }
f(3) = { 1 6 8 12 18 19 }
f(4) = { 2 4 5 6 8 13 18 }
f(5) = { 1 6 16 }
f(6) = { 1 3 7 11 12 17 }
f(7) = { 1 4 5 7 10 11 13 17 }
f(8) = { 2 5 6 7 9 16 20 }
f(9) = { 4 9 10 }
f(10) = { 2 6 7 8 9 11 14 15 16 17 20 }
f(11) = { 2 3 6 10 14 17 18 }
f(12) = { 3 4 6 9 18 }
f(13) = { 5 11 12 20 }
f(14) = { 1 4 8 9 12 15 16 19 20 }
f(15) = { 1 2 7 11 14 16 19 }
f(16) = { 1 2 3 5 8 9 10 17 }
f(17) = { 2 4 5 10 11 13 16 17 }
f(18) = { 1 6 9 13 14 16 20 }
f(19) = { 6 7 11 13 }
f(20) = { 2 7 13 14 15 18 }
...

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

g'

g(1) = { 9 10 12 14 15 20 }
g(2) = { 1 3 8 10 12 20 }
g(3) = { 1 2 9 11 12 14 17 19 }
g(4) = { 1 2 3 4 6 11 12 17 18 19 }
g(5) = { 1 8 12 13 18 }
g(6) = { 3 4 5 7 8 9 11 13 15 }
g(7) = { 7 10 11 20 }
g(8) = { 2 3 4 5 7 16 }
g(9) = { 5 8 9 10 14 17 18 19 }
g(10) = { 1 5 11 13 18 }
g(11) = { 3 }
g(12) = { 4 5 6 8 9 13 }
g(13) = { 1 2 4 5 7 9 11 12 18 }
g(14) = { 3 5 6 8 18 }
g(15) = { 1 8 9 10 14 17 }
g(16) = { 1 2 3 6 9 13 15 17 19 }
g(17) = { 1 3 4 5 10 11 14 17 }
g(18) = { 1 7 8 13 15 }
g(19) = { 3 4 6 10 17 }
g(20) = { 2 3 4 6 9 12 14 15 19 }
...

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

B' = {x in X | x is not an element of f'(X)}
= { 1 2 3 5 6 8 10 11 12 13 14 15 16 18 19 20 ...}

C' = {x in Y | x is not an element of f'(Y)}
= { 1 2 3 5 6 8 10 11 12 13 14 15 16 18 19 20 ...}


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

That is: the 'missing subset' of any' infinite set of subsets are the same,
depending on the enumeration not the data in the set.

Herc

[Graham Cooper]

> ---------------------------------
>
> B' = {x in X | x is not an element of f'(X)}
>    = { 1 2 3 5 6 8 10 11 12 13 14 15 16 18 19 20 ...}
>
> C' = {x in Y | x is not an element of f'(Y)}
>    = { 1 2 3 5 6 8 10 11 12 13 14 15 16 18 19 20 ...}
>
> ----------------------------------
>
> That is:  the 'missing subset' of any' infinite set of subsets are the same,
> depending on the enumeration not the data in the set.
>
> Herc

I changed X instead of f there!

C' = {x in X | x is not an element of f'(X)}

   = { 1 2 3 5 6 8 10 11 12 13 14 15 16 18 19 20 ...}

Same missing set for BOTH countable 'powersets'.

Herc

[Graham Cooper]

On Oct 29, 10:26 am, JRStern <JRSt...@foobar.invalid> wrote:
> On Sun, 28 Oct 2012 15:25:27 -0700 (PDT), Arturo Magidin

>
> <magi...@member.ams.org> wrote:
> >On Saturday, October 27, 2012 4:49:32 PM UTC-5, JRStern wrote:
> >> Are there any such published?
>
> >You said elsewhere you are interested in "peer-reviewed" criticisms to
> >Cantor's diagonal argument, but not from the point of view of intuitionism
> >or some other logical framework, but strictly within the context of ZFC.
>
> >There are no such things, because the argument is a valid argument
> >within ZF. It is in fact pretty short and clear.
>

> Then there can be a book taking on the objections one at a time and
> knocking them down.
>
>


Using onto properties of functions is just double-talk reiterating the
anti-diagonal method.

Proof. Let f:X->P(X) be a function.

This proves nothing about _enumerable_ sets!


This set of reals is _enumerable_ but it has 2 distinct _enumeration_
functions.

WHAT IS MISSING?
0.000000.. 0.110000.. 0.111010.. 0.111110.. ...
0.000111.. 0.101000.. 0.000010.. 0.001100.. ...
0.111000.. 0.010010.. 0.001010.. 0.101010.. ...
0.111100.. 0.001011.. 0.000011.. 0.111111.. ...
...


Try to use Cantor's method and you fail!

Herc

[Graham Cooper]

On Oct 29, 10:42 am, Arturo Magidin <magi...@member.ams.org> wrote:
> Who would waste his or her valuable time doing so
>
> --
> Arturo Magidin
>

TAKES 5 MINUTES YOU * C O W A R D *

[Graham Cooper]

>
> WHAT IS MISSING?
> 0.000000.. 0.110000.. 0.111010.. 0.111110.. ...
> 0.000111.. 0.101000.. 0.000010.. 0.001100.. ...
> 0.111000.. 0.010010.. 0.001010.. 0.101010.. ...
> 0.111100.. 0.001011.. 0.000011.. 0.111111.. ...
> ...
>

HINT:

[TONICO]
Then I choose the number 0.a_1a_2a_3...., where a_i = 0 if the i-th
number in your list had zero in its i-position, a_i = 1 otherwise.
TADAAAA!

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

YEH GO ON! DO THAT! THERE'S MY SET OF ALL REALS RIGHT THERE!


Herc

[Virgil]

In article <k6ki4c$18t$1...@dont-email.me>,

"INFINITY POWER" <infi...@limited.com> wrote:

> That is: the 'missing subset' of any' infinite set of subsets are the same,
> depending on the enumeration not the data in the set.
>
> Herc

That statement hides a contrary-to-fact assumption. Which Herc is no
doubt unable to see without help.

Given ANY listing of distinct subsets of N, there is no such thing as
"THE" missing subset, there are uncountably many of them missing from
any such listing.

There is, in fact, a totally mechanical and automatic way of
demonstrating AT LEAST as many sets missing from any such list as appear
listed in it.
--

[Graham Cooper]

On Oct 29, 10:52 am, Virgil <vir...@ligriv.com> wrote:
> In article <k6ki4c$18...@dont-email.me>,

>  "INFINITY POWER" <infin...@limited.com> wrote:
>
> > That is:  the 'missing subset' of any' infinite set of subsets are the same,
> > depending on the enumeration not the data in the set.
>
> > Herc
>
> That statement hides a contrary-to-fact assumption. Which Herc is no
> doubt unable to see without help.
>
> Given ANY listing of distinct subsets of N, there is no such thing as
> "THE" missing subset, there are uncountably many of them missing from
> any such listing.
>
> There is, in fact, a totally mechanical and automatic way of
> demonstrating AT LEAST as many sets missing from any such list as appear
> listed in it.
>

If your list of subsets
1/ contains all n=1,2,3... in infinitely many rows
2/ omits all n=1,2,3... in infinitely many rows

then any row can be swapped to either contain it's row number or not.

ROW7: {1,2, 7 , 9, 10..}

There are infinitely many rows to exchange with to arbitrarily decide
if 7eB.

Herc

[Graham Cooper]

On Oct 29, 10:42 am, Arturo Magidin <magi...@member.ams.org> wrote:

> Given that there is a straighforward, direct proof of Cantor's Theorem that holds in ZF, why would any reputable publishing house waste *anyone's* time peer-reviewing the arguments presented against?
> --
> Arturo Magidin


RIGHT! NO UNIVERSITY ON EARTH would be the 1st to admit they took 100
YEARS to stop ramming down everyone's throats

SIZE(SET) > SIZE({1,2,3...})
SIZE(SET) > INFINITY

which is BANNED WORD NOW!

Based on DEFINITIONS of NON-TERMINATING DIGITS {0,1,2,3,4,5,6,7,8,9}

1234..
7655..
3456..
9887..
..

DIAG = 1657...

X = 2768...

-----

ARE YOU SURE THAT PROVES SET SIZES BIGGER THAN INFINITY EXIST?

Because HALF THE PLANET CANT SEE IT and the 1000 ENYCLOPEDIAS on YOUR
ABSTRACT REAL MODELS prevents any FORMAL WORK FROM PROCEEDING!

Herc

[Graham Cooper]

On Oct 29, 11:00 am, Arturo Magidin <magi...@member.ams.org> wrote:

> On Sunday, October 28, 2012 7:47:12 PM UTC-5, Graham Cooper wrote:
> > On Oct 29, 10:42 am, Arturo Magidin <magi...@member.ams.org> wrote:
>
> > > Who would waste his or her valuable time doing so
>
> > > --
>
> > > Arturo Magidin
>
> > TAKES 5 MINUTES YOU  * C O W A R D *
>

> Sorry, but my time is too valuable to waste with the likes of ye. Now, if you are willing to pay my standard consulting fee, I'll be happy to consider it. Until I hear from your legal representative regarding a legally binding offer to do so, please consider my silence the only response you will get. Any further personal attacks from you will be evidence of your unwillingness to put your money where you mouth is.
>
> --
> Arturo Magidin


My mistake! I assumed by posting a powerset disproof directly under
my powerset algorithm you had something to say.

http://tinyurl.com/blueprints-POWERSET

a Universal Turing Machine to generate POWERSET(N).

Every <algorithm, input> pair that terminates populates a value into
P(N).

INPUT 1 2 3 4 5 6 7 8 9 10 11 ...
===========================================
TM1 H L H H H L L L L L H ...
TM2 H H H H H H H H H H H ...
TM3 H L L L L L L L L L L ...
TM4 H H H H H H H H H H H ...

...

If TM1(1) Halts then 1 e POWERSET_1
If TM1(2) Loops then 2 !e POWERSET_1
If TM2(1) Halts then 1 e POWERSET_2
If TM2(2) Halts then 2 e POWERSET_2
...

TAKEN FROM THE HALTING VALUES OF THIS EXAMPLE

INPUT 1 2 3 4 5 6 7 8 9 10 11 12 13 14...

TM1 H L H
TM2 H H H
...

POWERSET(N) = { {1,3,...} {1,2,3,...} ... }

[Hercules ofZeus]

On Oct 29, 12:20 pm, Arturo Magidin <magi...@member.ams.org> wrote:
> On Sunday, October 28, 2012 9:04:41 PM UTC-5, JRStern wrote:
> > On Sun, 28 Oct 2012 17:41:59 -0700 (PDT), Arturo Magidin
>
> > <magi...@member.ams.org> wrote:
>
> > >Huh?
>
> > >Sorry, but that statement makes absolute no sense
>
> > >whatsoever to me.  What exactly was the point you
>
> > >were attempting to make?
>
> > That I am not arguing with the conclusion that Cantor's Theorem is
>
> > true, I am questioning whether the diagonal argument is coherent.
>
> > How can this be unclear?
>
> Because I did not simply state the conclusion. I gave you the "diagonal argument". If you are questioning whether it is "coherent", then you should point to whatever point you find incoherent, rather than simply quote and then give a sentence fragment.
>
> The government doesn't like it when I read minds without a warrant, so I try not to do it, you see.
>
> I have you a complete proof of Cantor's Theorem; the diagonal argument is embedded in that theorem. What is it that you find incoherent?
>
> If there is nothing you find incoherent, then why is it that you continue to "question" its coherence?
>
> If you could not even tell that you were presented with the argument in the first place, then perhaps your problems arise much sooner than at Cantor's diagonal argument?
>
> If it is a *particular* presentation of the argument that concerns you, then it is incumbent upon you to specify which presentation it is you find yourself having doubts about, and stop nattering about "peer-review", books, and the like.
>
> --
> Arturo Magidin


Are you saying you just proved:

ALL(f):N->R E(r):R ALL(n):N f(n)=/=r

in 1ST ORDER LOGIC?

i.e. FOL = Quantifiers Over Arguments Not Functions.

G. Cooper (BInfTech)
--
http://tinyURL.com/BLUEPRINTS-MATHEMATICS
http://tinyURL.com/BLUEPRINTS-HYPERREALS
http://tinyURL.com/BLUEPRINTS-QUESTIONS
http://tinyURL.com/BLUEPRINTS-POWERSET
http://tinyURL.com/BLUEPRINTS-THEOREM
http://tinyURL.com/BLUEPRINTS-PROLOG
http://tinyURL.com/BLUEPRINTS-FORALL
http://tinyURL.com/BLUEPRINTS-TURING
http://tinyURL.com/BLUEPRINTS-GODEL
http://tinyURL.com/BLUEPRINTS-PROOF
http://tinyURL.com/BLUEPRINTS-MATHS
http://tinyURL.com/BLUEPRINTS-LOGIC
http://tinyURL.com/BLUEPRINTS-BRAIN
http://tinyURL.com/BLUEPRINTS-PERM
http://tinyURL.com/BLUEPRINTS-REAL
http://tinyURL.com/BLUEPRINTS-SETS
http://tinyURL.com/BLUEPRINTS-HALT
http://tinyURL.com/BLUEPRINTS-P-NP
http://tinyURL.com/BLUEPRINTS-GUT
http://tinyURL.com/BLUEPRINTS-BB
http://tinyURL.com/BLUEPRINTS-AI

[Hercules ofZeus]

On Oct 29, 1:29 pm, Arturo Magidin <magi...@member.ams.org> wrote:
> On Sunday, October 28, 2012 10:27:52 PM UTC-5, Hercules ofZeus wrote:
> > On Oct 29, 1:14 pm, Arturo Magidin <magi...@member.ams.org> wrote:

>
> > > On Sunday, October 28, 2012 10:01:31 PM UTC-5, Hercules ofZeus wrote:
>
> > > > On Oct 29, 12:20 pm, Arturo Magidin <magi...@member.ams.org> wrote:
>
> > > > > On Sunday, October 28, 2012 9:04:41 PM UTC-5, JRStern wrote:
>
> > > > > > On Sun, 28 Oct 2012 17:41:59 -0700 (PDT), Arturo Magidin
>
> > > > > > <magi...@member.ams.org> wrote:
>
> > > > > > >Huh?
>
> > > > > > >Sorry, but that statement makes absolute no sense
>
> > > > > > >whatsoever to me.  What exactly was the point you
>
> > > > > > >were attempting to make?
>
> > > > > > That I am not arguing with the conclusion that Cantor's Theorem is
>
> > > > > > true, I am questioning whether the diagonal argument is coherent.
>
> > > > > > How can this be unclear?
>
> > > > > Because I did not simply state the conclusion. I gave you the "diagonal argument". If you are questioning whether it is "coherent", then you should point to whatever point you find incoherent, rather than simply quote and then give a sentence fragment.
>
> > > > > The government doesn't like it when I read minds without a warrant, so I try not to do it, you see.
>
> > > > > I have you a complete proof of Cantor's Theorem; the diagonal argument is embedded in that theorem. What is it that you find incoherent?
>
> > > > > If there is nothing you find incoherent, then why is it that you continue to "question" its coherence?
>
> > > > > If you could not even tell that you were presented with the argument in the first place, then perhaps your problems arise much sooner than at Cantor's diagonal argument?
>
> > > > > If it is a *particular* presentation of the argument that concerns you, then it is incumbent upon you to specify which presentation it is you find yourself having doubts about, and stop nattering about "peer-review", books, and the like.
>
> > > > > --
>
> > > > > Arturo Magidin
>
> > > > Are you saying you just proved:
>
> > > > ALL(f):N->R  E(r):R  ALL(n):N  f(n)=/=r
>
> > > > in 1ST ORDER LOGIC?
>
> > > > i.e. FOL = Quantifiers Over Arguments Not Functions.
>

> > > In ZF, functions are sets; and the objects of the theory are sets. So the statement above is a perfectly fine first order statement in the language of ZF. Moreover, there is a *set* of functions from N to R, so I'm quantifying over the **objects** that are elements of that set.
>
> > > But surely, if you know about first order logic, then you knew that?
>
> > > And if you didn't... well, you are someone else whose problems come from much earlier than any objection you might by trying to raise here; your problem is the same as Sir Richard Phillips's.
>
> > > --
>
> > > Arturo Magidin
>
> > And you see no problem with using a logic composed of predicate
>
> > strings (functions) to construct sets to construct functions to range
>
> > over ALL functions, and calling it 1st order logic - No Quantifying
>
> > over functions here!
>
> The reason I see no problems is because I actually know what I'm talking about.
>
> You should try it some day. Until then, let me know when you are actually willing to pay for my insight. I'm tired of casting pearls before the hercs.
>
> --
> Arturo Magidin
>


I'll pay you $5000 to find this guys email address!

http://tinyurl.com/SetOfEverything

"It is extremely important to understand that the classical treatment
of
language interpretation parameterizes the universal quantifier over
collections. Moreover, the existential quantifier is interpreted only
with
respect to its status as a derivative concept relative to the
universal
quantifier. These are the underlying assumptions of construction that
allow
the self-inconsistency of a specific syntactic form to be extended to
a
metaphysical assertion of reality. "

so I can employ him to work on my Logic Solver!

