A Guide for Cantor Cranks
RussellE
there are so many known refutations available?
These are three tried and true arguments to
debunk Cantor's Anti-Diagonal proof.
1) Prove the anti-diagonal number is in
the list of real numbers.
The secret to this refutation is simply to
provide a list of real numbers in binary.
List of "all" real numbers:
1.000... (base 2)
0.000... (base 2)
0.000... (base 2)
0.000... (base 2)
...
Use the standard method for
creating the anti-diagonal number
to get this list's anti-diagonal:
0.1111... (base 2).
The anti-diagonal is obviously equal
to the first number in the list.
Cantor believers will argue the
anti-diagonal number won't be
on the list if we use two bits
at a time. While true, this doesn't
explain why the anti-diagonal argument
doesn't work in base 2 or base 3.
As we shall see, it doesn't work
is base 1, either.
2) Prove the set of all natural numbers
is uncountable.
This argument works really well with
Turing Machines.
Assume we have a list of all natural
numbers in the unary numbering system.
x = one
xx = two
xxx = three
...
Create an "anti-diagonal" as follows:
If the i-th natural number has n non-blank
positions, make the first n+1 positions of
the anti-diagonal non-blank.
Cantor believers will argue the resulting
anti-diagonal will be an infinitely long
string of non-blank positions.
This is provably false.
By construction,
the anti-diagonal is exactly one position
longer than some natural number in the list.
Since we assumed every unary number in
the list has a finite length, the anti-diagonal
will also be of finite length and it will not be in the list.
3) Claim 1.000... =/= 0.999...
This has nothing to do with Cantor, but will
always start a flame war. Generate the maximum
number of responses by claiming this proves
general relativity is wrong.
Cross post to as many groups as possible.
Russell
- Zeno was right. Motion is impossible.
/-\DAM
I prefer this one because all sci.math will just go
NO I DON'T HAVE TO ANSWER THAT BECAUSE THE OTHER PROOF WORKS!
----------------------------------------------------------
100 years of oversight!
[CARDINALITY VERY STRONG CLAIM - DIAGONALS OF PERMUTATIONS]
There is a procedure that given some diagonal(s) from any
permutation(s) of any list L, will output 1 anti-diagonal for
each given diagonal, all of which are not in L.
e.g.
DIAG(L) = 0.0555...
DIAG(L') = 0.1555...
DIAG(L'') = 0.2555...
DIAG(L''') = 0.3555...
DIAG(L'''') = 0.4555...
DIAG(L''''') = 0.5555...
DIAG(L'''''') = 0.6555...
DIAG(L''''''') = 0.7555...
DIAG(L'''''''') = 0.8555...
DIAG(L''''''''') = 0.9555...
->
AD(L) = 0.5000...
AD(L') = 0.6000...
AD(L'') = 0.7000...
AD(L''') = 0.8000...
AD(L'''') = 0.9000...
AD(L''''') = 0.0000...
AD(L'''''') = 0.1000...
AD(L''''''') = 0.2000...
AD(L'''''''') = 0.3000...
AD(L''''''''') = 0.4000...
The first digit of the given diagonal is arbitrary!
G. ADAM
--
Personally, it is the concrete examples that look
like twists and turns to me. Jim Burns (sci.math)
Virgil
<40e69d0c-eb77-4308...@j13g2000pro.googlegroups.com>,
RussellE <reas...@gmail.com> wrote:
> Why create another refutation of Cantor when
> there are so many known refutations available?
> These are three tried and true arguments to
> debunk Cantor's Anti-Diagonal proof.
>
> 1) Prove the anti-diagonal number is in
> the list of real numbers.
>
> The secret to this refutation is simply to
> provide a list of real numbers in binary.
>
> List of "all" real numbers:
>
> 1.000... (base 2)
> 0.000... (base 2)
> 0.000... (base 2)
> 0.000... (base 2)
> ...
>
> Use the standard method for
> creating the anti-diagonal number
> to get this list's anti-diagonal:
> 0.1111... (base 2).
>
> The anti-diagonal is obviously equal
> to the first number in the list.
But when one converts binary to base 4,
(with 00 -> 0, 01 -> 1, 10 -> 2, and 11 -> 3)
one can find an unlisted "antidiagonal" and when it is converted back to
binary it is still unlisted in the original list.
>
> Cantor believers will argue the
> anti-diagonal number won't be
> on the list if we use two bits
> at a time. While true, this doesn't
> explain why the anti-diagonal argument
> doesn't work in base 2 or base 3.
It only has to work in any one base to be valid, and it does work in
every base greater than 3. So it fails in only a few bases but works
properly in infintiely many bases.
> As we shall see, it doesn't work
> is base 1, either.
The point is not that there is some "base" in which it does not work,
but that there is some base in which it DOES work.
A proof of uncountability valid in any one base establishes the truth
for the set of numbers expressible in that base, which is all reals in
the unit interval. That there are bases in which the proof does appear
to not work does not invalidate the proofs that do work.
Androcles
"RussellE" <reas...@gmail.com> wrote in message
news:40e69d0c-eb77-4308...@j13g2000pro.googlegroups.com...
| 2) Prove the set of all natural numbers
| is uncountable.
Bwahahahahahahahaha!
"Countable" means there is a one-one mapping from the set to
be counted and the set of natural numbers!
A 60 piece dinner set is countable:
1 plate
2 plate
3 plate
4 plate
5 plate
6 plate
7 teacup
8 teacup
...
13 saucer
14 saucer
...
and so on.
The set of natural numbers is countable BY DEFINITION of
countable.
It does not mean there is no highest number, nor does it mean that
a number has to be reached, that is a different proposition altogether.
Put another way, no matter how many plates you have in your set,
there is a natural number that corresponds to every plate, so elements
of dinner sets are countable.
Real numbers are not countable, you cannot assign a natural number
to all those that lie between 0 and 1, never mind between 1 and 2.
Chose any two real numbers and there are an undefined number
of reals between them.
Barb Knox
<40e69d0c-eb77-4308...@j13g2000pro.googlegroups.com>,
RussellE <reas...@gmail.com> wrote:
Russell, I'd like your permission to use the above in a class on set
theory. My current thought is to use it as a homework assignment, with
20 marks given for identifying and refuting each provably incorrect item.
> 3) Claim 1.000... =/= 0.999...
> This has nothing to do with Cantor, but will
> always start a flame war. Generate the maximum
> number of responses by claiming this proves
> general relativity is wrong.
Didn't you get the memo? -- GR is fine, but this disproves the
Copenhagen interpretation of quantum mechanics.
> Cross post to as many groups as possible.
No no no! Only post to relevant ones. I suggest
alt.psychology.personality .
--
---------------------------
| BBB b \ Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum videtur.
| BBB aa a r bbb |
-----------------------------
Graham Cooper
> In article
> <40e69d0c-eb77-4308-996e-c7dfc1a0e...@j13g2000pro.googlegroups.com>,
How shameful to have taught so much lies all your professional
carreer.
Even Will has admitted the first 10 digits of the given diagonal for
Cantor's proof are arbitrary!