Unfortunately his ideas weren't welcome in sci.logic 9 years ago.

Apart from that, I have some spots available here:

www.CAMGIRLS.com

Pays $3 / minute???

Herc

[Hercules ofZeus]

On Oct 29, 1:43 pm, Arturo Magidin <magi...@member.ams.org> wrote:

> On Sunday, October 28, 2012 10:36:50 PM UTC-5, Hercules ofZeus wrote:
> > > You should try it some day. Until then, let me know when you are actually willing to pay for my insight. I'm tired of casting pearls before the hercs.
>

> > I'll pay you $5000 to find this guys email address!
>
> >http://tinyurl.com/SetOfEverything
>

> First, finding people's e-mail address is not my area of expertise; second, if you want my *professional* services, you'll have to do better than an Usenet promise: I'll want a legally binding offer presented to me by your authorized legal representative.


You mean this guy?


http://AustraliaMostWanted.com/images/4.jpg




>
> In the meantime, I'll take your reply as your way of demonstrating that you are not willing to put your money where you mouth is. Hardly surprising, given that you head is well past your sphincter. I wouldn't want to put anything there, either.
>
> --
> Arturo Magidin
>
>


Well I've said so much... and $3/min video conferenceing industry
being what it is...

Herc

[Graham Cooper]

On Oct 29, 2:26 pm, Virgil <vir...@ligriv.com> wrote:
> In article <sior88t2h05c7t3p0lhoa1g99uuq69i...@4ax.com>,

>
>
>  JRStern <JRSt...@foobar.invalid> wrote:
> > On Sun, 28 Oct 2012 17:41:59 -0700 (PDT), Arturo Magidin
> > <magi...@member.ams.org> wrote:
>
> > >Huh?
>
> > >Sorry, but that statement makes absolute no sense
> > >whatsoever to me.  What exactly was the point you
> > >were attempting to make?
>
> > That I am not arguing with the conclusion that Cantor's Theorem is
> > true, I am questioning whether the diagonal argument is coherent.
>
> > How can this be unclear?
>

> > J.
>
> In its original form (about sequences of letters from the set {m,w}) I
> do not see that Cantor's argument can possibly be either incoherent or
> unclear.
>



AN INFINITE ENUMER_ABLE SET OF BINARY STRINGS

WHAT IS MISSING?
mmmmmm.. wwmmmm.. wwwmwm.. wwwwwm.. ...
mmmwww.. wmwmmm.. mmmmwm.. mmwwmm.. ...
wwwmmm.. mwmmwm.. mmwmwm.. wmwmwm.. ...
wwwwmm.. mmwmww.. mmmmww.. wwwwww.. ...
...

Prove by extrapolation past 6 terms that some sequence is missing!



Herc

[Graham Cooper]

On Oct 29, 2:14 pm, Tonic...@yahoo.com wrote:

> On Monday, October 29, 2012 5:27:52 AM UTC+2, Hercules ofZeus wrote:
>
> > And you see no problem with using a logic composed of predicate
>
> > strings (functions) to construct sets to construct functions to range
>
> > over ALL functions, and calling it 1st order logic - No Quantifying
>
> > over functions here!
>

> > Herc
>
> Oh, dear hollie mollie! You have no pale idea what you're talking about, have you? You don't even know the basic of the basics of the theory you so deeply hate, the hell knows why.
>
> Ts,ts,ts....bad, my boy: very bad!

OK I'll give it another shot!

 [TONICO]
 Then I choose the number 0.a_1a_2a_3...., where a_i = 1 if the i-th
 number in your list had zero in its i-position, a_i = 0 otherwise.
 TADAAAA!


R1 = 0.111111..
R2 = 0.000000..
R3 = 0.101010..
R4 = 0.111000..
...


OK so a_1=0, a_2=1, a_3=0, a4=1..

0.0101...

TADAAAA!

UN-COUNTABLE!

Am I doing it right?

Let's check...

ROW1 a1 = 0 because that's a 1 so that's not it!
ROW2 a2 = 1 because that's a 0 so that's not it!
ROW3 a3 = 0 because that's a 1 so that's not it!

IT CHECKS OUT! MORE THAN SIZE({1,2,3..}) ITS A FACT!


Herc

[Peter Webb]

As you haven't told us what "w" and "m" are supposed to be, or the rule
that you are using to form the list, this cannot be done.

If you think you have a list of all Reals, post it here, and I will
happily prove its not a list of all Reals by finding a Real not on the
list for you.

[Graham Cooper]

On Oct 29, 4:05 pm, "Peter Webb"

What about an enumerable set of all reals?

The first one is 0.000000...

Is that OK?

Herc

[LudovicoVan]

"Peter Webb" <webbfamily...@optusnet.com.au> wrote in message
news:k6l6as$h0q$1...@news.albasani.net...

> As you haven't told us what "w" and "m" are supposed to be, or the rule
> that you are using to form the list, this cannot be done.

Parrot, at least read bloody Cantor's argument before pontificating your
dogma.

> If you think you have a list of all Reals, post it here, and I will
> happily prove its not a list of all Reals by finding a Real not on the
> list for you.

But you can do no such thing: the anti-diagonal is just not a real number.

-LV

[Peter Webb]

No. You have to provide a list of all Reals in [0,1)

Can't do it?

Do you wonder why?

[Graham Cooper]

> > What about an enumerable set of all reals?
>
> > The first one is 0.000000...
>
> > Is that OK?
>
> > Herc
>
> No. You have to provide a list of all Reals in [0,1)
>
> Can't do it?
>
> Do you wonder why?

TAKES 5 MINUTES YOU * C O W A R D *

WHAT IS MISSING?
0.000000.. 0.110000.. 0.111010.. 0.111110.. ...
0.000111.. 0.101000.. 0.000010.. 0.001100.. ...
0.111000.. 0.010010.. 0.001010.. 0.101010.. ...
0.111100.. 0.001011.. 0.000011.. 0.111111.. ...
...

[Graham Cooper]

On Oct 30, 6:30 am, Arturo Magidin <magi...@member.ams.org> wrote:
> On Monday, October 29, 2012 2:04:42 PM UTC-5, JRStern wrote:
> > On Mon, 29 Oct 2012 09:14:02 -0700 (PDT), Arturo Magidin

>
> > <magi...@member.ams.org> wrote:
>
> > >> >The government doesn't like it when I read minds without a
>
> > >> >warrant, so I try not to do it, you see.
>

> > >> There are constructivist objections to the whole enterprise.  If
>
> > >> nothing else, I was hoping to see something like these addressed
>
> > >> against the diagonal argument piecemeal.
>
> > >In a constructivist universe, you cannot have a function defined
>
> > >on the natural numbers, and you cannot have a set of real numbers at all.
>
> > >The very premise ("Suppose f:N-->(0,1) is a function") is considered
>
> > >nonsensical in the constructivist point of view.
>
> > >You can only talk about a "rule" that will allow you, in principle,
>
> > >to compute f(n) up to any degree of exactness that you care to specify.
>
> > >You can encode the procedure in the proof given assuming that the
>
> > >function f is given by Turing Machine which, when given n as an input,
>
> > >will give you the number of a Turing machine that computes f(n).
>
> > >Then you can establish the existence of a Turing machine r which,
>
> > >when given as input the number of a Turing machine that computes
>
> > >a function f as described above, will produce as an output the
>
> > >sequence b_n which is not equal to the output of any Turing machine
>
> > >whose number is produced by the Turing machine corresponding to f.
>
> > >That much is allowable within the computational universe.
>
> > Doesn't this sound rather like what some of the "cranks" hereabouts
>
> > are trying to say?
>
> Which "cranks", and what specifically do they say that you find "rather like" exactly which part of the above?
>
> Yet again: one cannot give you an exact answer if you insist on presenting nothing but vague statements that have little or no actual content. I certainly have no desire to waste my time discussing phantoms and ephemerals, so perhaps you can stop being vague and wishy-washy, and give some specifics? If you can't, then stop trying to think about math. Now. Stop. Yes. You. Stop it. There is absolutely no point in discussing mathematics on the basis of vague pronouncements, vague statements, vague "feelings", and vague impressions; because, whatever it is you end up doing, it's not mathematics.
>
> --
> Arturo Magidin


Cantor's Proof is like looking between 2 parallel mirrors and seeing
the infinite sequence of fools reflected within!

size(SET) > size( {1,2,3...} ) #term of which is now duckspeaked

based on...

inverting the diagonal of non terminating sequences!

Changing 1 digit of the diagonal does NOTHING!
Changing 1 digit at a time of the diagonal does NOTHING!
Changing all digits of the diagonal (of a non terminating list)

DOES NOTHING!



Herc

[Peter Webb]

I can't see what the "..." is supposed to mean. Could you provide the
formula which tells me the Real in position n for all n, so I can step
through the list and see what is missing?

[Graham Cooper]

On Oct 30, 11:52 am, "Peter Webb"

WHAT IS MISSING?
0.000000..
0.000111..
0.111000..
0.111100..
...

Is that better maaate?

Herc

[Peter Webb]

What is the nth Real in your list?

[Peter Webb]

LudovicoVan wrote:

> "Peter Webb" <webbfamily...@optusnet.com.au> wrote in message
> news:k6l6as$h0q$1...@news.albasani.net...
>
> > As you haven't told us what "w" and "m" are supposed to be, or the
> > rule that you are using to form the list, this cannot be done.
>
> Parrot, at least read bloody Cantor's argument before pontificating
> your dogma.
>

I have. And its not dogma.

> > If you think you have a list of all Reals, post it here, and I will
> > happily prove its not a list of all Reals by finding a Real not on
> > the list for you.
>
> But you can do no such thing: the anti-diagonal is just not a real
> number.
>
> -LV

Of course it is a Real number. It is the limit of a Cauchy sequence.
You do know that convergent Cauchy sequences define unique Reals, right?


(I just bagged somebody for providing a proof of the anti-diagonal was
Real using Cauchy sequences, on the basis that no crank disputes
whether the anti-diagonal is Real. And then you prove me wrong by
raising exactly this isuue).

So, where is your list of all the Reals?

[Peter Webb]

A definition of what the nth Real in your list is.

> Is that better maaate?
>
> Herc

No.

Do you claim to have a list of all Reals?

If so, please post the rule which determines which Real is in the nth
position.

[LudovicoVan]

"Peter Webb" <webbfamily...@optusnet.com.au> wrote in message

news:k6nfnq$dre$1...@news.albasani.net...

> LudovicoVan wrote:
>> "Peter Webb" <webbfamily...@optusnet.com.au> wrote in message
>> news:k6l6as$h0q$1...@news.albasani.net...
>>
>> > As you haven't told us what "w" and "m" are supposed to be, or the
>> > rule that you are using to form the list, this cannot be done.
>>
>> Parrot, at least read bloody Cantor's argument before pontificating
>> your dogma.
>
> I have. And its not dogma.

But you have just asked what "w" and "m" are supposed to mean, remember?
That's why I call you (and co., don't take it personally as it isn't)
parrots etc.: because you are just right, whatever kind of bollocks you
might say, in the end you are still right. By dogma, and the guns.

>> > If you think you have a list of all Reals, post it here, and I will
>> > happily prove its not a list of all Reals by finding a Real not on
>> > the list for you.
>>
>> But you can do no such thing: the anti-diagonal is just not a real
>> number.
>

> Of course it is a Real number. It is the limit of a Cauchy sequence.
> You do know that convergent Cauchy sequences define unique Reals, right?
>
> (I just bagged somebody for providing a proof of the anti-diagonal was
> Real using Cauchy sequences, on the basis that no crank disputes
> whether the anti-diagonal is Real. And then you prove me wrong by
> raising exactly this isuue).

Indeed, to be precise, you and co. excel in not even being wrong.

> So, where is your list of all the Reals?

The real numbers are a subset of the surreals. But I had said that already.
Of course they are countable: there just is no such thing as a number that
is not a number: a number is all we can do with it seriously (you won't get
this, but never mind).

Well, maybe it's not dogma, it's just that you cannot read... But no, I am
not optimistic about anybody's intellectual honesty around here: except
maybe for Prof. Magidin, who at least shows some good math.

-LV

[Peter Webb]

LudovicoVan wrote:

> "Peter Webb" <webbfamily...@optusnet.com.au> wrote in message
> news:k6nfnq$dre$1...@news.albasani.net...
> > LudovicoVan wrote:
> >>"Peter Webb" <webbfamily...@optusnet.com.au> wrote in message
> > > news:k6l6as$h0q$1...@news.albasani.net...
> > >
> >>> As you haven't told us what "w" and "m" are supposed to be, or the
> >>> rule that you are using to form the list, this cannot be done.
> > >
> > > Parrot, at least read bloody Cantor's argument before
> > > pontificating your dogma.
> >
> > I have. And its not dogma.
>
> But you have just asked what "w" and "m" are supposed to mean,
> remember? That's why I call you (and co., don't take it personally as
> it isn't) parrots etc.: because you are just right, whatever kind of
> bollocks you might say, in the end you are still right. By dogma,
> and the guns.
>

You actually have to identify what I said which you think is "bollocks".

Somehow you forgot to say what it is.

> >>> If you think you have a list of all Reals, post it here, and I
> will >>> happily prove its not a list of all Reals by finding a Real
> not on >>> the list for you.
> > >
> > > But you can do no such thing: the anti-diagonal is just not a real
> > > number.
> >
> > Of course it is a Real number. It is the limit of a Cauchy sequence.
> > You do know that convergent Cauchy sequences define unique Reals,
> > right?
> >
> > (I just bagged somebody for providing a proof of the anti-diagonal
> > was Real using Cauchy sequences, on the basis that no crank disputes
> > whether the anti-diagonal is Real. And then you prove me wrong by
> > raising exactly this isuue).
>
> Indeed, to be precise, you and co. excel in not even being wrong.
>
> > So, where is your list of all the Reals?
>
> The real numbers are a subset of the surreals. But I had said that
> already. Of course they are countable:

Terrific. Provide a surjection from N to R.

> there just is no such thing as
> a number that is not a number: a number is all we can do with it
> seriously (you won't get this, but never mind).
>

You are supposed to either say what you think I said which is bollocks,
or provide a counter-example to Cantor.

As it is, you are just mumbling.

> Well, maybe it's not dogma, it's just that you cannot read... But
> no, I am not optimistic about anybody's intellectual honesty around
> here: except maybe for Prof. Magidin, who at least shows some good
> math.
>
> -LV

The cranks should be pretty obvious. Clearly Madigin is not a crank,
but there are many others who are also not cranks. Anybody who claims
that Cantor's proof is false without identifying an error in the proof
is a crank.

[LudovicoVan]

"Peter Webb" <webbfamily...@optusnet.com.au> wrote in message

news:k6nk8v$lht$1...@news.albasani.net...

> LudovicoVan wrote:
>> "Peter Webb" <webbfamily...@optusnet.com.au> wrote in message
>> news:k6nfnq$dre$1...@news.albasani.net...
>> > LudovicoVan wrote:
>> >>"Peter Webb" <webbfamily...@optusnet.com.au> wrote in message
>> > > news:k6l6as$h0q$1...@news.albasani.net...
>> > >
>> >>> As you haven't told us what "w" and "m" are supposed to be, or the
>> >>> rule that you are using to form the list, this cannot be done.
>> > >
>> > > Parrot, at least read bloody Cantor's argument before
>> > > pontificating your dogma.
>> >
>> > I have. And its not dogma.
>>
>> But you have just asked what "w" and "m" are supposed to mean,
>> remember? That's why I call you (and co., don't take it personally as
>> it isn't) parrots etc.: because you are just right, whatever kind of
>> bollocks you might say, in the end you are still right. By dogma,
>> and the guns.
>
> You actually have to identify what I said which you think is "bollocks".
>
> Somehow you forgot to say what it is.

I have just said it... Unless you mean a full list of all the bollocks
ever stated in sci.math: of course, just forget it.

>> >>> If you think you have a list of all Reals, post it here, and I
>> will >>> happily prove its not a list of all Reals by finding a Real
>> not on >>> the list for you.
>> > >
>> > > But you can do no such thing: the anti-diagonal is just not a real
>> > > number.
>> >
>> > Of course it is a Real number. It is the limit of a Cauchy sequence.
>> > You do know that convergent Cauchy sequences define unique Reals,
>> > right?
>> >
>> > (I just bagged somebody for providing a proof of the anti-diagonal
>> > was Real using Cauchy sequences, on the basis that no crank disputes
>> > whether the anti-diagonal is Real. And then you prove me wrong by
>> > raising exactly this isuue).
>>
>> Indeed, to be precise, you and co. excel in not even being wrong.
>>
>> > So, where is your list of all the Reals?
>>
>> The real numbers are a subset of the surreals. But I had said that
>> already. Of course they are countable:
>
> Terrific. Provide a surjection from N to R.

Ever heard of the surreals at all? But you are correct in that I was
imprecise: all infinite sets are *extended-countable*, i.e. all infinite
sets biject with N*, the extended naturals.

>> there just is no such thing as
>> a number that is not a number: a number is all we can do with it
>> seriously (you won't get this, but never mind).
>
> You are supposed to either say what you think I said which is bollocks,
> or provide a counter-example to Cantor.
>
> As it is, you are just mumbling.
>
>> Well, maybe it's not dogma, it's just that you cannot read... But
>> no, I am not optimistic about anybody's intellectual honesty around
>> here: except maybe for Prof. Magidin, who at least shows some good
>> math.
>

> The cranks should be pretty obvious. Clearly Madigin is not a crank,
> but there are many others who are also not cranks. Anybody who claims
> that Cantor's proof is false without identifying an error in the proof
> is a crank.

In fact, your denial is integral to your dogma.

-LV

[Graham Cooper]

On Oct 30, 1:47 pm, Arturo Magidin <magi...@member.ams.org> wrote:
> On Monday, October 29, 2012 6:49:41 PM UTC-5, JRStern wrote:
> > On Mon, 29 Oct 2012 13:30:43 -0700 (PDT), Arturo Magidin

>
> > <magi...@member.ams.org> wrote:
>
> > >> Doesn't this sound rather like what some of the "cranks" hereabouts
>
> > >> are trying to say?
>
> > >Which "cranks", and what specifically do they say that you find
>
> > >"rather like" exactly which part of the above?
>
> > >Yet again: one cannot give you an exact answer if you
>
> > >insist on presenting nothing but vague statements that have
>
> > >little or no actual content. I certainly have no desire
>
> > >to waste my time discussing phantoms and ephemerals,
>
> > >so perhaps you can stop being vague and wishy-washy,
>
> > >and give some specifics? If you can't, then stop trying
>
> > >to think about math. Now. Stop. Yes. You. Stop it.
>
> > >There is absolutely no point in discussing mathematics
>
> > >on the basis of vague pronouncements, vague statements,
>
> > >vague "feelings", and vague impressions; because, whatever
>
> > >it is you end up doing, it's not mathematics.
>

> > Thank you for trying, although I have clearly not been able to phrase
>
> > a question that you feel has a useful answer.
>
> I find your entire participation now entirely frustrating; now you are actively *refusing* to engage in the discussion in any reasonable way, prefering instead to bow out as soon as specifics are requested from you.
>
> That makes the entire thing nothing but an empty mental exercise on your part, or a successful attempt at having other people waste their time. For which I cannot find any reason to thank you. I do hope, however, that you will be on your way and stop wasting everyone's time, unless you actually want to start engaging in a reasonable manner.
>
> --
> Arturo Magidin


but you answered his Question beautifully, 'no University would sink
to peer review any critique on Cantor's theory'.

This is synonymous with - 'if Cantor's theory was full of holes, we
wouldn't know it!'

Thank you for your input!

Herc

[Graham Cooper]

On Oct 30, 1:51 pm, "Peter Webb"

<webbfamilyDIEspam...@optusnet.com.au> wrote:
>
> No.
>
> Do you claim to have a list of all Reals?

I have a listable set of all Reals.

Do you claim to be able to examine infinite lists?

If so, specify the infinite stream protocol of your choice.

Herc

[Arturo Magidin]

On Tuesday, October 30, 2012 12:15:26 AM UTC-5, Graham Cooper wrote:

> but you answered his Question beautifully, 'no University would sink
>
> to peer review any critique on Cantor's theory'.

When you use quotation marks, you are implying that you are making a literal quote.

Above, you use quotation marks, and attribute to me a statement that I did not make.

As such, you are guilty of libel and deliberate falsehood.

Not that this is a surprise, given your (utter lack of) character.

Kindly point your pathetic attempts at a rejoinder in a different direction; your mind is too feeble to deal with me.

--
Arturo Magidin

[Graham Cooper]

On Oct 30, 3:20 pm, Arturo Magidin <magi...@member.ams.org> wrote:
> On Tuesday, October 30, 2012 12:15:26 AM UTC-5, Graham Cooper wrote:
>

> > but you answered his Question beautifully, 'no University would sink
>
> > to peer review any critique on Cantor's theory'.
>

> You use quotation  marks to surround a sentence I never wrote and attribute it to me. As such, you are guilty of libel and deliberate falsehood.
>
> Not a surprise, given your (utter lack of) character and principles.
>
> Do kindly try to point your rejoinders in a different direction. You lack the intellectual acumen necessary to tangle with me.
>
> In other words: don't try engaging in a battle of wits; you are terminally unarmed for it.
>
> --
> Arturo Magidin


I think the Doc Madagan has been saving it up for me for quite some
while!!

OK I will leave this thread for the Academics to chew on this
computable list self defence mechanism..



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

1 IF YOU CHANGE 1 DIGIT OF THE DIAGONAL
THERE IS 0 EFFECT!

LIST OF ALL REALS
R1 0.00000..
R2 0.11211..
R3 0.22222..
R4 0.33333..
...

DIAGONAL 0.0123..
R1 0.*0000.. \
R2 0.1*211.. \
R3 0.22*22.. \
R4 0.333*3.. \
...

NEW-DIAGONAL 0.0223..
R1 0.*0000.. \
R2 0.11*11.. ^
R3 0.2*222.. v
R4 0.333*3.. \
...

NEW-PERMUTATION
R1 0.*0000.. \
R3 0.2*222.. v
R2 0.11*11.. ^
R4 0.333*3.. \

SAME *SET* OF REALS
R1 0.00000.. \
R3 0.22222.. \
R2 0.11211.. \
R4 0.33333.. \



ORIGINAL DIAGONAL = 0.0123...
CHANGED DIAGONAL = 0.0223..

No change to the underlying set, 1 changed diagonal digit at a time!

Herc

[Peter Webb]

Graham Cooper wrote:

> On Oct 30, 1:51 pm, "Peter Webb"
> <webbfamilyDIEspam...@optusnet.com.au> wrote:
> >
> > No.
> >
> > Do you claim to have a list of all Reals?
>
> I have a listable set of all Reals.
>

Great. Send it to me. I will tell you at least one missing Real.

> Do you claim to be able to examine infinite lists?
>

No, I claim to be able to examine rules which apply to infinite
sequences. For example, I can tell you the sequence of Reals 1, 2, 3,
... where the Real in position n is n interpreted as a Real number ...
does not contain 0.5, without explicitly examining every element in the
list.

> If so, specify the infinite stream protocol of your choice.
>
> Herc

Not needed.

Just tell us which Real appears in position n for all n.

I will happily find a Real not on the list, indicating that it is not a
list of all Reals as you claim.

[Peter Webb]

No, I want to know what I said which you think is "bollocks".

Why won't you tell us?

>
> >>>>> If you think you have a list of all Reals, post it here, and I
> >>will >>> happily prove its not a list of all Reals by finding a Real
> >>not on >>> the list for you.
> >>> >
> >>> > But you can do no such thing: the anti-diagonal is just not a
> real >>> > number.
> > > >
> >>> Of course it is a Real number. It is the limit of a Cauchy
> sequence. >>> You do know that convergent Cauchy sequences define
> unique Reals, >>> right?
> > > >
> >>> (I just bagged somebody for providing a proof of the anti-diagonal
> >>> was Real using Cauchy sequences, on the basis that no crank
> disputes >>> whether the anti-diagonal is Real. And then you prove me
> wrong by >>> raising exactly this isuue).
> > >
> > > Indeed, to be precise, you and co. excel in not even being wrong.
> > >
> >>> So, where is your list of all the Reals?
> > >
> > > The real numbers are a subset of the surreals. But I had said
> > > that already. Of course they are countable:
> >
> > Terrific. Provide a surjection from N to R.
>
> Ever heard of the surreals at all?

Of course. But we are discussing Reals, not Surreals.

> But you are correct in that I was
> imprecise: all infinite sets are *extended-countable*, i.e. all
> infinite sets biject with N*, the extended naturals.

The "extended Naturals, huh"? Do you mean the naturals extended with
transfinite elements represented with "non-standard numbers", as for
example in
http://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic ?

If so, your comment is meaningless, as these theories only add a
countably infinite number of new "numbers", so the cardinality of N
extended in this manner is identical to the cardinality of N alone.

If you mean something else by the "extended naturals", you should tell
us what it is.

>
> >> there just is no such thing as
> > > a number that is not a number: a number is all we can do with it
> > > seriously (you won't get this, but never mind).
> >
> > You are supposed to either say what you think I said which is
> > bollocks, or provide a counter-example to Cantor.
> >
> > As it is, you are just mumbling.
> >
> > > Well, maybe it's not dogma, it's just that you cannot read... But
> > > no, I am not optimistic about anybody's intellectual honesty
> > > around here: except maybe for Prof. Magidin, who at least shows
> > > some good math.
> >
> > The cranks should be pretty obvious. Clearly Madigin is not a crank,
> > but there are many others who are also not cranks. Anybody who
> > claims that Cantor's proof is false without identifying an error in
> > the proof is a crank.
>
> In fact, your denial is integral to your dogma.

In fact, you really need to find something I have said which you
consider false. Only then can I explain it to you.


>
> -LV

[LudovicoVan]

"Peter Webb" <webbfamily...@optusnet.com.au> wrote in message

news:k6nqvf$lc9$1...@news.albasani.net...

> LudovicoVan wrote:
>> "Peter Webb" <webbfamily...@optusnet.com.au> wrote in message
>> news:k6nk8v$lht$1...@news.albasani.net...
>> > LudovicoVan wrote:

<snip>

>> > > But you have just asked what "w" and "m" are supposed to mean,
>> > > remember? That's why I call you (and co., don't take it
>> > > personally as it isn't) parrots etc.: because you are just right,
>> > > whatever kind of bollocks you might say, in the end you are still
>> > > right. By dogma, and the guns.
>> >
>> > You actually have to identify what I said which you think is
>> > "bollocks".
>> >
>> > Somehow you forgot to say what it is.
>>
>> I have just said it... Unless you mean a full list of all the
>> bollocks ever stated in sci.math: of course, just forget it.
>
> No, I want to know what I said which you think is "bollocks".
>
> Why won't you tell us?

I won't say it again: shut your mouth up and learn to read and think. If
you like.

>> But you are correct in that I was
>> imprecise: all infinite sets are *extended-countable*, i.e. all
>> infinite sets biject with N*, the extended naturals.
>
> The "extended Naturals, huh"? Do you mean the naturals extended with
> transfinite elements represented with "non-standard numbers", as for
> example in
> http://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic ?
>
> If so, your comment is meaningless

The meaninglessness is all yours, as usual. N* := N u {oo}, and succ(oo) :=
oo, but I have posted links up-thread already: not to mention that you are a
regular around here, which is why you are still either a liar or cannot
read, or both.

> In fact, you really need to find something I have said which you
> consider false. Only then can I explain it to you.

Not false: bollocks, which here translates as not even wrong. Anyway, what
is it that you'd like to explain? Please go on, I'm all ears!