William Hughes
And I am sure that Barb Knox (modulo your idiosyncratic nomenclature
and
unstated assumptions) would agree. She would also note that the
diagonal
is not arbitrary.
- William Hughes
LudovicoVan
On Mar 10, 8:10 am, Virgil <V...@gil.Gil> wrote:
> In article
> <40e69d0c-eb77-4308-996e-c7dfc1a0e...@j13g2000pro.googlegroups.com>,
I haven't tried it, but unless the operations in question are not
reversible, it should simply not be the case.
> > Cantor believers will argue the
> > anti-diagonal number won't be
> > on the list if we use two bits
> > at a time. While true, this doesn't
> > explain why the anti-diagonal argument
> > doesn't work in base 2 or base 3.
>
> It only has to work in any one base to be valid, and it does work in
> every base greater than 3. So it fails in only a few bases but works
> properly in infintiely many bases.
>
> > As we shall see, it doesn't work
> > is base 1, either.
>
> The point is not that there is some "base" in which it does not work,
> but that there is some base in which it DOES work.
>
> A proof of uncountability valid in any one base establishes the truth
> for the set of numbers expressible in that base, which is all reals in
> the unit interval. That there are bases in which the proof does appear
> to not work does not invalidate the proofs that do work.
Hmm, that looks too weak to be true: one single counter-example *is*
enough to invalidate a proof...
OTOH, what is not obvious to me is that one can assume that 1.0 == 0.
(1): if I am not mistaken, that is assuming some properties of the
reals (so it is "circular"), while the two binary strings are of
course different.
-LV
LudovicoVan
Actually, a single counter-example is enough to disprove a theorem.
Gee, you guys are picky... ;)
-LV
Michael Stemper
>"RussellE" <reas...@gmail.com> wrote in message news:40e69d0c-eb77-4308...@j13g2000pro.googlegroups.com...
>| 2) Prove the set of all natural numbers
>| is uncountable.
>
>Bwahahahahahahahaha!
>
>"Countable" means there is a one-one mapping from the set to
>be counted and the set of natural numbers!
s/and the/and a subset of the/
--
Michael F. Stemper
#include <Standard_Disclaimer>
A preposition is something that you should never end a sentence with.
Michael Stemper
>In article <CM%dp.25415$Pz1....@newsfe21.ams2>, "Androcles" <Headm...@Hogwarts.physics_2011march> writes:
>>"RussellE" <reas...@gmail.com> wrote in message news:40e69d0c-eb77-4308...@j13g2000pro.googlegroups.com...
>
>>| 2) Prove the set of all natural numbers
>>| is uncountable.
>>
>>Bwahahahahahahahaha!
>>
>>"Countable" means there is a one-one mapping from the set to
>>be counted and the set of natural numbers!
>
>s/and the/and a subset of the/
I retract this. You said "one to one mapping" not "one to one
correspondence" or "one to one and onto", so my modification
was unnecessary.
--
Michael F. Stemper
#include <Standard_Disclaimer>
Outside of a dog, a book is man's best friend.
Inside of a dog, it's too dark to read.
RussellE
> Russell, I'd like your permission to use the above in a class on set
> theory. My current thought is to use it as a homework assignment, with
> 20 marks given for identifying and refuting each provably incorrect item.
Please do.
Of course, you can't give credit for answers like
"it works in base 4". And, your students aren't
allowed to use induction. Induction assumes
there is a set of all natural numbers.
As my second argument proves, no such set exists.
> > 3) Claim 1.000... =/= 0.999...
> > This has nothing to do with Cantor, but will
> > always start a flame war. Generate the maximum
> > number of responses by claiming this proves
> > general relativity is wrong.
>
> Didn't you get the memo? -- GR is fine, but this disproves the
> Copenhagen interpretation of quantum mechanics.
What?
Isn't it well known God has a gambling problem?
And hates cats?
> > Cross post to as many groups as possible.
>
> No no no! Only post to relevant ones. I suggest
> alt.psychology.personality .
OK
>
> --
> ---------------------------
> | BBB b \ Barbara at LivingHistory stop co stop uk
> | B B aa rrr b |
> | BBB a a r bbb | Quidquid latine dictum sit,
> | B B a a r b b | altum videtur.
> | BBB aa a r bbb |
> ------------------------------
Russell
- The universe is one dimensional
Androcles
>
> Russell, I'd like your permission to use the above in a class on set
> theory. My current thought is to use it as a homework assignment, with
> 20 marks given for identifying and refuting each provably incorrect item.
Please do.
Of course, you can't give credit for answers like
"it works in base 4". And, your students aren't
allowed to use induction. Induction assumes
there is a set of all natural numbers.
As my second argument proves, no such set exists.
> > 3) Claim 1.000... =/= 0.999...
> > This has nothing to do with Cantor, but will
> > always start a flame war. Generate the maximum
> > number of responses by claiming this proves
> > general relativity is wrong.
>
> Didn't you get the memo? -- GR is fine, but this disproves the
> Copenhagen interpretation of quantum mechanics.
What?
Isn't it well known God has a gambling problem?
======================================
No, it was Einstein that said "God does not play dice".
God said "Yes I do, Einstein is just a poor loser; he's the
one that won't play."
RussellE
<Headmas...@Hogwarts.physics_2011march> wrote:
> "RussellE" <reaste...@gmail.com> wrote in message
>
> news:40e69d0c-eb77-4308...@j13g2000pro.googlegroups.com...
>
> | 2) Prove the set of all natural numbers
> | is uncountable.
>
> Bwahahahahahahahaha!
>
> "Countable" means there is a one-one mapping from the set to
> be counted and the set of natural numbers!
What makes you think the set of natural numbers has
a one to one mapping to the set of natural numbers?
Russell
- Integers are an illusion
Androcles
=======================================
What makes you hallucinate you can think?
a) alcohol
b) cocaine
c) blatant stupidity
d) all of the above.
JimmyJohn
"Androcles" <Headm...@Hogwarts.physics_2011march> wrote:
> "RussellE" <reas...@gmail.com> wrote in message
> news:4d587e90-db62-470d...@k15g2000prk.googlegroups.com...
> On Mar 10, 12:15 am, "Androcles"
> <Headmas...@Hogwarts.physics_2011march> wrote:
> > "RussellE" <reaste...@gmail.com> wrote in message
> >
> > news:40e69d0c-eb77-4308...@j13g2000pro.googlegroups.com...
> >
> > | 2) Prove the set of all natural numbers
> > | is uncountable.
> >
> > Bwahahahahahahahaha!
> >
> > "Countable" means there is a one-one mapping from the set to
> > be counted and the set of natural numbers!
>
> What makes you think the set of natural numbers has
> a one to one mapping to the set of natural numbers?
Every set can be put in one-to-one correspondence with itself via an
identity mapping.
--
Science is based directly on objective physical evidence,
and nothing that is not based directly on objective physical evidence
can be science.