-LV

[Graham Cooper]

On Oct 30, 4:01 pm, "Peter Webb"

<webbfamilyDIEspam...@optusnet.com.au> wrote:
> Graham Cooper wrote:
> > On Oct 30, 1:51 pm, "Peter Webb"
> > <webbfamilyDIEspam...@optusnet.com.au> wrote:
>
> > > No.
>
> > > Do you claim to have a list of all Reals?
>
> > I have a listable set of all Reals.
>
> Great. Send it to me. I will tell you at least one missing Real.
>
> > Do you claim to be able to examine infinite lists?
>
> No, I claim to be able to examine rules which apply to infinite
> sequences. For example, I can tell you the sequence of Reals 1, 2, 3,
> ... where the Real in position n is n interpreted as a Real number ...
> does not contain 0.5, without explicitly examining every element in the
> list.
>
> > If so, specify the infinite stream protocol of your choice.
>
> > Herc
>
> Not needed.
>
> Just tell us which Real appears in position n for all n.
>

You want me to 'tell you' an infinite sequence?

I told you which digit appears in position 1,2,3,4,5 & 6
for reals in positions

R11 R12 R13 R14
R21 R22 R23 R24
R31 R32 R33 R34
R41 R42 R43 R44

Here are those digits again.

0.000000.. 0.110000.. 0.111010.. 0.111110.. ...
0.000111.. 0.101000.. 0.000010.. 0.001100.. ...
0.111000.. 0.010010.. 0.001010.. 0.101010.. ...
0.111100.. 0.001011.. 0.000011.. 0.111111.. ...
...

Should you require any further digits then specify which ones from
this enumerable_set_of_reals.

Since you have FAILED 5 times now to provide any hint of a missing
real from this enumerable_set_of_reals I think your claim is clearly
debunked.

If you wish to REFINE YOUR CLAIM that given an Enumeration or
specification thereof you have something to prove then do so.

Otherwise your claim to prove anything given any ENUMER-ABLE SET OF
REALS clearly appears to have failed on the set provided 5 times
already.

Herc

[Peter Webb]

Graham Cooper wrote:

> On Oct 30, 4:01 pm, "Peter Webb"
> <webbfamilyDIEspam...@optusnet.com.au> wrote:
> > Graham Cooper wrote:
> > > On Oct 30, 1:51 pm, "Peter Webb"
> > > <webbfamilyDIEspam...@optusnet.com.au> wrote:
> >
> > > > No.
> >
> > > > Do you claim to have a list of all Reals?
> >
> > > I have a listable set of all Reals.
> >
> > Great. Send it to me. I will tell you at least one missing Real.
> >
> > > Do you claim to be able to examine infinite lists?
> >
> > No, I claim to be able to examine rules which apply to infinite
> > sequences. For example, I can tell you the sequence of Reals 1, 2,
> > 3, ... where the Real in position n is n interpreted as a Real
> > number ... does not contain 0.5, without explicitly examining
> > every element in the list.
> >
> > > If so, specify the infinite stream protocol of your choice.
> >
> > > Herc
> >
> > Not needed.
> >
> > Just tell us which Real appears in position n for all n.
> >
>
>
> You want me to 'tell you' an infinite sequence?
>

No, I want you to tell me what Real appears at position n on your list.

You do claim to have a list of all Reals, don't you?

> I told you which digit appears in position 1,2,3,4,5 & 6
> for reals in positions
>
> R11 R12 R13 R14
> R21 R22 R23 R24
> R31 R32 R33 R34
> R41 R42 R43 R44
>
> Here are those digits again.
>
> 0.000000.. 0.110000.. 0.111010.. 0.111110.. ...
> 0.000111.. 0.101000.. 0.000010.. 0.001100.. ...
> 0.111000.. 0.010010.. 0.001010.. 0.101010.. ...
> 0.111100.. 0.001011.. 0.000011.. 0.111111.. ...
> ...
>
> Should you require any further digits then specify which ones from
> this enumerable_set_of_reals.
>

No, I want the Reals in your list.

You haven't even specified the first Real in your list. What is it?
What is the second Real on your list? What is the nth Real on your list?

> Since you have FAILED 5 times now to provide any hint of a missing
> real from this enumerable_set_of_reals I think your claim is clearly
> debunked.
>

You haven't specified the list yet.

You need to tell us what Real is in the first position, what Real is in
the second position, and so on. Otherwise how do we know what Reals are
already on your list?

> If you wish to REFINE YOUR CLAIM that given an Enumeration or
> specification thereof you have something to prove then do so.
>
> Otherwise your claim to prove anything given any ENUMER-ABLE SET OF
> REALS clearly appears to have failed on the set provided 5 times
> already.
>
> Herc

I made no such claim. To start off with, its nonsense.

You claim to have a list of all Reals. Produce it.

[Peter Webb]

LudovicoVan wrote:

> "Peter Webb" <webbfamily...@optusnet.com.au> wrote in message
> news:k6nqvf$lc9$1...@news.albasani.net...
> > LudovicoVan wrote:
> >>"Peter Webb" <webbfamily...@optusnet.com.au> wrote in message
> > > news:k6nk8v$lht$1...@news.albasani.net...
> >>> LudovicoVan wrote:
> <snip>
>
> >>> > But you have just asked what "w" and "m" are supposed to mean,
> >>> > remember? That's why I call you (and co., don't take it
> >>> > personally as it isn't) parrots etc.: because you are just
> right, >>> > whatever kind of bollocks you might say, in the end you
> are still >>> > right. By dogma, and the guns.
> > > >
> >>> You actually have to identify what I said which you think is
> >>> "bollocks".
> > > >
> >>> Somehow you forgot to say what it is.
> > >
> > > I have just said it... Unless you mean a full list of all the
> > > bollocks ever stated in sci.math: of course, just forget it.
> >
> > No, I want to know what I said which you think is "bollocks".
> >
> > Why won't you tell us?
>
> I won't say it again: shut your mouth up and learn to read and think.
> If you like.

Correction: you won't say it at all. Because I didn't say anything
which was bollocks.

>
> >> But you are correct in that I was
> > > imprecise: all infinite sets are *extended-countable*, i.e. all
> > > infinite sets biject with N*, the extended naturals.
> >
> > The "extended Naturals, huh"? Do you mean the naturals extended with
> > transfinite elements represented with "non-standard numbers", as for
> > example in
> > http://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic ?
> >
> > If so, your comment is meaningless
>
> The meaninglessness is all yours, as usual. N* := N u {oo},

Ummm, you mean that there is one element added to N to make N* ? And
you think this makes it uncoutable?

Wrong.

Clearly N* is countable. Here is a bijection with N.

N N*
1 oo
2 1
3 2
4 3
. .
. .

N* has exactly the same cardinality as N. Introducing N* instead of N
doesn't change a thing.


> succ(oo) := oo,

No. oo is not normally considered a number with a successor. Perhaps
you mean w, but succ(w) = w+1 so that's not correct either.

> but I have posted links up-thread already: not to
> mention that you are a regular around here, which is why you are
> still either a liar or cannot read, or both.
>

What lie did I tell. Or is this like your "bollocks" claim, which
turned out to be bullshit?

> > In fact, you really need to find something I have said which you
> > consider false. Only then can I explain it to you.
>
> Not false: bollocks, which here translates as not even wrong.

So, what was it?

> Anyway, what is it that you'd like to explain? Please go on, I'm all
> ears!
>

I am explaining you are wrong, and why.


> -LV

[Graham Cooper]

On Oct 30, 6:24 pm, "Peter Webb" wrote:
>
> You do claim to have a list of all Reals, don't you?

I have a listable set of all reals.

0.000000.. 0.110000.. 0.111010.. 0.111110.. ...
0.000111.. 0.101000.. 0.000010.. 0.001100.. ...
0.111000.. 0.010010.. 0.001010.. 0.101010.. ...
0.111100.. 0.001011.. 0.000011.. 0.111111.. ...
...


>

> You haven't specified the list yet.
>

AHA! Peter is on the ball!

No enumeration function! Just a good ole countable SET of all reals!

The order is arbitrary isn't it?

> > Otherwise your claim to prove anything given any ENUMER-ABLE SET OF
> > REALS clearly appears to have failed on the set provided 5 times
> > already.
>
> > Herc
>
> I made no such claim. To start off with, its nonsense.

Nonetheless it was the topic before you took over from Moeblee.


On Oct 30, 9:18 am, MoeBlee <modem...@gmail.com> wrote:
> I just proved that given any enumerable set S of denumerable binary
> sequences there is denumerable binary sequence not in S.


If you do not wish to apply any mathematical approach to the given
countable set of reals,

0.000000.. 0.110000.. 0.111010.. 0.111110.. ...
0.000111.. 0.101000.. 0.000010.. 0.001100.. ...
0.111000.. 0.010010.. 0.001010.. 0.101010.. ...
0.111100.. 0.001011.. 0.000011.. 0.111111.. ...
...

then I have nothing more to add about countable complete sets of
reals.

Herc

[Jesse F. Hughes]

"Peter Webb" <webbfamily...@optusnet.com.au> writes:

>> succ(oo) := oo,
>
> No. oo is not normally considered a number with a successor. Perhaps
> you mean w, but succ(w) = w+1 so that's not correct either.

The structure N* = N u {oo} and we define a function s on it (which we
call successor) so that s(n) = n + 1 and s(oo) = oo.

There's nothing wrong with the structure N*.

Now, of course, I don't see that N* has a damned thing to do with the
countability of the reals, but LV's description of it is fairly
standard, right down to his use of the term "successor" for the function
described.

--
"[I]n mathematics there are two types of integers: primes and
composites. [...] It's like how in the world there are mostly two
kinds of people: male and female [...] and lots of reasons for
interest in the differences." -- JSH on math/biology

[MoeBlee]

On Oct 30, 3:46 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On Oct 30, 9:18 am, MoeBlee <modem...@gmail.com> wrote:
>
> > I just proved that given any enumerable set S of denumerable binary
> > sequences there is denumerable binary sequence not in S.
>
> If you do not wish to apply any mathematical approach to the given
> countable set of reals,
>
>  0.000000.. 0.110000.. 0.111010.. 0.111110.. ...
>  0.000111.. 0.101000.. 0.000010.. 0.001100.. ...
>  0.111000.. 0.010010.. 0.001010.. 0.101010.. ...
>  0.111100.. 0.001011.. 0.000011.. 0.111111.. ...
>  ...
>
> then I have nothing more to add about countable complete sets of
> reals.

Good! Indeed, good that you have no more to say about it.

MoeBlee

[Richard Tobin]

In article <ef46bd8a-882b-4ed8...@uc4g2000pbc.googlegroups.com>,

Graham Cooper <graham...@gmail.com> wrote:

>I have a listable set of all Reals.

You are lying, of course.

But let's pretend that you aren't. If you find some way to convey it,
we will tell you some reals that are missing.

-- Richard

[Graham Cooper]

On Oct 31, 5:30 am, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
> In article <ef46bd8a-882b-4ed8-8aff-8c902f0d1...@uc4g2000pbc.googlegroups.com>,

> Graham Cooper  <grahamcoop...@gmail.com> wrote:
>
> >I have a listable set of all Reals.
>
> You are lying, of course.
>
> But let's pretend that you aren't.  If you find some way to convey it,
> we will tell you some reals that are missing.
>
> -- Richard
>

This is not difficult! This is a maths problem.


Given a list say:

0.11000..
0.00111..
0.11000..
0.00011..
...

YOU THEN SAY

0.1001.. is the DIAGAONAL

So BY EXTRAPOLATION TO ALL ROWS 0.0110.. is not equal to any row of
the LIST.

----------

But when faced with a COUNTABLE SET
(NOT AN ENUMERATION - AN ENUMER-ABLE SET)

0.000000.. 0.110000.. 0.111010.. 0.111110.. ...
0.000111.. 0.101000.. 0.000010.. 0.001100.. ...
0.111000.. 0.010010.. 0.001010.. 0.101010.. ...
0.111100.. 0.001011.. 0.000011.. 0.111111.. ...
...

Your tails go between your legs and instead of action you're all
words!

HENCE MY PROOF:

CLAIM: GIVEN AN ENUMER-ABLE SET OF REALS
YOU CAN'T PROVE SQUAT!

PROOF: HERE IS AN ENUMERABLE SET OF REALS!

COROLLARY: WATCH MOEBLEE RUN! RUN MOEBLEE! RUN!

Herc

[MoeBlee]

On Oct 30, 3:20 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:

> CLAIM:  GIVEN AN ENUMER-ABLE SET OF REALS
>           YOU CAN'T PROVE SQUAT!

For sake of simplicity, let's refer to the denumerable binary
sequences rather than to reals themselves. (We understand that there
is a 1-1 correspondence between the real interval [0 1] and the set of
denumerable binary sequences.)

By definition, an enumerable set S of reals is one for which there
exists an enumeration f of S.

So let f be an enumeration of S. Consider the denumerable binary
sequence g defined by g(n)=0 if f(n)(n)=1 and g(n)=1 if f(n)(n)=0. We
see easily that g is not in the range of f.

/

My claim is not that given an enumerable set S of denumeragble binary
sequences we can constructively produce a denumerable binary sequence
not in S.

Rather, the claim is that given an enumeration f of a set S of
denumerable binary sequences we can constructively produce a

denumerable binary sequence not in S.

Again, what we claim is that there is no enumeration of the set of
real numbers. And that is proven by proving, as we do (and we do it
constructively), that given an enumeration of a set S of denumerable
binary sequences there is a denumerable binary sequence not in the
range of the enumeration, i.e., not in S.

MoeBlee

[Graham Cooper]

On Oct 31, 6:41 am, MoeBlee <modem...@gmail.com> wrote:
> My claim is not that given an enumerable set S of denumeragble binary
> sequences we can constructively produce a denumerable binary sequence
> not in S.

> On Oct 30, 9:18 am, MoeBlee <modem...@gmail.com> wrote:

> > I just proved that given any enumerable set S of denumerable binary
> > sequences there is denumerable binary sequence not in S


F&~F |- Omega

From a contradiction, any bullshit follows.

>
> Again, what we claim is that there is no enumeration of the set of
> real numbers. And that is proven by proving, as we do (and we do it
> constructively), that given an enumeration of a set S of denumerable

Fine! We'll use a countable chart of ALL REAL NUMBERS instead!

0.000000.. 0.110000.. 0.111010.. 0.111110.. ...
0.000111.. 0.101000.. 0.000010.. 0.001100.. ...
0.111000.. 0.010010.. 0.001010.. 0.101010.. ...
0.111100.. 0.001011.. 0.000011.. 0.111111.. ...
...

Herc

[MoeBlee]

On Oct 30, 3:54 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On Oct 31, 6:41 am, MoeBlee <modem...@gmail.com> wrote:
>
> > My claim is not that given an enumerable set S of denumeragble binary
> > sequences we can constructively produce a denumerable binary sequence
> > not in S.
> > On Oct 30, 9:18 am, MoeBlee <modem...@gmail.com> wrote:
> > > I just proved that given any enumerable set S of denumerable binary
> > > sequences there is denumerable binary sequence not in S
>
> F&~F |- Omega
>
> From a contradiction, any bullshit follows.

Yes, so?