Androcles
"JimmyJohn" <N...@Jtheist.net> wrote in message
news:4d79cdd5$0$19316$ec3e...@unlimited.usenetmonster.com...
| In article <L6iep.157785$ud6.1...@newsfe19.ams2>,
| > "RussellE" <reas...@gmail.com> wrote in message
| >
news:4d587e90-db62-470d...@k15g2000prk.googlegroups.com...
| > On Mar 10, 12:15 am, "Androcles"
| > <Headmas...@Hogwarts.physics_2011march> wrote:
| > > "RussellE" <reaste...@gmail.com> wrote in message
| > >
| > >
news:40e69d0c-eb77-4308...@j13g2000pro.googlegroups.com...
| > >
| > > | 2) Prove the set of all natural numbers
| > > | is uncountable.
| > >
| > > Bwahahahahahahahaha!
| > >
| > > "Countable" means there is a one-one mapping from the set to
| > > be counted and the set of natural numbers!
| >
| > What makes you think the set of natural numbers has
| > a one to one mapping to the set of natural numbers?
|
| Every set can be put in one-to-one correspondence with itself via an
| identity mapping.
|
MoeBlee
> Induction assumes
> there is a set of all natural numbers.
There is induction on finite ordinals even if there is not a set of
all the finite ordinals (natural numbers). See, e.g., Suppes
'Axiomatic Set Theory'.
MoeBlee
MoeBlee
> What makes you think the set of natural numbers has
> a one to one mapping to the set of natural numbers?
In Z set theory we prove there exists such a mapping, trivially, as
the identity function on any set is a 1-1 mapping from the set onto
itself.
MoeBlee
MoeBlee
> List of "all" real numbers:
>
> 1.000... (base 2)
> 0.000... (base 2)
> 0.000... (base 2)
> 0.000... (base 2)
That's not a list of all the real numbers. The only real numbers in
that list are 1 and 0.
MoeBlee
Patricia Shanahan
I think RussellE understands that. "all" was in quotes. He correctly
claims that the Cantor diagonal construction fails to construct a number
that does not already appear on the list if it is applied to that list
using the binary representation one bit at a time.
His basic mistake is that he seems to be assuming that the list cannot
be proved incomplete by diagonalization unless all ways of constructing
an element from the diagonal, no matter how messed up, succeed in
finding a new number. Obviously, a single way of constructing an element
that is not on the list, such as taking the bits in pairs, is sufficient
to prove that the list is incomplete.
Patricia
Marshall
Whatever system or logic we are using, we always have
available the technique of case analysis.
In a universe where the only values that exist are those
that derive from constructors (as is the case with algebraic
data types,) we can use exhaustive case analysis on
constructors as a proof technique, with each case
parameterized by the arguments of the particular
constructor; this is called "structural induction." I haven't
seen it described as a special case of case analysis,
but it so qualifies.
The usual two constructors for the natural numbers are "zero"
and "successor to some natural number". If we take
structural induction on these two constructors and partially
apply them to structural induction, we get (ordinary)
induction.
Thus, induction is just a special case of structural induction,
which is itself just a form of case analysis. These things
all work whether or not there is a set of all natural numbers,
or whether there is any distinction between "potential" and
"actual", or what-have-you.
Marshall
PS. It's possible that I've used terms from sufficiently many
disciplines as to render this post unreadable.
Transfer Principle
> Why create another refutation of Cantor when
> there are so many known refutations available?
Welcome back, RE!
The return of RE means that there are now several opponents of
Cantor here in a single thread. But unfortunately, those who
support Cantor (the ones you call "Cantor believers") are still
in the majority, since about three or so of them who haven't
posted in a while have also jumped into this thread.
> 3) Claim 1.000... =/= 0.999...
> This has nothing to do with Cantor, but will
> always start a flame war. Generate the maximum
> number of responses by claiming this proves
> general relativity is wrong.
I'm curious about the relationship between 0.999... and GR.
I think someone's alluded to this before -- how a mathematical
model for GR depends on classical analysis with 0.999...=1.000
being part of it. Please (not necessarily RE, but whoever was
discussing it) remind me of the details.
MoeBlee
> I think RussellE understands that.
Okay.
Here's how I would put it to him:
The interval of real numbers between (including) 0 and (not including)
1 is uncountable by this argument:
For every real number in the interval [0 1) there is a unique
denumerable decimal sequence (and one not with a tail of 9's unending)
that represents that real number in the ordinary system of decimal
representations. And every denumerable decimal sequence (not with a
tail of 9's unending) represents a unique real number in the interval
[0 1). (Proofs of the two previous assertions may be found in various
basic textbooks of set theory or real analysis, e.g., Suppes
'Axiomatic Set Theory'.)
So, if given any (not just a particular one, but any arbitrary one)
enumeration of aforementioned decimal representations of real numbers
in the interval [0 1), there is a real number in the interval [0 1)
that has a decimal representation that is not in the enumeration, then
the interval [0 1) is uncountable.
Let f be any (not just a particular one, by any arbitrary one)
enumeration of such decimal representations. Define the sequence t by:
t(n) = 7 if f(n)(n) not= 7
t(n) = 8 if f(n)(n) = 7
Now, t is a decimal representation (and not one with a tail of 9's
unending) of a real number in the interval [0 1) but t is not a term
in the enumeration f.
Thus [0 1) is uncountable, thus, a fortiori, the set of real numbers
is uncountable.
Of course, if one does not accept ordinary mathematical reasoning (and
the only reasoning used in the above argument is intuitionistically
valid, let alone classically valid) or some basic mathematical
principles about functions, etc., then one doesn't have to accept the
conclusion above. Also, if one denies there are infinite sets, then,
of course, there is no set of all the natural numbers or of all the
real numbers, thus, a fortiori, no functions on such sets. But
arguments showing that there are certain lists of real numbers such
that in some representation system we still get back the anti-diagonal
do NOT refute the argument given here. The argument is NOT that EVERY
representation system results in an anti-diagonal for a real number
that has some other representation in the list. Rather, the argument
is as I gave it above: It suffices that there is at least one
representation system such that for any given list of such
representations there is a real number in that representation system
that is not in the list. One can make all kinds of methods of
construction, but they do not refute that there is the specific method
of construction I mentioned in the proof and that, to prove
uncountability, it suffices to show just one such method of
construction.
MoeBlee
MoeBlee
> Welcome back, RE!
And, Transfer Principle, please feel free to explain to him why his
arguments are illogical and irrelevant to the matter of the
uncountability of the reals.
MoeBlee
MoeBlee
> please feel free to explain to him why his
> arguments are illogical and irrelevant to the matter of the
> uncountability of the reals.
Oh, wait, sorry I forgot that he's in the minority so that his
position and arguments must be worth defending for that reason alone.
MoeBlee
MoeBlee
> those who
> support Cantor (the ones you call "Cantor believers")
To speak for myself, I am clear that Cantor did not himself propose a
formal theory. I don't opine as to whether we should regard Cantor's
work itself as satisfactory. However, there is good reason to think
that the formal system ZFC does provide a reasonable formalization of
the basics of Cantor's conception. But even if ZFC is not faithful in
some way to Cantor, then still ZFC is a formal theory in which we
derive the uncountability of the reals with arguments that do
formalize those of Cantor's. In other words, in terms of formal
systems and to a definitive proof (in formal classical mathematics) of
the uncountability of the reals, I don't appeal to Cantor AT ALL.
MoeBlee
LudovicoVan
How did you dare post some mathematics!! :)
FWIW, I find your post very nice and readable: that connection between
structural induction and case analysis is pretty interesting as well.
-LV
LudovicoVan
On Mar 11, 5:15 pm, Patricia Shanahan <p...@acm.org> wrote:
> [...] he seems to be assuming that the list cannot
> be proved incomplete by diagonalization unless all ways of constructing
> an element from the diagonal, no matter how messed up, succeed in
> finding a new number. Obviously, a single way of constructing an element
> that is not on the list, such as taking the bits in pairs, is sufficient
> to prove that the list is incomplete.
Thanks for the clear explanation on that point.
-LV
Jesse F. Hughes
> The return of RE means that there are now several opponents of
> Cantor here in a single thread. But unfortunately, those who
> support Cantor (the ones you call "Cantor believers") are still
> in the majority, since about three or so of them who haven't
> posted in a while have also jumped into this thread.
Unfortunately?
If that weren't the case, then you would (by your own silly criterion)
have to tell Russell that he's wrong.
--
Jesse F. Hughes
"There's a thrill that's gone that I'll probably not have in quite the
same way again. After all, FLT was a unique animal, and we had a
great dance." -J.S. Harris on "proving" Fermat's last theorem
David R Tribble
> please feel free to explain to him why his
> arguments are illogical and irrelevant to the matter of the
> uncountability of the reals.
>
> Oh, wait, sorry I forgot that he's in the minority so that his
> position and arguments must be worth defending for that reason alone.
After reading the latest issue of 'The Skeptical Inquirer',
it occurs to me that Transfer Principle operates like the
UFOlogists, who support as true, as their default position,
stories of alien abduction.
Just as the UFOlogists have not directly experienced abduction
by extraterrestrial aliens, neither has TP put forth an original
anti-Cantorian or other crank-like theory; yet just as the
UFOlogists assume such abduction tales are true from the
get-go, likewise TP assumes that such theories must have
mathematical merit.
David Bernier
> Transfer Principle<lwa...@lausd.net> writes:
>
>> The return of RE means that there are now several opponents of
>> Cantor here in a single thread. But unfortunately, those who
>> support Cantor (the ones you call "Cantor believers") are still
>> in the majority, since about three or so of them who haven't
>> posted in a while have also jumped into this thread.
>
> Unfortunately?
>
> If that weren't the case, then you would (by your own silly criterion)
> have to tell Russell that he's wrong.
>
a dictum: When inquiring into which side in a debate represents the
majority sentiment, it is best to use unorthodox methods.
-- Anonymous
Transfer Principle
> MoeBlee wrote:
> > Oh, wait, sorry I forgot that he's in the minority so that his
> > position and arguments must be worth defending for that reason alone.
> After reading the latest issue of 'The Skeptical Inquirer',
> it occurs to me that Transfer Principle operates like the
> UFOlogists
Note that while I don't always accept mainstream _mathematical_
theories, I almost always prefer mainstream _scientific_ theories.
UFO's aren't _mainstream_ science. If at some future date the
mainstream accepts that an abduction has occurred, I'll accept it.
Atom Totality isn't _mainstream_ science, so I reject it.
But General Relativity _is_ mainstream science. This is why I
inquired about GR and 0.999... earlier.
So why do I accept mainstream _science_ but not math? All I have
to do to make nonstream math appear is choose different axioms,
but no axiom can make UFO's suddenly appear.
Richard Harter
<marshal...@gmail.com> wrote:
>On Mar 11, 8:32=A0am, MoeBlee <jazzm...@hotmail.com> wrote:
>> On Mar 10, 9:53=A0pm, RussellE <reaste...@gmail.com> wrote:
>>
>> > Induction assumes
>> > there is a set of all natural numbers.
>>
>> There is induction on finite ordinals even if there is not a set of
>> all the finite ordinals (natural numbers). See, e.g., Suppes
>> 'Axiomatic Set Theory'.
>
>Whatever system or logic we are using, we always have
>available the technique of case analysis.
>
>In a universe where the only values that exist are those
>that derive from constructors (as is the case with algebraic
>data types,) ...
Well, that is the sticking point, isn't it. How do you propose to
establish "the only values that exist are those..." within axiomatic
set theory?
Richard Harter
<jazz...@hotmail.com> wrote:
I hope you are aware axiomatic set theories have countable models.
Just because a set is provably uncountable within an axiomatic theory
does not mean that the set is uncountable; what has been established
is that a mapping cannot be established within the theory.
Jesse F. Hughes
So far, your attempts at making crank mathematics appear respectable
have been no more successful than the UFO parties trying to flag down
passing saucers by waving flashlights.
In fact, I think I see *less* of a difference between the two activities
after your post than before.
--
Jesse F. Hughes
"I can't tell you how many times she left me.
I lost count the very first time that she did."
-- The Flatlanders, "I Thought the Wreck Was Over"
MoeBlee
> On Fri, 11 Mar 2011 10:33:52 -0800 (PST), MoeBlee
> >To speak for myself, I am clear that Cantor did not himself propose a
> >formal theory. I don't opine as to whether we should regard Cantor's
> >work itself as satisfactory. However, there is good reason to think
> >that the formal system ZFC does provide a reasonable formalization of
> >the basics of Cantor's conception. But even if ZFC is not faithful in
> >some way to Cantor, then still ZFC is a formal theory in which we
> >derive the uncountability of the reals with arguments that do
> >formalize those of Cantor's. In other words, in terms of formal
> >systems and to a definitive proof (in formal classical mathematics) of
> >the uncountability of the reals, I don't appeal to Cantor AT ALL.
>
> I hope you are aware axiomatic set theories have countable models.
Of course. And that doesn't vitiate anything I've said.
> Just because a set is provably uncountable within an axiomatic theory
> does not mean that the set is uncountable;
When I say "x is uncountable," I mean that in some appropritate theory
(such as Z set theory) there is a proof of the theorem "x is
uncountable". However, even though a consistent first order theory has
a countable model, and aside from formal theories anyway, in ordinary
mathematical reasoning we show that there is no bijection between the
naturals and the reals.
> what has been established
> is that a mapping cannot be established within the theory
Of course. And aside from theories, by ordinary mathematical
reasoning, we see that there is no bijection between the naturals and
the reals.
In any case, crank refutations of uncountability show NONE of these:
(1) Our expositions of reasoning that would be formalized in Z set
theory are flawed so that our claim of a formal proof is incorrect.
(2) Our informal mathematical reasoning is flawed so that our claimed
informal proof is not a proof by ordinary mathematical principles.
(3) ZFC is inconsistent.
(4) Our informal mathematical principles are inconsistent.
However, as I noted, if one does not accept certain formal axioms and
rules of inference, then one is not obligated to accept such theorems
(but that does not show that such theorems are not indeed theorems
from said axioms and rules of inference), and if one does not accept
certain informal mathematical principles, then one is not obligated to
accept our informal arguments regarding uncountability (but that does
not show that our arguments are logically flawed given our
mathematical principles nor that our mathematical principles are
inconsistent).