> > Again, what we claim is that there is no enumeration of the set of
> > real numbers. And that is proven by proving, as we do (and we do it
> > constructively), that given an enumeration of a set S of denumerable
>
> Fine!  We'll use a countable chart of ALL REAL NUMBERS instead!

So what? Whatever charts you make in your day room wherever you are,
the fact remains that there is no enumeration of the set of real
numbers.

MoeBlee

[Graham Cooper]

[~F]

> > On Oct 31, 6:41 am, MoeBlee <modem...@gmail.com> wrote:
> > > My claim is not that given an enumerable set S of denumeragble binary
> > > sequences we can constructively produce a denumerable binary sequence
> > > not in S.

[F]

> > > On Oct 30, 9:18 am, MoeBlee <modem...@gmail.com> wrote:
> > > > I just proved that given any enumerable set S of denumerable binary
> > > > sequences there is denumerable binary sequence not in S
>
> > F&~F |- Omega
>
> > From a contradiction, any bullshit follows.
>
> Yes, so?

> So what? Whatever charts you make in your day room wherever you are,
> the fact remains that there is no enumeration of the set of real
> numbers.
>
> MoeBlee

How would you know? You say 1 thing 1 day and the opposite the next!

An ENUMERATION is a BIGGER DATA STRUCTURE than an ENUMERABLE SET!

ENUMERATION = COUNTABLE_SET + INDEX

You haven't proven anything about ENUMERABLE (COUNTABLE SETS)

Herc

[Peter Webb]

That's not a given countable set of Reals.

Its not even a list of a few Reals, or even a single Real.

Lets start with the first Real on your list. Just the first one. I
don't care about the others for the time being. What is the first Real
on your list?

[Graham Cooper]

On Oct 31, 1:37 pm, "Peter Webb"

0! Top left corner!

Maybe you should DEFINE what you mean by countable set before asking
for one.

>
> > then I have nothing more to add about countable complete sets of
> > reals.
>
> > Herc

Herc

[Peter Webb]

Jesse F. Hughes wrote:

> "Peter Webb" <webbfamily...@optusnet.com.au> writes:
>
> >> succ(oo) := oo,
> >
> > No. oo is not normally considered a number with a successor. Perhaps
> > you mean w, but succ(w) = w+1 so that's not correct either.
>
> The structure N* = N u {oo} and we define a function s on it (which we
> call successor) so that s(n) = n + 1 and s(oo) = oo.
>
> There's nothing wrong with the structure N*.
>

Well, there are a couple of things wrong with it. It doesn't satisfy
Peano's axioms, which means we have lost much of the underlying
structure that we use in proofs over N.

The other problem is that in this treatment oo is undefined and hence
is treated as an Ur-element, which means we are also taking a small
step outside standard ZF.

> Now, of course, I don't see that N* has a damned thing to do with the
> countability of the reals, but LV's description of it is fairly
> standard, right down to his use of the term "successor" for the
> function described.

We are discussing Cardinality. N* clearly has exactly the same
cardinality as N, and its introduction was a complete red-herring.

[Peter Webb]

OK, what is the second Real on the list?

> Maybe you should DEFINE what you mean by countable set before asking
> for one.
>

You don't know what a countable set is?

For the purposes of this thread, you can treat it as a set which can be
bijected with N.

Which means you have to specify which values in R correspond to which
values in N.

You have already told us that the Real corresponding to n=1 is 0.

Now you can tell us the Real corresponding to n=2.

Eventually you will need to specify the Reals corresponding to all
natural numbers n, but I figure the values for n=1 (given) and n=2
(next) will be a good start.

[Graham Cooper]

On Oct 31, 2:34 pm, "Peter Webb"

This set can be bijected with N

1/1 1/2 1/3 1/4 ...
2/1 2/2 2/3 2/4 ...
3/1 3/2 3/3 3/4 ...
4/1 4/2 4/3 4/4 ...
...


similarly this set of reals CAN-BE bijected with N

0.000000.. 0.110000.. 0.111010.. 0.111110.. ...
0.000111.. 0.101000.. 0.000010.. 0.001100.. ...
0.111000.. 0.010010.. 0.001010.. 0.101010.. ...
0.111100.. 0.001011.. 0.000011.. 0.111111.. ...
...

Hence, you CANNOT prove a COUNTABLE SET OF REALS is incomplete.

PROOF: This is the 10th time asking.

In fact, I call this THE UNANSWERABLE QUESTION for a reason!


Herc

[Peter Webb]

> > are actively refusing to engage in the discussion in any reasonable

> > way, prefering instead to bow out as soon as specifics are
> > requested from you.
> >
> > That makes the entire thing nothing but an empty mental exercise on
> > your part, or a successful attempt at having other people waste
> > their time. For which I cannot find any reason to thank you. I do
> > hope, however, that you will be on your way and stop wasting
> > everyone's time, unless you actually want to start engaging in a
> > reasonable manner.
> >
> > --
> > Arturo Magidin
>
>
> but you answered his Question beautifully, 'no University would sink
> to peer review any critique on Cantor's theory'.
>
> This is synonymous with - 'if Cantor's theory was full of holes, we
> wouldn't know it!'
>

No its not.

We would know its full of holes because the proof is about 5 lines
long, is sufficently elementary to be accessible to an intelligent 12
year old, and in over 100 years nobody has raised a plausible objection
to it or produced a counter-example.

[Peter Webb]

Graham Cooper wrote:

> On Oct 30, 4:01 pm, "Peter Webb"
> <webbfamilyDIEspam...@optusnet.com.au> wrote:
> > Graham Cooper wrote:
> > > On Oct 30, 1:51 pm, "Peter Webb"
> > > <webbfamilyDIEspam...@optusnet.com.au> wrote:
> >
> > > > No.
> >
> > > > Do you claim to have a list of all Reals?
> >
> > > I have a listable set of all Reals.
> >
> > Great. Send it to me. I will tell you at least one missing Real.
> >
> > > Do you claim to be able to examine infinite lists?
> >
> > No, I claim to be able to examine rules which apply to infinite
> > sequences. For example, I can tell you the sequence of Reals 1, 2,
> > 3, ... where the Real in position n is n interpreted as a Real
> > number ... does not contain 0.5, without explicitly examining
> > every element in the list.
> >
> > > If so, specify the infinite stream protocol of your choice.
> >
> > > Herc
> >
> > Not needed.
> >
> > Just tell us which Real appears in position n for all n.
> >
>
>
> You want me to 'tell you' an infinite sequence?
>
> I told you which digit appears in position 1,2,3,4,5 & 6
> for reals in positions
>

Yes. But you haven't told us what the Real numbers in your list are.

What is the second Real in your list?

> R11 R12 R13 R14
> R21 R22 R23 R24
> R31 R32 R33 R34
> R41 R42 R43 R44
>
> Here are those digits again.
>
> 0.000000.. 0.110000.. 0.111010.. 0.111110.. ...
> 0.000111.. 0.101000.. 0.000010.. 0.001100.. ...
> 0.111000.. 0.010010.. 0.001010.. 0.101010.. ...
> 0.111100.. 0.001011.. 0.000011.. 0.111111.. ...
> ...
>
> Should you require any further digits then specify which ones from
> this enumerable_set_of_reals.

Its not an enumerable set of Reals. Its not even a set of Reals. It is
the first few decimal places of a few unspecified Real numbers.

What is the second Real on your list?

>
> Since you have FAILED 5 times now to provide any hint of a missing
> real from this enumerable_set_of_reals I think your claim is clearly
> debunked.

You haven't specified a single Real other than the first (0).

So you clearly haven't provided a list of all Reals. You have provided
(so far) a list with a single Real on it, 0 at posn 1.

Lets see if we can push this to two Reals ....

What is the second Real on your list?

>
> If you wish to REFINE YOUR CLAIM that given an Enumeration or
> specification thereof you have something to prove then do so.
>

Nope.

> Otherwise your claim to prove anything given any ENUMER-ABLE SET OF
> REALS clearly appears to have failed on the set provided 5 times
> already.
>
> Herc

You haven't specified an infinite list of Reals.

You haven't even specified the second Real on your list.

What is it?

[Peter Webb]

Yes.

>
> similarly this set of reals CAN-BE bijected with N
>
> 0.000000.. 0.110000.. 0.111010.. 0.111110.. ...
> 0.000111.. 0.101000.. 0.000010.. 0.001100.. ...
> 0.111000.. 0.010010.. 0.001010.. 0.101010.. ...
> 0.111100.. 0.001011.. 0.000011.. 0.111111.. ...
> ...

But this isn't a list of Reals.

It is a list of the first few decimal digits of a few unspecified Reals.

You need to produce a list of Reals.

Lets do this in steps.

What is the second Real on your list?

>
>
> Hence, you CANNOT prove a COUNTABLE SET OF REALS is incomplete.
>
> PROOF: This is the 10th time asking.
>
> In fact, I call this THE UNANSWERABLE QUESTION for a reason!
>
>
> Herc

And the second Real on your list is?

[Graham Cooper]

On Oct 31, 5:39 pm, "Peter Webb"

OK use this countable set of reals.

1/10  1/20  1/30  1/40 ...
2/11  2/21  2/31  2/41 ...
3/31  3/32  3/33  3/34 ...
4/17  4/27  4/37  4/47 ...
...

what's missing?

Herc

[Richard Tobin]

In article <6ab0167f-5eb4-44df...@pb2g2000pbc.googlegroups.com>,

Graham Cooper <graham...@gmail.com> wrote:

>But when faced with a COUNTABLE SET

If the set is countable, by definition it has a bijection with the
integers, and that bijection gives us a list.

If you want us to construct the list, you'd better give us a
constructive proof that the set is countable.

-- Richard

[Jesse F. Hughes]

"Peter Webb" <webbfamily...@optusnet.com.au> writes:

> Jesse F. Hughes wrote:
>
>> "Peter Webb" <webbfamily...@optusnet.com.au> writes:
>>
>> >> succ(oo) := oo,
>> >
>> > No. oo is not normally considered a number with a successor. Perhaps
>> > you mean w, but succ(w) = w+1 so that's not correct either.
>>
>> The structure N* = N u {oo} and we define a function s on it (which we
>> call successor) so that s(n) = n + 1 and s(oo) = oo.
>>
>> There's nothing wrong with the structure N*.
>>
>
> Well, there are a couple of things wrong with it. It doesn't satisfy
> Peano's axioms, which means we have lost much of the underlying
> structure that we use in proofs over N.

Of *course* it doesn't satisfy those axioms. No one said it did.

Nonetheless, there's nothing wrong with it.

> The other problem is that in this treatment oo is undefined and hence
> is treated as an Ur-element, which means we are also taking a small
> step outside standard ZF.

It doesn't matter what the element oo is defined as in ZFC. That's a
triviality. Pick any set not in N and define oo as that set.

I first studied this structure in category theory, where it is the final
coalgebra for the successor functor. A category theorist really does
not give a damn which set you pick to be oo.

>
>> Now, of course, I don't see that N* has a damned thing to do with the
>> countability of the reals, but LV's description of it is fairly
>> standard, right down to his use of the term "successor" for the
>> function described.
>
> We are discussing Cardinality. N* clearly has exactly the same
> cardinality as N, and its introduction was a complete red-herring.

Probably so, but your response regarding N* was also misplaced. It's a
simple mathematical structure.

--
Jesse F. Hughes

"/Monster Ballads/ is packed with pure hits from the artists who taught
us how to love." -- As seen on TV

[Graham Cooper]

On Oct 31, 8:00 pm, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
> In article <6ab0167f-5eb4-44df-96aa-e0066f034...@pb2g2000pbc.googlegroups.com>,

> Graham Cooper  <grahamcoop...@gmail.com> wrote:
>
> >But when faced with a COUNTABLE SET
>
> If the set is countable, by definition it has a bijection with the
> integers, and that bijection gives us a list.

Right! In SQL you might have the relation/set:

REAL
--------
0.343434...
arctan(1)*4/10
utm(143)
(0+1)/2
...

and a DIFFERENT RELATION might have an INDEX

POS REAL
--------------
4 0.343434...
2 arctan(1)*4/10
9 utm(143)
77 (0+1)/2
...

And THAT INFINITE INDEX GIVES YOU A LIST!

If you WINZIP a SET (order unimportant) you get a 99KB file!
If you WINZIP a LIST (order specified) you get a 100KB file!

(hypothetical illustration)

>
> If you want us to construct the list, you'd better give us a
> constructive proof that the set is countable.
>

R11 R12 R13 ...
R21 R22 R23 ...
R31 R32 R33 ...
...

Proof:
Each R index has 2 natnum indexes that can all be tallied trivially.

This is a COUNT-ABLE SET.

If you want a COUNTED SET that is a bigger data structure with MORE
INFORMATION IN IT!

*information that you confuse with having relevance to the unordered
set*

Herc

[Graham Cooper]

> If you WINZIP a SET (order unimportant) you get a 99KB file!
> If you WINZIP a LIST (order specified) you get a 100KB file!
>
> (hypothetical illustration)

This is IMPORTANT and TRIVIAL to computer scientists but you cannot
even comment on it in 10 years!

Maybe not WINZIP.

Assume an ENCRYPTION SYSTEM EXISTS that compresses SETS.


S-COMPRESS( <1,4,6,3,9,10,22,53> )= 3434

S-UNCOMPRESS(3434) = <10,22,53,9,1,4,3,6>


There is MORE ROOM FOR COMPRESSION if you don't have to worry about
the order of the elements!

The ENTROPY of a LIST > ENTROPY of a SET

HERE IS A LOWER ENTROPY COUNTABLE SET OF REALS

0.000000.. 0.110000.. 0.111010.. 0.111110.. ...
0.000111.. 0.101000.. 0.000010.. 0.001100.. ...
0.111000.. 0.010010.. 0.001010.. 0.101010.. ...
0.111100.. 0.001011.. 0.000011.. 0.111111.. ...
...

If you S-COMPRESS that SET - it might decompress in one order or
another order!

Meaning it has MORE COMPRESSION OPTIONS. (technically 1 bit smaller
zip file!)

0 = transverse sort LR
1 = transverse sort RL

You are using the REPRESENTATION IMPLIED ORDER of SETS because PAPER
is 2 DIMENSIONAL and you can't visualise what a set is!

THIS IS A SET {
1
4
3
6
}

not a list, if you DESIGN A FUNCTION AROUND THE ORDER you are
INVENTING infinite streams that are just noise.


Herc

[Richard Tobin]

In article <623781e5-4542-4a3d...@b4g2000pby.googlegroups.com>,

Graham Cooper <graham...@gmail.com> wrote:

>> If you want us to construct the list, you'd better give us a
>> constructive proof that the set is countable.

>R11 R12 R13 ...
>R21 R22 R23 ...
>R31 R32 R33 ...
>...
>
>Proof:
>Each R index has 2 natnum indexes that can all be tallied trivially.
>
>This is a COUNT-ABLE SET.

So consider the list

R11
R12
R21
R13
R22
R31
...

and use the diagonal to get a real that is different from all the Rxy.
Your set is not complete. QED.