What cranks do show is that they are ignorant, self-misinformed,
illogical, confused, dogmatic, and pathologically mulish.
MoeBlee
MoeBlee
On Mar 11, 4:07 pm, c...@tiac.net (Richard Harter) wrote:
> what has been established
> is that a mapping cannot be established within the theory.
To be more precise: What is established is that the theory has the
theorem that there does not exist such a mapping.
We don't just show that the axioms fail to prove that there is such a
mapping, but rather we show that axioms prove that there is no such
mapping.
MoeBlee
MoeBlee
> So far, your attempts at making crank mathematics appear respectable
> have been no more successful than the UFO parties trying to flag down
> passing saucers by waving flashlights.
We use kites and banners in my village.
MoeBlee
RussellE
Changing the algorithm is cheating.
I use exactly the same algorithm given by Wikipedia:
http://en.wikipedia.org/wiki/Cantor's_diagonal_argument
I give a set of real numbers such that the stated
method fails to produce a real number not in the list.
Changing the algorithm after the fact allows us
to prove anything.
Moblee gives the following algorithm for real
numbers in decimal:
t(n) = 7 if f(n)(n) not= 7
t(n) = 8 if f(n)(n) = 7
I am going to change this to:
t(n) = 0 if f(n)(n) not= 0
t(n) = 9 if f(n)(n) = 0
Consider the following list:
1.000... (base 10)
0.000... (base 10)
0.000... (base 10)
....
The anti-diagonal is 0.999...
Moblee and others have complained
my list doesn't contain all of the real numbers.
If the diagonal method fails for a list with only
two real numbers, why would I assume it works
for a list with an infinite number of reals?
Russell
- Never never means never in set theory
RussellE
The identity function may be a trivial mapping,
but there is nothing trivial about identifying the
"set of all natural numbers".
I have already shown no list contains every natural number.
How can a set that doesn't exist have an identity function?
I will give my proof again.
Prove the set of all natural numbers is uncountable.
Assume we have a list of all natural
numbers in the unary numbering system.
x = one
xx = two
xxx = three
...
Create an "anti-diagonal" as follows:
If the i-th natural number has n non-blank
positions, make the first n+1 positions of
the anti-diagonal non-blank.
By construction, the diagonal is one
position longer than some unary number
in the list. Since the successor of a
natural number is also a natural number,
the anti-diagonal is a natural number
not in the list.
This proves no list can contain every
natural number.
Russell
- 2 many 2 count
Marshall
Clearly I am having some sort of off day. Perhaps it was the
amount of wine I drank at dinner last night. ;-)
> FWIW, I find your post very nice and readable: that connection between
> structural induction and case analysis is pretty interesting as well.
Thanks.
Marshall
Marshall
> On Fri, 11 Mar 2011 09:56:18 -0800 (PST), Marshall
> <marshall.spi...@gmail.com> wrote:
>
> >> > Induction assumes there is a set of all natural numbers.
>
> >> There is induction on finite ordinals even if there is not a set of
> >> all the finite ordinals (natural numbers). See, e.g., Suppes
> >> 'Axiomatic Set Theory'.
>
> >Whatever system or logic we are using, we always have
> >available the technique of case analysis.
>
> >In a universe where the only values that exist are those
> >that derive from constructors (as is the case with algebraic
> >data types,) ...
>
> Well, that is the sticking point, isn't it.
It is indeed, which is why I specifically called it out.
> How do you propose to
> establish "the only values that exist are those..." within axiomatic
> set theory?
I do not so propose, especially since, if I understand correctly,
such is impossible. Do you have any ideas?
On the other hand, this difficulty seems to me to be just
another of the difficulties that arise out of thinking of
theories as coming before models, when it seems to me
it is better to think of them as "above" or abstracted from
models.
Marshall
Graham Cooper
Can anyone explain why there are a half dozen (mainstream) branches of
mathematics, none of which have produced a single practical useful
formula?
These are some of the foundations of various branches of mathematics.
A program that examines itself, if it halts then continues?
A number that is different to all other listed numbers, only by an
explicit infinite clause that splices the opposing digit of all
listed infinite numbers?
A statement that asserts "you can't prove me"?
A *finite* sized algorithm that gives the maximum output length of
*any* sized algorithm?
A set that doesn't contain itself, yet contains all sets that don't
contain themselves?
----------
Maybe it's Jim's Lemma that's at fault?
For all x in X, y not= x. Therefore, y not in X.
It's certainly trivial to find a counterexample working only in N with
X=N.
Tonico
> On Mar 9, 11:03 pm, RussellE <reaste...@gmail.com> wrote:
>
> > Why create another refutation of Cantor when
> > there are so many known refutations available?
>
> Welcome back, RE!
>
> The return of RE means that there are now several opponents of
> Cantor here in a single thread. But unfortunately, those who
> support Cantor (the ones you call "Cantor believers") are still
> in the majority, since about three or so of them who haven't
> posted in a while have also jumped into this thread.
>
> > 3) Claim 1.000... =/= 0.999...
> > This has nothing to do with Cantor, but will
> > always start a flame war. Generate the maximum
> > number of responses by claiming this proves
> > general relativity is wrong.
>
> I'm curious about the relationship between 0.999... and GR.
>
What I am curious about is how come so many people haven't yet
realized that Russell is having a big laugh on trolls/cranks/
anticantorians, etc.
For me it is plain he's being sarcastic and ironic from his very first
post...:)....though, of course, I could be wrong.
Tonio
Ben Bacarisse
The fact that you make this change means, presumably, that you accept
that Moblee's argument (as originally presented) is sound. That's a big
step forward.
I get the feeling that you don't really believe what you are presenting
here. If you thought the usual argument was wrong, you'd say why
instead of swapping parts of it about until it's not valid. There are
few proofs that would survive the deliberate substitution of symbols
that you've chosen to make to this one. An argument stands or falls as
presented, not as you'd like it to have been presented.
<snip>
--
Ben.
Graham Cooper
> What I am curious about is how come so many people haven't yet
> realized that Russell is having a big laugh on trolls/cranks/
> anticantorians, etc.
>
> For me it is plain he's being sarcastic and ironic from his very first
> post...:)....though, of course, I could be wrong.
>
> Tonio
You are wrong about a great number of things.
None of you seem to wish to continue the Cantor Dialog.
HERE IS MY LIST
123..
456..
789..
..
CANTOR
DIAG=159..
AD = 260..
AD is missing form the list.
WHY DID YOU CHOOSE THAT PARTICULAR ORDERING OF MY LIST?
Anyone?
RussellE
> The fact that you make this change means, presumably, that you accept
> that Moblee's argument (as originally presented) is sound. That's a big
> step forward.
>
No. I am an ultra-finitist. I don't think infinite sets exist.
Since real numbers are infinite by definition,
I don't think they exist either.
I am demonstrating that the diagonal argument
creates contradictory results when applied to
infinite sets.
> I get the feeling that you don't really believe what you are presenting
> here. If you thought the usual argument was wrong, you'd say why
> instead of swapping parts of it about until it's not valid. There are
> few proofs that would survive the deliberate substitution of symbols
> that you've chosen to make to this one. An argument stands or falls as
> presented, not as you'd like it to have been presented.
I use an algorithm from Wikipedia and produce a list
of real numbers containing the anti-diagonal.
No one has given any explaination why a standard
diagonal algorithm fails to produce a unique real number
not on my list.
Saying some other algorithm works is irrelevent.
Why doesn't the algorithm specified by Wikipedia
work on my set?
The proof given on Wikipedia is very similar to
Cantor's original proof showing P(N) > N.
http://en.wikipedia.org/wiki/Cantor's_theorem
Ben Bacarisse
<snip>
> None of you seem to wish to continue the Cantor Dialog.
OK, I'll exchange a few posts or two...
> HERE IS MY LIST
> 123..
> 456..
> 789..
> ..
>
> CANTOR
> DIAG=159..
> AD = 260..
> AD is missing form the list.
>
> WHY DID YOU CHOOSE THAT PARTICULAR ORDERING OF MY LIST?
*You* chose it. The anti-diagonal is taken from the list in the order
you present it.
To refute the claim that a set like [0,1) is uncountable, an ordered
list of its elements must be given[1] and it is this list (in the order
given) that is used to construct at least one element that is not on the
list. Since the construction applies to any purported list, there is no
point in trying to produce one.
It is not unlike James Randi's Million Dollar Challenge. If you make
some extraordinary claim that you can do X, you must agree on the test
that will be used to determine if you can indeed do X, and you must sign
a declaration beforehand that you accept that you've failed if you don't
pass the test. If you claim that [0,1) is countable, you must produce a
definitive ordered list of it's members and sign a declaration that if
anyone can demonstrate a real in [0,1) that is not on it, you accept
that you've failed (this time) to show the countability of the set.
[1] Of course, it will have to be a procedure, formula or method since
the set is not finite.
--
Ben.
Ben Bacarisse
So why do you care about the un-countability of the reals? It's a
theorem in another formal system that you don't accept, so your beef
should be with the axioms that permit the reals to exit, not with the
argument about the set's cardinality.
Whether you think a set exists or not has nothing to do with the
theorems that can be proved from some set of axioms. It may have some
effect on what axioms you are prepared admit, but the validity of the
argument is independent of your thoughts about the sets.
> I am demonstrating that the diagonal argument
> creates contradictory results when applied to
> infinite sets.
In the post I replied to, you were demonstrating that an incorrect
argument can be made by altering a correct one. No one should be
surprised by that.
<snip>
--
Ben.
Peter Webb
No. I am an ultra-finitist. I don't think infinite sets exist.
Since real numbers are infinite by definition,
I don't think they exist either.
___________________________________
Real numbers are not "infinite by definition". To an ultra-finitist, they
are no more infinite than natural numbers. If you accept the existence of
the Natural number 3, then you accept the existence of the Real number 3.0
JimmyJohn
<3d1555f8-26cd-4d46...@f36g2000pri.googlegroups.com>,
RussellE <reas...@gmail.com> wrote:
Not to me. Nor to anyone with his head straight.
--
Science is based directly on objective physical evidence,
and nothing that is not based directly on objective physical evidence
can be science.
RussellE
I don't really. Talking about real numbers requires
far too many implicit assumptions.
I was just providing an example for Cantor doubters.
I much prefer my second argument with unary numbers.
Its hard to get simpler than unary. Marks on a tape.
> Whether you think a set exists or not has nothing to do with the
> theorems that can be proved from some set of axioms. It may have some
> effect on what axioms you are prepared admit, but the validity of the
> argument is independent of your thoughts about the sets.
In ZF without the Axiom of Infinity, we can show
any set that is provably an ordinal has
a largest element.
This is basically my unary proof.
I prove all natural numbers are finite
(all ordinals have a largest member).
> > I am demonstrating that the diagonal argument
> > creates contradictory results when applied to
> > infinite sets.
>
> In the post I replied to, you were demonstrating that an incorrect
> argument can be made by altering a correct one. No one should be
> surprised by that.
Why is my argument incorrect?
Is my algorithm incorrect?
I followed the algorithm given on Wikipedia
and used by Cantor. My list contains
the anti-diagonal number.
Are you claiming 0.111... (base 2)
is not the anti-diagonal number?
Are you claimng 1.000... =/= 0.111...???
Which part of my proof are you claiming
is incorrect?
If both of our proofs are correct then we
have shown a contradiction.
JimmyJohn
<ee4aeac0-f793-4316...@a21g2000prj.googlegroups.com>,
RussellE <reas...@gmail.com> wrote:
The fact that your particular antidiagonal fails does not mean that
anyone else's has to.
If one adds a rule that the list of numbers be represented in base n,
with n > 3 and each antidiagonal digit excludes 0, n-1 and the digit
being replaced, then the antidiagonal number produced from a list will
NEVER be among those listed.
JimmyJohn
<9cdebb8c-c95f-4742...@n2g2000prj.googlegroups.com>,
RussellE <reas...@gmail.com> wrote:
> No. I am an ultra-finitist. I don't think infinite sets exist.
> Since real numbers are infinite by definition,
> I don't think they exist either.
Then you think that there aren't any real numbers at all?
>
> I am demonstrating that the diagonal argument
> creates contradictory results when applied to
> infinite sets.
You are only demonstrating your own lack of understanding.
RussellE
> The fact that your particular antidiagonal fails does not mean that
> anyone else's has to.
I never claimed that.
I claim my list contains its antidiagonal.
I use a standard algorithm.to compute
the anti-diagonal. In fact, it is pretty much
the only algorithm that can be used on
individual bits of a binary number.
> If one adds a rule that the list of numbers be represented in base n,
> with n > 3 and each antidiagonal digit excludes 0, n-1 and the digit
> being replaced, then the antidiagonal number produced from a list will
> NEVER be among those listed.
>
OK. I have no argument with that.
Why do we need to make all of these additional assumptions?
What do you have against binary numbers?
> --
> Science is based directly on objective physical evidence,
> and nothing that is not based directly on objective physical evidence
> can be science
Graham Cooper
> Graham Cooper <grahamcoop...@gmail.com> writes:
>
> <snip>
>
> > None of you seem to wish to continue the Cantor Dialog.
>
> OK, I'll exchange a few posts or two...
>
> > HERE IS MY LIST
> > 123..
> > 456..
> > 789..
> > ..
>
> > CANTOR
> > DIAG=159..
> > AD = 260..
> > AD is missing form the list.
>
> > WHY DID YOU CHOOSE THAT PARTICULAR ORDERING OF MY LIST?
>
> *You* chose it. The anti-diagonal is taken from the list in the order
> you present it.
>
> To refute the claim that a set like [0,1) is uncountable, an ordered
> list of its elements must be given[1] and it is this list (in the order
> given) that is used to construct at least one element that is not on the
> list. Since the construction applies to any purported list, there is no
> point in trying to produce one.