-- Richard

[Graham Cooper]

> HERE IS A LOWER ENTROPY COUNTABLE SET OF REALS
>
>   0.000000.. 0.110000.. 0.111010.. 0.111110.. ...
>   0.000111.. 0.101000.. 0.000010.. 0.001100.. ...
>   0.111000.. 0.010010.. 0.001010.. 0.101010.. ...
>   0.111100.. 0.001011.. 0.000011.. 0.111111.. ...
>   ...
>

Look you can dodge every question and claim

CANTORS PROOF HAS STOOD UP TO EVERY ATTACK

but you've got your head in the sand!

Cantors proof fails every attack, logic, data structure, formal proof,
resolution, definitions, semantics, probability, .. , ... ,...

...

THESE NEVER GET ANSWERED!


tinyurl.com/blueprints-questions

*****************************************************
*****************************************************
Q4
How can there be uncountable many GODEL NUMBERS like this?

20130415
a01(0,1)
MIDPOINT(0,1)
A CHOICE FUNCTION

*****************************************************
*****************************************************
Q5
Which 1 of these does not hold?

a) N <-BIJECTS-> GODEL NUMBERS
b) GODEL NUMBERS <-BIJECT-> FUNCTIONS
c) FUNCTIONS <-BIJECT-> CHIOCE FUNCTIONS
d) CHOICE FUNCTIONS <-BIJECT-> SETS
e) |SETS| > |N|

????


*****************************************************
*****************************************************
Q6
Does this Anti-Diagonal Method produce any unique digit segment not
listed?

AD METHOD
Choose the number 0.a_1a_2a_3...., where a_i = 1 if the i-th
number in your list had zero in its i-position, a_i = 0 otherwise.

LIST
R1= < <314><15><926><535><8979><323> ... >
R2= < <27><18281828><459045><235360> ... >
R3= < <333><333><333><333><333><333> ... >
R4= < <888888888888888888888><8><88> ... >
R5= < <0123456789><0123456789><01234 ... >
R6= < <1><414><21356><2373095><0488> ... >
....


you just place arbitrary FINITE SEGMENTS (same real model)

and the AD CANNOT make a NEW STRING!!! Finite Or Infinite!


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

And my 25TH DISPROOF I made this week!

1/ CHANGE 1 DIGIT OF THE DIAGONAL = 0 EFFECT!
2/ CHANGE 1 DIGIT OF THE DIAGONAL AT A TIME = 0 EFFECT!

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

1 IF YOU CHANGE 1 DIGIT OF THE DIAGONAL
THERE IS 0 EFFECT!

LIST OF ALL REALS
R1 0.00000..
R2 0.11211..
R3 0.22222..
R4 0.33333..
...

DIAGONAL 0.0123..
R1 0.*0000.. \
R2 0.1*211.. \
R3 0.22*22.. \
R4 0.333*3.. \
...

NEW-DIAGONAL 0.0223..
R1 0.*0000.. \
R2 0.11*11.. ^
R3 0.2*222.. v
R4 0.333*3.. \
...

NEW-PERMUTATION
R1 0.*0000.. \
R3 0.2*222.. v
R2 0.11*11.. ^
R4 0.333*3.. \
...

SAME *SET* OF REALS
R1 0.00000.. \
R3 0.22222.. \
R2 0.11211.. \
R4 0.33333.. \
...

ORIGINAL DIAGONAL = 0.0123...
CHANGED DIAGONAL = 0.0223..



--------

Now all pretend to roll your eyes like

HE JUST DOESN'T GET SUPER-INFINITY!

Anyway I'm finished with Cantor, if you don't address Hard
Contradictions nothing I can do.

PROLOG MODUS PONENS is more interesting!



G. Cooper (BInfTech)

--
if( if(t(S),f(R)) , if(t(R),f(S)) ).
if the sun's out then it's not raining
ergo
if it's raining then the sun's not out

[Graham Cooper]

On Oct 31, 9:45 pm, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
> In article <623781e5-4542-4a3d-98d9-a299ec236...@b4g2000pby.googlegroups.com>,

> Graham Cooper  <grahamcoop...@gmail.com> wrote:
>
> >> If you want us to construct the list, you'd better give us a
> >> constructive proof that the set is countable.
> >R11 R12 R13 ...
> >R21 R22 R23 ...
> >R31 R32 R33 ...
> >...
>
> >Proof:
> >Each R index has 2 natnum indexes that can all be tallied trivially.
>
> >This is a COUNT-ABLE SET.
>
> So consider the list
>
> R11
> R12
> R21
> R13
> R22
> R31
> ...
>
> and use the diagonal to get a real that is different from all the Rxy.
> Your set is not complete.  QED.
>

by ... 'use the diagonal' 0.a1a2a3...

a1 = 0 if the first real on YOUR list is 1, otherwise 1.
a2 = 0 if the first real on YOUR list is 1, otherwise 0.
...

proves nothing. chose a different enumeration f, and the values
arbitrarily change depending on your particular f.

Herc

[Shmuel Metz]

In <877gq8u...@phiwumbda.org>, on 10/30/2012

at 08:28 AM, "Jesse F. Hughes" <je...@phiwumbda.org> said:

>Now, of course, I don't see that N* has a damned thing to do with
>the countability of the reals, but LV's description of it is
>fairly standard,

No. "N* = N u {N}" is fairly standard; "N* = N u {oo}" is nonsense.

>right down to his use of the term "successor" for the function
>described.

"s(oo) = oo" is not standard; what is standard is "s(n) = n u {n}" for
all ordinals[1], not just for finite ordinals.

[1] Or something similar if you're not using the standard
definition of the ordinals.

--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>

Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org

[Shmuel Metz]

In <k6ns3h$ulk$1...@dont-email.me>, on 10/30/2012
at 06:28 AM, "LudovicoVan" <ju...@diegidio.name> said:

>The meaninglessness is all yours, as usual. N* := N u {oo},

Sorry, tonto, but that has the same cardinality as N. Of course, you
haven't defined the symbol "oo", but presumably you meant N u {N} and
are too dumb to understand the difference.

[Shmuel Metz]

In <k6nkql$1gh$1...@dont-email.me>, on 10/30/2012
at 04:24 AM, "LudovicoVan" <ju...@diegidio.name> said:

>Ever heard of the surreals at all? But you are correct in that I was
> imprecise: all infinite sets are *extended-countable*, i.e. all
>infinite sets biject with N*, the extended naturals.

If by N* you mean N \union {N} (N+1), that's demonstrably wrong. If by
N* you are referring to NSA, that has no relevance to Cantor's
results, nor do surreals.

[Shmuel Metz]

In <k6nj52$oq1$1...@dont-email.me>, on 10/30/2012
at 03:56 AM, "LudovicoVan" <ju...@diegidio.name> said:

>But no, I am
>not optimistic about anybody's intellectual honesty around here

PKB.

[Graham Cooper]

On Oct 31, 8:00 pm, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
> In article

> Graham Cooper  <grahamcoop...@gmail.com> wrote:
>
> >But when faced with a COUNTABLE SET
>
> If the set is countable, by definition it has a bijection with the
> integers, and that bijection gives us a list.
>

You guys should think about

CHANGING THE DIAGONAL from a different perspective.

What can you CHANGE THE DIAGONAL into by re-ordering an infinite List?


Herc

[Jesse F. Hughes]

Shmuel (Seymour J.) Metz <spam...@library.lspace.org.invalid> writes:

> In <877gq8u...@phiwumbda.org>, on 10/30/2012
> at 08:28 AM, "Jesse F. Hughes" <je...@phiwumbda.org> said:
>
>>Now, of course, I don't see that N* has a damned thing to do with
>>the countability of the reals, but LV's description of it is
>>fairly standard,
>
> No. "N* = N u {N}" is fairly standard; "N* = N u {oo}" is nonsense.

Context matters.

In category theory, we have no qualms with writing N u {oo}, because
*what set oo actually is* matters not one whit -- as long as it's not an
element of N.

>>right down to his use of the term "successor" for the function
>>described.
>
> "s(oo) = oo" is not standard; what is standard is "s(n) = n u {n}" for
> all ordinals[1], not just for finite ordinals.

For the structure N*, s(oo) = oo is perfectly standard.

You know, I'm not making this up. You'll find N* described essentially
as above[1] in, say, my doctoral thesis, among other locations.

N is interesting because it's the initial algebra for the successor
functor.

N* is interesting because it's the final coalgebra for the same functor.

> [1] Or something similar if you're not using the standard
> definition of the ordinals.

Footnotes:
[1] Not quite as above, since the predecessor function is more basic to
a coalgebraic picture than successor, but this is a mere detail.

--
"Tempted and tried we're oft made to wonder
Why it should be thus all the day long
When there are others living about us
Never molested though in the wrong." -- Bad Livers, "Farther Along"

[Jesse F. Hughes]

Shmuel (Seymour J.) Metz <spam...@library.lspace.org.invalid> writes:

> In <k6ns3h$ulk$1...@dont-email.me>, on 10/30/2012
> at 06:28 AM, "LudovicoVan" <ju...@diegidio.name> said:
>
>>The meaninglessness is all yours, as usual. N* := N u {oo},
>
> Sorry, tonto, but that has the same cardinality as N. Of course, you
> haven't defined the symbol "oo", but presumably you meant N u {N} and
> are too dumb to understand the difference.

You're of course right that |N*| = |N| and I won't speak to whether or
not LudovicoVan has any point in mentioning N* here.

But your arrogance in dismissing a fairly standard use of the oo symbol
is misplaced.

It's not a notation you may have encountered, but it's certainly not
LudovicoVan's invention.

--
Jesse F. Hughes
"Mathematicians don't fit in with a consistent view, unless you accept
that to a strangely large extent they are acting under the influence
of something very powerful, dark, and negative." -- James S. Harris

[Richard Tobin]

In article <a4ec801c-15d5-4d4d...@jl13g2000pbb.googlegroups.com>,

Graham Cooper <graham...@gmail.com> wrote:

>by ... 'use the diagonal' 0.a1a2a3...
>
>a1 = 0 if the first real on YOUR list is 1, otherwise 1.
>a2 = 0 if the first real on YOUR list is 1, otherwise 0.
>...
>
>proves nothing. chose a different enumeration f, and the values
>arbitrarily change depending on your particular f.

Yes, and *every one of them* gives you a number that isn't in the set.
There isn't just one number missing from your set of reals, almost
all of them are missing!

-- Richard

[Graham Cooper]

On Nov 1, 12:35 am, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
> In article <a4ec801c-15d5-4d4d-aa02-e30552568...@jl13g2000pbb.googlegroups.com>,

> Graham Cooper  <grahamcoop...@gmail.com> wrote:
>
> >by ... 'use the diagonal' 0.a1a2a3...
>
> >a1 = 0 if the first real on YOUR list is 1, otherwise 1.
> >a2 = 0 if the first real on YOUR list is 1, otherwise 0.
> >...
>
> >proves nothing.  chose a different enumeration f,  and the values
> >arbitrarily change depending on your particular f.
>
> Yes, and *every one of them* gives you a number that isn't in the set.
> There isn't just one number missing from your set of reals, almost
> all of them are missing!
>
> -- Richard

That contradicts hard facts if you work out Q6, or simply try it with
the 16 reals given and the 2 possible enumerations!!

Your missing real segmented 0.a1a2a3a4 a5a6a7 a8a9 a10 a11 a12a13a14 .

has neither a new digit segment of any length anywhere or a new
sequence of those sements anywhere!

NO NEW DIGIT SEQUENCE - IF YOU DO THE GROUND WORK!

[Graham Cooper]

On Nov 1, 8:38 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> "LudovicoVan" <ju...@diegidio.name> writes:
> > "Jesse F. Hughes" <je...@phiwumbda.org> wrote in message
> >news:87mwz2q...@phiwumbda.org...
> >> "LudovicoVan" <ju...@diegidio.name> writes:
> >>> "Jesse F. Hughes" <je...@phiwumbda.org> wrote in message
> >>>news:87txtaq...@phiwumbda.org...
> > <snip>
>
> >>>> I never said you thought that set theory was a root of evil, but, near
> >>>> as I can figger, you said that it was a symptom of a lying culture which
> >>>> lies just 'cause it can.
>
> >>> You could say because it wants, not because it can: anyway, you rephrase
> >>> it
> >>> as a 13 year old would, but yes, let's say you almost got it, son, though
> >>> not quite.  OTOH, I am pretty sure you could do better, if only you could
> >>> be
> >>> any little more honest.
>
> >> Sorry, I've studied too much set theory to be honest, I guess.
>
> > Set theory is not responsible for your honesty, big boy.
>
> >>>>  In an honest culture, we would all admit that
> >>>> set theory is a plain falsehood.
>
> >>> No, I have never said that: there are indeed things that I find are
> >>> patently
> >>> wrong, the standard theory of cardinality being one of them, but that
> >>> does
> >>> not mean I'd discard the baby too.  Not to mention that we all have
> >>> "search"
> >>> strategies, and a world of fools and criminals means just do not expect
> >>> that
> >>> I be a gentlemen.  It's a war, mate.
>
> >> See, here's the weird thing.  The theorems of ZFC can be confirmed by
> >> anyone.
>
> > Apart from the fact that proof by consensus is not a valid argument, that's
> > not even true.
>
> Who the fuck said anything about proof by consensus?
>
> And, surely, if the argument is invalid, perhaps you can point out the
> invalid step.
>
> For that, of course, we should be clear on what argument we are
> discussing.  There are various arguments that go by the name "Cantor's
> theorem".  The easiest to analyze, of course, is the proof that, for all
> sets X, |X| < |PX|.  Are you prepared to show me how that argument is
> invalid?  If so, we can discuss it.
>
> But I'm not going on some vague, meandering and conspiracy-tinged
> rantfest.  If you want to claim that the proof is invalid, you have to
> show me the step which is invalid.
>
> >>  At best, you can complain that either the axioms are false
> >> (I'm sure I don't know what that would mean)
>
> > At best?  Anyway, try and ask Aatu about that: to you he might even
> > reply.
>
> >> or that the logic we use is
> >> mistaken (and that's a mighty hard sell).  But it is undeniable that ZFC
> >> proves for all X, |X| < |PX|.  Anyone can confirm that the proof is a
> >> valid argument.
>
> > Again, proof by consensus is not a proof, but that is not even true: as you
> > should know even too well, not anyone would confirm, and this is not just
> > the cranks.
>
> And, again, to say that "anyone can confirm the validity" is not proof
> by consensus, you tedious twat.
>
> And, as far as non-cranks "not confirming" the validity, well, that is
> the subject of this discussion.  Can you name a single, reputable source
> that disputes whether ZFC proves Cantor's theorem?  (NOTE: I'm talking
> about a particular formal theory here, so the various mathematicians who
> gave philosophical disputes over Cantor's informal argument are
> irrelevant to our purposes here, unless those disputes can explicitly
> show an invalid step in this very simple proof.)
>


Can you state explicitly what it proves?

I don't see how MODUS PONENS might make this deduction.

LHS->RHS & LHS -> RHS

where RHS = "X > size({1,2,3...})"

nor how the enumeration of a set and it's index inclusion or not has
anything to do what's in the superset.