Equally there is no point in choosing the ordering I merely presented
my set in.
So why did you *SPECIFICALLY* choose that ordering?
I am not presenting a FUNCTION 1 to 1 with elements of my set,
I am presenting an ordinary LIST like a shopping list.
Stop using a mathemtical defn of a list here!
It's just a PRESENTED FORMAT of an unordered set of reals.
>
> It is not unlike James Randi's Million Dollar Challenge. If you make
> some extraordinary claim that you can do X, you must agree on the test
> that will be used to determine if you can indeed do X, and you must sign
> a declaration beforehand that you accept that you've failed if you don't
> pass the test. If you claim that [0,1) is countable, you must produce a
> definitive ordered list of it's members and sign a declaration that if
> anyone can demonstrate a real in [0,1) that is not on it, you accept
> that you've failed (this time) to show the countability of the set.
>
I'm not the one making extraordiary claims, you are!
I don't have to prove reals are countable, I'm challgening YOUR CLAIM
of HIGHER INFINITY SETS.
2/2 WRONG!
Any other takers?
Peter Webb
I followed the algorithm given on Wikipedia
and used by Cantor. My list contains
the anti-diagonal number.
_________________________________
Neither Cantor nor Wikipedia do this in base 2, so you clearly *aren't*
using their algorithm. If you want to use Canor on a base 2 Real, group the
digits into pairs (giving the base 4 representation 0 and use the
substitution 0->1, 1->2, 2->1, 3->1, then reconvert into base 2 and you have
a Real not on the list.
Or simply convert your base 2 list into base 10, calculate the antidiagonal,
then convert this into base 2. Again a Real not on base 2 list.
HTH
Graham Cooper
> > list of its elements must be given[1]
>
>
>
> > some extraordinary claim that you can do X, you must agree on the test
> > that will be used to determine if you can indeed do X, and you must sign
> > a declaration beforehand that you accept that you've failed if you don't
> > pass the test. If you claim that [0,1) is countable, you must produce a
> > definitive ordered list of it's members and sign a declaration that if
> > anyone can demonstrate a real in [0,1) that is not on it, you accept
> > that you've failed (this time) to show the countability of the set.
>
> I'm not the one making extraordiary claims, you are!
>
> I don't have to prove reals are countable, I'm challgening YOUR CLAIM
> of HIGHER INFINITY SETS.
Admittedly you DID PASS THE PRELIMINARY TEST!
You made an ad hoc proof of supersized sets bigger than infinity!
KUDOS TO YOU!
NOW IT THE FINAL ROUND FOR THE $1,000,000
AND I HAVE ADDED SOME CONSTRAINTS TO MAKE SURE YOU DON'T CHEAT!
.
Justify why you ONLY apply diagonalisation to the single ordering
given.
LET US REDEFINE:
SET: UNORDERED BAG OF ITEMS
LIST: ORDERED PRESENTATION OF SET
SEQUENCE: ORDERED (NUMBERED) LIST
Although a LIST and a SEQUENCE look equivalent, let us define them as
different here.
Now justify why Cantor's proof would work using any permuation of the
given LIST.
HERE IS WHAT YOU ARE DOING.
PROOF THAT ANY SET HAS A FIRST ELEMENT!
GIVEN ANY SET, I CAN GIVE THE FIRST ELEMENT
OK, MY SET IS
DOG
CAT
FOX
THE FIRST ELEMENT IS DOG!
THAT PROVES ALL SETS HAVE A FIRST ELEMENT!
.
THAT IS YOUR PROOF, NOW SHOW IT WORKS FOR ANY PRESENTATION OF THE
GIVEN SET.
Richard Harter
<marshal...@gmail.com> wrote:
>On Mar 11, 1:54=A0pm, c...@tiac.net (Richard Harter) wrote:
>> On Fri, 11 Mar 2011 09:56:18 -0800 (PST), Marshall
>> <marshall.spi...@gmail.com> wrote:
>>
>> >> > Induction assumes there is a set of all natural numbers.
>>
>> >> There is induction on finite ordinals even if there is not a set of
>> >> all the finite ordinals (natural numbers). See, e.g., Suppes
>> >> 'Axiomatic Set Theory'.
>>
>> >Whatever system or logic we are using, we always have
>> >available the technique of case analysis.
>>
>> >In a universe where the only values that exist are those
>> >that derive from constructors (as is the case with algebraic
>> >data types,) ...
>>
>> Well, that is the sticking point, isn't it.
>
>It is indeed, which is why I specifically called it out.
>
>
>> How do you propose to
>> establish "the only values that exist are those..." within axiomatic
>> set theory?
>
>I do not so propose, especially since, if I understand correctly,
>such is impossible. Do you have any ideas?
You are right, it is impossible.
>
>On the other hand, this difficulty seems to me to be just
>another of the difficulties that arise out of thinking of
>theories as coming before models, when it seems to me
>it is better to think of them as "above" or abstracted from
>models.
I'm not sure that either of us understands what you are saying in that
paragraph. Be that as it may, the concept of the set of all subsets
of the set of integers has problems. The notion is seductive - after
all, we feel that there is no difficulty with the notion of the power
set of a finite set - there is a simple procedure for constructing
said power set. The continuum, however, is a different matter. That
"all" is irretrievably fuzzy.
Graham Cooper
> paragraph. Be that as it may, the concept of the set of all subsets
> of the set of integers has problems. The notion is seductive - after
> all, we feel that there is no difficulty with the notion of the power
> set of a finite set - there is a simple procedure for constructing
> said power set. The continuum, however, is a different matter. That
> "all" is irretrievably fuzzy
What pattern can the set of all computer programs miss?
The Omega-Halt sequence?
Define a computer program that examines it's own halt value
and keeps looping if it says halt and vice versa?
That's the ONLY obstacle to computing infinite powersets.
n e P(m) IFF UTM(m,n)=1
UTM(m,n) is the mth computer program with input tape n.
Richard Harter
<jazz...@hotmail.com> wrote:
>P.S.
Not quite. We can show that for any model M satisfying the axioms of
a set theory S there is no such mapping among the sets of M. In turn,
for each countable model M it is straightforward to produce such
mappings; however the mapping is not a model within M. What is not
possible is to construct a model M that is actually uncountable.
Tim Little
> In ZF without the Axiom of Infinity, we can show any set that is
> provably an ordinal has a largest element.
Oh, this should be good: go ahead and prove it then.
Hint: you can't. Every theorem of ZF-Infinity is also a theorem of
ZF.
--
Tim
fishfry
c...@tiac.net (Richard Harter) wrote:
This has nothing to do with "procedures." The power set of N exists by
the power set axiom. Constructability in the sense of algorithms and
computer science is not required by set theory.
In fact the term "constructable" in set theory means something else, as
in Godel's constructible universe. Given a set such as N, its power set
exists by the power set axiom; and therefore all the members of the
power set are constructible by definition. Note that this is a totally
different meaning of "constructible" than people are thinking about when
they talk about being able to define a set via an algorithm or procedure.
fishfry
fishfry <BLOCKSPA...@your-mailbox.com> wrote:
I just realized that I may be confused about the constructible universe.