[Graham Cooper]

Let's construct the missing set.

Here's is my purported P(N) 1st 20 subsets.


f(1) = { 2 3 4 5 6 7 10 12 14 16 17 18 20 }
f(2) = { 1 2 3 4 6 7 9 11 12 13 14 16 18 }
f(3) = { 1 4 5 6 10 12 13 18 }
f(4) = { 1 2 3 9 10 14 15 18 19 }
f(5) = { 4 5 6 9 14 }
f(6) = { 2 3 5 7 8 10 12 19 20 }
f(7) = { 1 2 5 6 7 8 14 }
f(8) = { 1 2 3 4 6 7 10 11 12 13 }
f(9) = { 2 6 8 10 15 }
f(10) = { 5 7 15 17 19 20 }
f(11) = { 1 2 7 8 10 12 19 }
f(12) = { 5 6 8 13 19 }
f(13) = { 1 2 4 5 8 11 }
f(14) = { 1 3 11 15 20 }
f(15) = { 2 4 5 6 9 12 }
f(16) = { 1 3 4 8 12 14 15 19 }
f(17) = { 9 13 15 }
f(18) = { 2 3 4 5 7 9 13 17 }
f(19) = { 1 2 3 5 7 12 13 19 }
f(20) = { 1 2 4 5 8 20 }


HERE'S THE MISSING SET! { x | ~x e f(n) }

{ 1 3 4 6 8 9 10 11 12 13 14 15 16 17 18 }

BWAHAHAHAHAHA!!

MISSING!???

We don't need to show a flaw in stupidity!

Herc

[Peter Webb]

Yes.

>
> similarly this set of reals CAN-BE bijected with N
>
> 0.000000.. 0.110000.. 0.111010.. 0.111110.. ...
> 0.000111.. 0.101000.. 0.000010.. 0.001100.. ...
> 0.111000.. 0.010010.. 0.001010.. 0.101010.. ...
> 0.111100.. 0.001011.. 0.000011.. 0.111111.. ...
> ...
>

Its not a set of Reals. It is set of the 7 leading digits of 16
different Reals.

We know the first Real is 0, because you told us.

What is the value of the second Real?

>
> Hence, you CANNOT prove a COUNTABLE SET OF REALS is incomplete.
>
> PROOF: This is the 10th time asking.
>
> In fact, I call this THE UNANSWERABLE QUESTION for a reason!
>
>
> Herc

And the second Real on your list is what, exactly?

[Hercules ofZeus]

On Nov 1, 5:41 pm, "Peter Webb" <webbfamilyDIEspam...@optusnet.com.au>

[Hercules ofZeus]

On Nov 1, 8:38 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> > irrelevant to our purposes here, unless those disputes can explicitly
> > show an invalid step in this very simple proof.)
>

it DOES NOT HOLD UP TO INDUCTION!!!!!


Examine the lower level Decimal Proof Technique...


ROW1_1=/=AD_1 -> AD=/=ROW1
ROW2_2=/=AD_2 -> AD=/=ROW2
ROW3_3=/=AD_3 -> AD=/=ROW3
...
AND SO ON!


THIS IS THE INDUCTIVE STEP

P(n) -> P(S(n))


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

But P(1) DOESNT HOLD BY ITSELF!!!

Proviging a single digit of the diagonal is WORTHLESS, even for ROW 1!

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

If you claim this is a proof in PREDICATE CALCULUS
then the only Proof Method in PREDICATE CALCULUS
about infinite sets is INDUCTION!

p(1) & p(n)->p(s(n))
-> ALL(n) p(n)

[Peter Webb]

Jesse F. Hughes wrote:

> "Peter Webb" <webbfamily...@optusnet.com.au> writes:
>
> > Jesse F. Hughes wrote:
> >
> >> "Peter Webb" <webbfamily...@optusnet.com.au> writes:
> >>
> >> >> succ(oo) := oo,
> >> >
> >> > No. oo is not normally considered a number with a successor.
> Perhaps >> > you mean w, but succ(w) = w+1 so that's not correct
> either. >>
> >> The structure N* = N u {oo} and we define a function s on it
> (which we >> call successor) so that s(n) = n + 1 and s(oo) = oo.
> >>
> >> There's nothing wrong with the structure N*.
> >>
> >
> > Well, there are a couple of things wrong with it. It doesn't satisfy
> > Peano's axioms, which means we have lost much of the underlying
> > structure that we use in proofs over N.
>

> Of course it doesn't satisfy those axioms. No one said it did.

>
> Nonetheless, there's nothing wrong with it.

Other than it doesn't satisfy Peano's axioms, which means we have lost

much of the underlying structure that we use in proofs over N.


>

> > The other problem is that in this treatment oo is undefined and
> > hence is treated as an Ur-element, which means we are also taking a
> > small step outside standard ZF.
>
> It doesn't matter what the element oo is defined as in ZFC. That's a
> triviality. Pick any set not in N and define oo as that set.
>

You don't think that equating the ordered pair {2,{2,3}} with infinity
might not be a problem if you want to use N* to develop function theory?

Or how about oo = {N} ? I guess if you want to drop the von Neumann
construction of numbers with ordinals well ordered by set inclusion
this isn't a problem, but again its a bit of a compromise.

> I first studied this structure in category theory, where it is the
> final coalgebra for the successor functor. A category theorist
> really does not give a damn which set you pick to be oo.
> >
> >> Now, of course, I don't see that N* has a damned thing to do with
> the >> countability of the reals, but LV's description of it is fairly
> >> standard, right down to his use of the term "successor" for the
> >> function described.
> >
> > We are discussing Cardinality. N* clearly has exactly the same
> > cardinality as N, and its introduction was a complete red-herring.
>
> Probably so, but your response regarding N* was also misplaced. It's
> a simple mathematical structure.

Its a completly useless structure for this question. It adds a single
value to N (and hence does not change its cardinality) but in the
process provides a substantially weaker basis for analysis and proof.

[Jesse F. Hughes]

"Peter Webb" <webbfamily...@optusnet.com.au> writes:

> Jesse F. Hughes wrote:
>
>> "Peter Webb" <webbfamily...@optusnet.com.au> writes:
>>
>> > Jesse F. Hughes wrote:
>> >
>> >> "Peter Webb" <webbfamily...@optusnet.com.au> writes:
>> >>
>> >> >> succ(oo) := oo,
>> >> >
>> >> > No. oo is not normally considered a number with a successor.
>> Perhaps >> > you mean w, but succ(w) = w+1 so that's not correct
>> either. >>
>> >> The structure N* = N u {oo} and we define a function s on it
>> (which we >> call successor) so that s(n) = n + 1 and s(oo) = oo.
>> >>
>> >> There's nothing wrong with the structure N*.
>> >>
>> >
>> > Well, there are a couple of things wrong with it. It doesn't satisfy
>> > Peano's axioms, which means we have lost much of the underlying
>> > structure that we use in proofs over N.
>>
>> Of course it doesn't satisfy those axioms. No one said it did.
>>
>> Nonetheless, there's nothing wrong with it.
>
>
> Other than it doesn't satisfy Peano's axioms, which means we have lost
> much of the underlying structure that we use in proofs over N.

It isn't N, you silly person.

Look, you might as well complain that there's something wrong with Q as
well, since it doesn't satisfy Peano's axioms either.

There is not a single thing "wrong" or suspect with N*. Of course, if
LudovicoVan decides to use induction on N*, then he will be using an
invalid step. But you keep pretending that this shows there's something
dubious about N*, and this just ain't so.

>
>
>>
>> > The other problem is that in this treatment oo is undefined and
>> > hence is treated as an Ur-element, which means we are also taking a
>> > small step outside standard ZF.
>>
>> It doesn't matter what the element oo is defined as in ZFC. That's a
>> triviality. Pick any set not in N and define oo as that set.
>>
>
> You don't think that equating the ordered pair {2,{2,3}} with infinity
> might not be a problem if you want to use N* to develop function
> theory?

You're confused. oo in this context is simply a term. Sure, it is
suggestive of infinity, but the definition of N* tells you what we mean
by oo.

> Or how about oo = {N} ? I guess if you want to drop the von Neumann
> construction of numbers with ordinals well ordered by set inclusion
> this isn't a problem, but again its a bit of a compromise.

Huh?

There's no "compromise" here. The structure N* comes with a particular
function s:N* -> N* satisfying a particular set of equations. In this
context, we don't give a damn about von Neumann's ordinals or any other
tedious details of particular constructions of N. All of this is beside
the point.

The behavior of the set N* under the function s is all that is really
relevant here.

The same is true, by the way, of N. From a categorical perspective, who
cares what the elements of N are? All that matters is <N,s> satisfies
the defining principle of a natural numbers object (technically, that it
is an initial algebra for the successor functor). Any countably
infinite set will do for this purpose.

>> I first studied this structure in category theory, where it is the
>> final coalgebra for the successor functor. A category theorist
>> really does not give a damn which set you pick to be oo.
>> >
>> >> Now, of course, I don't see that N* has a damned thing to do with
>> the >> countability of the reals, but LV's description of it is fairly
>> >> standard, right down to his use of the term "successor" for the
>> >> function described.
>> >
>> > We are discussing Cardinality. N* clearly has exactly the same
>> > cardinality as N, and its introduction was a complete red-herring.
>>
>> Probably so, but your response regarding N* was also misplaced. It's
>> a simple mathematical structure.
>
> Its a completly useless structure for this question. It adds a single
> value to N (and hence does not change its cardinality) but in the
> process provides a substantially weaker basis for analysis and proof.
>

I'm not defending any argument LV is making with N*. I'm pointing out
that your view that something is "wrong" with N* is a simple symptom of
mathematical immaturity.

I'm frankly surprised by it.
--
"Many argue that its programmers have turned out shoddy programs, but
[their] objective is to make profit, not superlative programs per
se. By the profit criterion, Microsoft has been one of the greatest
companies in the history of this country." -- ADTI defends Microsoft

[Jesse F. Hughes]

Graham Cooper <graham...@gmail.com> writes:

> Can you state explicitly what it proves?
>
> I don't see how MODUS PONENS might make this deduction.
>
> LHS->RHS & LHS -> RHS
>
> where RHS = "X > size({1,2,3...})"
>
> nor how the enumeration of a set and it's index inclusion or not has
> anything to do what's in the superset.

Sorry, Graham, I was hoping to have a conversation with someone a bit
more coherent today.

Some other time, perhaps.

--
Jesse F. Hughes
"I will admit I can get giddy over these forays into ideas at the
extreme edge. Being wrong can just add to the fun, oddly enough."
--James S. Harris just wants to have fun.

[LudovicoVan]

"Peter Webb" <webbfamily...@optusnet.com.au> wrote in message
news:k6nqvf$lc9$1...@news.albasani.net...

> If you mean something else by the "extended naturals", you should tell
> us what it is.

Jesse is of course already explaining this much better than I can, anyway be
reassured that the qualifier "extended" is pretty customary: here infinities
are just the projective or the affine infinites (as the result of division
by zero) and, maybe you haven't though of it, but floating-point arithmetic
is an example of an extended system already (i.e. extended pseudo-reals, as
they are in fact a subset of the rationals).

-LV

[Jesse F. Hughes]

Do you plan on responding to my suggestion that you point out the
invalid step in Cantor's theorem?

You say that Cantor's theorem is invalid, right? (You are using the
term "invalid" in its customary sense, I assume.)

This means that there is some step in the argument which is invalid,
right?

Would you like to step through the argument with me and show me which
step is invalid?

If so, how about I present the proof that, for any set X, there is no
surjection X -> PX, and you show me where that argument goes wrong?

--
Quincy (age 5): Baba, play some [computer games].
Mama: Quincy, if you want [Baba] to live, don't make those
suggestions.
Quincy: Make those suggestions. Got it.

[Shmuel Metz]

In <k6tnql$q2i$1...@dont-email.me>, on 11/01/2012

at 11:52 AM, "LudovicoVan" <ju...@diegidio.name> said:

>Jesse is of course already explaining this much better than I can,
>anyway be reassured that the qualifier "extended" is pretty
>customary: here infinities are just the projective or the affine
>infinites

N is not a field, so that is meaningless.

>(as the result of division by zero

No.

[Shmuel Metz]

In <87wqy6s...@phiwumbda.org>, on 10/31/2012

at 09:59 AM, "Jesse F. Hughes" <je...@phiwumbda.org> said:

>But your arrogance in dismissing a fairly standard use of the oo
>symbol is misplaced.

Standard usage is to provide a definition, even if that definition is
simply something like "additional element".

BTW, one could construe your calling him an idiot and a tedious twat
as arrogance. Not that I disagree.

[Jesse F. Hughes]

Shmuel (Seymour J.) Metz <spam...@library.lspace.org.invalid> writes:

> In <87wqy6s...@phiwumbda.org>, on 10/31/2012
> at 09:59 AM, "Jesse F. Hughes" <je...@phiwumbda.org> said:
>
>>But your arrogance in dismissing a fairly standard use of the oo
>>symbol is misplaced.
>
> Standard usage is to provide a definition, even if that definition is
> simply something like "additional element".

Well, what counts as a necessary explanation varies from field to field.
Not that, I assume, LV comes from category theory, but in category
theory, one would quite casually write N* = N u {oo} (or, better, N* = N
+ {oo}) and give the successor (or, perhaps, predecessor) function.

> BTW, one could construe your calling him an idiot and a tedious twat
> as arrogance. Not that I disagree.

Oh, I've never said that I'm free from arrogance.
--
Jesse F. Hughes
"So, either I've found the world's first perfect, non-quantum, random
number generator, or there's a way to make this [factorization] method
work." -- James S. Harris: It's Win-Win, Baby!

[LudovicoVan]

"Jesse F. Hughes" <je...@phiwumbda.org> wrote in message
news:87y5ilp...@phiwumbda.org...

> There is not a single thing "wrong" or suspect with N*. Of course, if
> LudovicoVan decides to use induction on N*, then he will be using an
> invalid step.

Of course we extend induction too, and there is no problem there either.
Look up transfinite induction and structural induction.

-LV

[Jesse F. Hughes]

Son, I know about induction. This structure is not inductive in the
usual sense. (It is coinductive.)

The best you can do is this:

If P(0) and Pn -> P(n+1) *and* P(oo) then An in N* Pn.

But that's a god-awful hack made by adding the condition that P holds of
oo. It's not a proper inductive principle at all (that is, it is not
induction because N* is not an initial algebra).

Say, you were telling me yesterday that Cantor's theorem is invalid.
Let's discuss that. Show me the invalid step already, will you?

--
"And God Himself won't help you if this goes bad as despite your
beliefs I can assure you that angry people will against all law if
necessary tear their rage out of your hides if it goes badly."
-- James S. Harris, on the dangers of criticizing his mathematics

[LudovicoVan]

"Jesse F. Hughes" <je...@phiwumbda.org> wrote in message

news:87objgo...@phiwumbda.org...

> "LudovicoVan" <ju...@diegidio.name> writes:
>> "Jesse F. Hughes" <je...@phiwumbda.org> wrote in message
>> news:87y5ilp...@phiwumbda.org...
>>
>>> There is not a single thing "wrong" or suspect with N*. Of course, if
>>> LudovicoVan decides to use induction on N*, then he will be using an
>>> invalid step.
>>
>> Of course we extend induction too, and there is no problem there either.
>> Look up transfinite induction and structural induction.
>
> Son, I know about induction. This structure is not inductive in the
> usual sense. (It is coinductive.)

Maybe because you use a predecessor, not a successor?

Yeah, there is co-induction too, still no troubles in sight.

> The best you can do is this:
>
> If P(0) and Pn -> P(n+1) *and* P(oo) then An in N* Pn.

Look up articles on Wikipedia for a starter.

> But that's a god-awful hack made by adding the condition that P holds of
> oo. It's not a proper inductive principle at all (that is, it is not
> induction because N* is not an initial algebra).
>
> Say, you were telling me yesterday that Cantor's theorem is invalid.
> Let's discuss that. Show me the invalid step already, will you?

Suck my socks, your math is over. And the lies too.

-LV

[Jesse F. Hughes]

"LudovicoVan" <ju...@diegidio.name> writes:

> "Jesse F. Hughes" <je...@phiwumbda.org> wrote in message
> news:87objgo...@phiwumbda.org...
>> "LudovicoVan" <ju...@diegidio.name> writes:
>>> "Jesse F. Hughes" <je...@phiwumbda.org> wrote in message
>>> news:87y5ilp...@phiwumbda.org...
>>>
>>>> There is not a single thing "wrong" or suspect with N*. Of course, if
>>>> LudovicoVan decides to use induction on N*, then he will be using an
>>>> invalid step.
>>>
>>> Of course we extend induction too, and there is no problem there either.
>>> Look up transfinite induction and structural induction.
>>
>> Son, I know about induction. This structure is not inductive in the
>> usual sense. (It is coinductive.)
>
> Maybe because you use a predecessor, not a successor?
>
> Yeah, there is co-induction too, still no troubles in sight.
>
>> The best you can do is this:
>>
>> If P(0) and Pn -> P(n+1) *and* P(oo) then An in N* Pn.
>
> Look up articles on Wikipedia for a starter.

I don't need to look up articles. What I said above is correct.

If you have some other principle in mind, state it here and we can
discuss it.

>> But that's a god-awful hack made by adding the condition that P holds of
>> oo. It's not a proper inductive principle at all (that is, it is not
>> induction because N* is not an initial algebra).
>>
>> Say, you were telling me yesterday that Cantor's theorem is invalid.
>> Let's discuss that. Show me the invalid step already, will you?
>
> Suck my socks, your math is over. And the lies too.

I really don't understand this response. The argument is invalid,
right? That's what you said, right?

So, let's discuss this claim. We can give a clear statement of the
proof that |X| < |PX| and you can point out the invalid step.

That would be a perfectly natural way to defend your claim, wouldn't it?

Why on earth won't you do that?

--
"So I speak before a crowd of the damned, cursed to be unloved
throughout time, with only their hatred and bile to comfort them now,
having betrayed what should have been their one true lover:
Mathematics." -- James Harris reaches a bit

[LudovicoVan]

"Jesse F. Hughes" <je...@phiwumbda.org> wrote in message

news:87ip9oo...@phiwumbda.org...

> "LudovicoVan" <ju...@diegidio.name> writes:
>> "Jesse F. Hughes" <je...@phiwumbda.org> wrote in message
>> news:87objgo...@phiwumbda.org...
>>> "LudovicoVan" <ju...@diegidio.name> writes:
>>>> "Jesse F. Hughes" <je...@phiwumbda.org> wrote in message
>>>> news:87y5ilp...@phiwumbda.org...
>>>>
>>>>> There is not a single thing "wrong" or suspect with N*. Of course, if
>>>>> LudovicoVan decides to use induction on N*, then he will be using an
>>>>> invalid step.
>>>>
>>>> Of course we extend induction too, and there is no problem there
>>>> either.
>>>> Look up transfinite induction and structural induction.
>>>
>>> Son, I know about induction. This structure is not inductive in the
>>> usual sense. (It is coinductive.)
>>
>> Maybe because you use a predecessor, not a successor?
>>
>> Yeah, there is co-induction too, still no troubles in sight.
>>
>>> The best you can do is this:
>>>
>>> If P(0) and Pn -> P(n+1) *and* P(oo) then An in N* Pn.
>>
>> Look up articles on Wikipedia for a starter.
>
> I don't need to look up articles. What I said above is correct.

It isn't, patently.

Bye bye,

-LV

[Peter Webb]

Jesse F. Hughes wrote:

> "Peter Webb" <webbfamily...@optusnet.com.au> writes:
>
> > Jesse F. Hughes wrote:
> >
> >> "Peter Webb" <webbfamily...@optusnet.com.au> writes:
> >>
> >> > Jesse F. Hughes wrote:
> >> >
> >> >> "Peter Webb" <webbfamily...@optusnet.com.au> writes:
> >> >>
> >> >> >> succ(oo) := oo,
> >> >> >
> >> >> > No. oo is not normally considered a number with a successor.
> >> Perhaps >> > you mean w, but succ(w) = w+1 so that's not correct
> >> either. >>
> >> >> The structure N* = N u {oo} and we define a function s on it
> >> (which we >> call successor) so that s(n) = n + 1 and s(oo) = oo.
> >> >>
> >> >> There's nothing wrong with the structure N*.
> >> >>
> >> >
> >> > Well, there are a couple of things wrong with it. It doesn't
> satisfy >> > Peano's axioms, which means we have lost much of the
> underlying >> > structure that we use in proofs over N.
> >>
> >> Of course it doesn't satisfy those axioms. No one said it did.
> >>
> >> Nonetheless, there's nothing wrong with it.
> >
> >
> > Other than it doesn't satisfy Peano's axioms, which means we have
> > lost much of the underlying structure that we use in proofs over N.
>
> It isn't N, you silly person.
>
> Look, you might as well complain that there's something wrong with Q
> as well, since it doesn't satisfy Peano's axioms either.
>

No, Q has many other structures we can use in proofs. And we can use
proper arithmetic in it. And it fills an important role in the
construction of the Reals and other structures.

Nothing wrong with Q. I use it all the time.

N* = N U {oo} is useless. THe oo isn't even defined, and as you point
out could be anything not in N. This doesn't make it remotely
comparable to Q, which has significant and consistent internal
structure.

> There is not a single thing "wrong" or suspect with N*.

Other than its completely useless for this intended purpose, and as far
as I can see for any other purpose. (Where else have you seen N* = N U
{oo} used in maths, and for what purpose?).

> Of course, if
> LudovicoVan decides to use induction on N*, then he will be using an
> invalid step. But you keep pretending that this shows there's
> something dubious about N*, and this just ain't so.
>

I don't recall using the word "dubious".

I do recall the word "useless".

Can you tell us the use of N* = N U {oo} other than in incorrect proofs
by cranks?

> >
> >
> >>
> >> > The other problem is that in this treatment oo is undefined and
> >> > hence is treated as an Ur-element, which means we are also
> taking a >> > small step outside standard ZF.
> >>
> >> It doesn't matter what the element oo is defined as in ZFC.
> That's a >> triviality. Pick any set not in N and define oo as that
> set. >>
> >
> > You don't think that equating the ordered pair {2,{2,3}} with
> > infinity might not be a problem if you want to use N* to develop
> > function theory?
>
> You're confused. oo in this context is simply a term. Sure, it is
> suggestive of infinity, but the definition of N* tells you what we
> mean by oo.

Which is nothing.

Or is arithmetic defined in N* ?

>
> > Or how about oo = {N} ? I guess if you want to drop the von Neumann
> > construction of numbers with ordinals well ordered by set inclusion
> > this isn't a problem, but again its a bit of a compromise.
>
> Huh?
>
> There's no "compromise" here. The structure N* comes with a
> particular function s:N* -> N* satisfying a particular set of
> equations. In this context, we don't give a damn about von Neumann's
> ordinals or any other tedious details of particular constructions of
> N. All of this is beside the point.

Not, I would have thought, if you are trying to make proofs about
infinite sets, which is what is happening here.

I note that you are no longer talking about the set N* but rather the
structure. Can you tell us what structure Ludicovan is imposing on N* ?
In particular, can you describe the rules of arithmetic which apply in
N* ?

It accomplishes and achieves nothing at all in this context. The thing
which is wrong with it is that it is completely useless as a means of
considering this problem.

[LudovicoVan]

[Follow-up to sci.math, sci.logic only.]

"Peter Webb" <webbfamily...@optusnet.com.au> wrote in message

news:k6vm94$et5$1...@news.albasani.net...

> Jesse F. Hughes wrote:

>> I'm not defending any argument LV is making with N*. I'm pointing out
>> that your view that something is "wrong" with N* is a simple symptom
>> of mathematical immaturity.
>
> It accomplishes and achieves nothing at all in this context. The thing
> which is wrong with it is that it is completely useless as a means of
> considering this problem.

It isn't useless in the counter-arguments I have linked up-thread (the
"objections").

-LV

[Graham Cooper]

On Nov 1, 8:43 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> Graham Cooper <grahamcoop...@gmail.com> writes:
> > Can you state explicitly what it proves?
>
> > I don't see how MODUS PONENS might make this deduction.
>
> > LHS->RHS & LHS -> RHS
>
> > where RHS = "X > size({1,2,3...})"
>
> > nor how the enumeration of a set and it's index inclusion or not has
> > anything to do what's in the superset.
>
> Sorry, Graham, I was hoping to have a conversation with someone a bit
> more coherent today.
>
> Some other time, perhaps.

Ahhah!

I thought up a New System of Logic today!

Some people like C. Boo think if you're using Natural Deduction anyway
then there need not be this Huge Platonic Web of RULES of Set Theory
to abide by... just use Naive Set Theory anyway.

So it is really true that from a contradiction you can prove anything?

Only if you keep MODUS PONENS!

LHS->RHS ^ LHS -> RHS

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

Nve. Set THEORY |- RSeRS, ~RSeRS


Now with

THEORY |- FALSE

INDUCTION RULE : LHS->RHS

INDUCTION CHECK IF IT APPLI:ES : LHS? (MP)

LHS -> RHS

NOT(LHS) or RHS

This version of IMPLIES means: if the LHS applies (is true) then the
RHS must apply
i.e. if the LHS is false, the induction rule doesn't MATCH any fact
(with the bindings in use)
so it has no effect on the RHS.

Sp.... back to my previous derivation from MP.

LHS->RHS ^ LHS -> RHS
(!LHS or RHS) ^ LHS -> RHS
(!LHS ^ LHS) v (RHS^LHS) -> RHS

So if the theory is inconsistent... there is 'likely' a inference
rule LHS->RHS
where LHS MATCHES the predicate pattern of RSeRS..

So
(~RSeRS) & (RSeRS) -> RHS

i.e. a contradictory system proves anything!

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

The QUICK FIX!!??

(LHS->RHS) ^ (LHS is true in some model) ^ (LHS is not false in any
model)
-> RHS

It's very slow to check for errors with every deduction, which is how
humans work with Natural Deductive logic!

Herc

[Peter Webb]

Jesse F. Hughes wrote:

> Do you plan on responding to my suggestion that you point out the
> invalid step in Cantor's theorem?
>
> You say that Cantor's theorem is invalid, right? (You are using the
> term "invalid" in its customary sense, I assume.)
>
> This means that there is some step in the argument which is invalid,
> right?
>
> Would you like to step through the argument with me and show me which
> step is invalid?
>

You want some sport, but LV won't play?

> If so, how about I present the proof that, for any set X, there is no
> surjection X -> PX, and you show me where that argument goes wrong?

Let me take his part.

The proof moves from the assertion that there can be no list of Real
numbers to the assertion they are not countable. These are not quite
the same thing. There are countable sets (eg computable numbers) which
cannot be explicitly listed. Given that the inability to list a set
even in principle is not proof that it is uncountable, how does the
proof go from 'there is no list' to 'the set is uncountable'.

[LudovicoVan]

"Peter Webb" <webbfamily...@optusnet.com.au> wrote in message

news:k6vmqk$ka7$1...@news.albasani.net...

You seem frustrated. I have already referred to the links I have (already)
posted up-thread. What is it that you still do not understand?

-LV

[Graham Cooper]

On Nov 2, 3:48 pm, "Peter Webb" <webbfamilyDIEspam...@optusnet.com.au>
wrote:

> Jesse F. Hughes wrote:
> > Do you plan on responding to my suggestion that you point out the
> > invalid step in Cantor's theorem?
>
> > You say that Cantor's theorem is invalid, right?  (You are using the
> > term "invalid" in its customary sense, I assume.)
>
> > This means that there is some step in the argument which is invalid,
> > right?
>
> > Would you like to step through the argument with me and show me which
> > step is invalid?
>
> You want some sport, but LV won't play?
>
> > If so, how about I present the proof that, for any set X, there is no
> > surjection X -> PX, and you show me where that argument goes wrong?
>
> Let me take his part.

OK show us the base step.

[GEORGE GREENE]

BASE: 1 e f(1) then ~1 e MISS
~1 e f(1) then 1 e MISS
ergo ~MISS = f(1)
By induction, ~MISS=f(any k)

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

Is this how you prove some CARDINALITY > size( {1,2,3,....} )

Herc

[Peter Webb]

Graham Cooper wrote:

> On Oct 31, 5:39�pm, "Peter Webb"

> <webbfamilyDIEspam...@optusnet.com.au> wrote:
> > Graham Cooper wrote:

> > > On Oct 31, 2:34�pm, "Peter Webb"

> > > <webbfamilyDIEspam...@optusnet.com.au> wrote:
> > > > Graham Cooper wrote:

> > > > > On Oct 31, 1:37 pm, "Peter Webb"

> > But this isn't a list of Reals.
> >
> > It is a list of the first few decimal digits of a few unspecified
> > Reals.
> >
> > You need to produce a list of Reals.
> >
> > Lets do this in steps.
> >
> > What is the second Real on your list?

> >
> >
> >
> > > Hence, you CANNOT prove a COUNTABLE SET OF REALS is incomplete.
> >
> > > PROOF: �This is the 10th time asking.
> >
> > > In fact, I call this �THE UNANSWERABLE QUESTION �for a reason!
> >
> > > Herc
> >

> > And the second Real on your list is?
>
> OK use this countable set of reals.
>
> 1/10 �1/20 �1/30 �1/40 ...
> 2/11 �2/21 �2/31 �2/41 ...
> 3/31 �3/32 �3/33 �3/34 ...
> 4/17 �4/27 �4/37 �4/47 ...
> ...
>
> what's missing?
>
> Herc

pi - 3

So your list of all Reals is missing pi - 3. Turned out it wasn't a
list of all Reals after all.

So at least that is cleared up; you now know your list of all Reals
doesn't include all Reals you can stop going on about it.

[Peter Webb]

pi - 3

Turns out you don't have a list of all Reals after all. And I didn't
even need Cantor to find it.

Surely you know that the proposed set contains only rational numbers,
and some numbers are irrational?