Evidently given a set, the power set in the constructible universe
consists of only those subsets definable by a formula. So that doesn't
include the entire conceptual power set.
I wonder if someone can straighten me out about this.
Transfer Principle
> Transfer Principle <lwal...@lausd.net> writes:
> > So why do I accept mainstream _science_ but not math? All I have
> > to do to make nonstream math appear is choose different axioms,
> > but no axiom can make UFO's suddenly appear.
> So far, your attempts at making [insult] mathematics appear respectable
Let's discuss this new UFO analogy in further detail.
So far, your UFO analogy appears to be like this: The mainstream
doesn't accept the existence of UFO's. Therefore, when someone
claims to have seen a UFO, the only rational response is to
assume that the claim is mistaken or a hoax and keep in line with
the mainstream theory, in which UFO's don't exist.
Similarly, the mainstream accepts Cantor. Therefore, when someone
claims to have disproved Cantor, the only rational response is to
assume that the claim is mistaken and keep in line with the
mainstream theory, ZFC, which proves Cantor.
If this is an accurate depiction of the UFO analogy (if not, then
please correct me), here are my comments about it. First of all,
there are several counterexamples to Cantor. We have theories
such as NFU (in which non-Cantorian sets exist), zuhair's theory
ND, as well as Lowenheim-Skolem (which isn't a theory, rather it
proves the existence of countable models).
These aren't "UFO's" -- i.e., objects (theories) that some claim
to have seen, but the mainstream doesn't accept as existing. NFU
is an _actual_ theory. One can see the _actual_ post in which
zuhair states his theory ND. L-S proves the existence of _actual_
countable models of ZFC -- this is being discussed by others
elsewhere in the thread.
> have been no more successful than the UFO parties trying to flag down
At this point, it appears that you are referring to my attempts
to rigorize not the "Cantor deniers" in this thread, but those
posters like AP, TO, tommy1729, and possibly even MR. Each of
these posters have not just one, but several desiderata that he
demands be satisfied in his theory. Trying to find a rigorous
theory in which _all_ of the desiderata are satisfied proves to
be rather difficult. And so, returning to the analogy, one says
that it's more likely that one will see a real live UFO than a
rigorous theory in which all of the desiderata are satisfied.
To emphasize this analogy, let's call a rigorous theory where
all of a single such posters's desiderata are satisfied an
"Unidentified Flying _Theory_" or UFT.
Now, what should a poster do if he simply does not like a
certain result of ZFC or mainstream theory? It seems as if
there should be an alternative besides just simply accepting
ZFC/mainstream theory and calling out several desiderata that
amounts to a UFT.
How about this -- a theory in which just _one_ of a poster's
desiderata is satisfied (and isn't satisfied by ZFC or the
mainstream theory)? And the theory should be a theory that is
already known -- and I include zuhair's theories here since
he's better at making theories than I am. This means that for
the opponent of ZFC, the theory would be preferable to ZFC as
the former proves one desideratum which fails in ZFC. But for
those on the standard side, the theory would be preferable to
a UFT, since this is an _actual_ theory that can be confirmed
by looking up the theory in a book (posters on this side
usually have books) or one of zuhair's posts. So this would
end up being a compromise theory.
Examples of the above:
For the Cantor deniers in this thread, the three examples
mentioned earlier (NFU, ND, L-S) should suffice.
For posters who want infinitesimals, Robinson's hyperreals
are a compromise theory. For the infinitesimalist, even
though Robinson's infinitesimals aren't adjacent, surely
having any nonzero infinitesimals at all is preferable to
classical analysis with _no_ positive infinitesimals. For
the mainstream, Robinson's hyperreals are known, while
adjacent infinitesimals are a UFT.
For posters like tommy1729, a mereology theory as given
at the Stanford site. For tommy1729, surely any mereology
at all is better than no mereology. For the mainstream,
Stanford mereology is known since anyone can look up that
website, while tommy1729's nonstandard three-valued logic
makes his theory a UFT.
Then, once a known theory has been established, one can
attempt to satisfy another desideratum, one at a time,
until the task becomes impossible (i.e., a UFT), and see
what develops. I'd much rather do this then just simply
accept only mainstream mathematical theories and insult
anyone who tries to challenge them.
Transfer Principle
<webbfam...@DIESPAMDIEoptusnet.com.au> wrote:
> RE wrote:
> > No. I am an ultra-finitist. I don't think infinite sets exist.
> > Since real numbers are infinite by definition,
> > I don't think they exist either.
What is the definition of real number here? Is it a Dedekind cut,
or an equivalence class of Cauchy sequences?
If the former, then D-cuts contain infinitely many rationals, and
so are infinte.
If the latter, then these classes contain infinitely many
C-sequences, and so are infinite.
> you accept the existence of the Real number 3.0
For example, the real number 3.0, seen as a D-cut, is infinite as
it contains all of the following rationals:
2, 5/2, 8/3, 11/4, 14/5, 17/6, ...
Seen as an equivalence class of C-sequences, the real number 3.0
is infinite as it contains all of the following C-sequences:
{2, 5/2, 8/3, 11/4, 14/5, 17/6, ...}
{5/2, 11/4, 17/6, 23/8, 29/10, ...}
{8/3, 17/6, 26/9, 35/12, 44/15, ...}
{11/4, 23/8, 35/12, 47/16, ...}
{14/5, 29/10, 44/15, 59/20, ...}
{17/6, 35/12, 53/18, 71/24, ...}
...
All known proofs in ZF that the real numbers exist require the
Axiom of Infinity. So far, I haven't seen a proof in ZF-Infinity
that the real numbers exist.
I think Megill says it the best:
http://us.metamath.org/mpegif/omex.html
"The finitist could still develop natural number, integer, and
rational number arithmetic but would be denied the real numbers
(as well as much of the rest of mathematics)."
And of course, if a mere _finitist_ is denied the real numbers,
how much more so is an _ultra_finitist?
Graham Cooper
> what develops. I'd much rather do this then just simply
> accept only mainstream mathematical theories and insult
> anyone who tries to challenge them.
Mathematics is one million men deciding how to cross a river bank for
100 years, when there is no water.
Here are TWO PERMUTATIONS of my list in base 4.
List
000001...
111101...
222201...
333301...
000000...
111111...
..
DIAG=012301..
AD = 230123..
Permutation
222201...
333301...
000001...
111101...
000000...
111111...
..
DIAG=230101..
AD = 012323..
-------------
So BOTH 230101.. AND 012323...
are missing right?
Earth to sci.math! Real list with no missing real!
En,
LIST(1,1) = ADn(1)
FOR CHRISTS SAKE CANTOR'S PROOF ONLY HOLDS IF YOU DO THE ROW BY ROW
PROOF 1 PERMUTATION AT A TIME!
YOU'RE ALL DAFT!
Graham Cooper
> PROOF 1 PERMUTATION AT A TIME!
Don't believe me! Ask Rupert!
> It's completely different to the weak claim, when used in conjuction
> with the fact that your claim is to provide missing reals for ANY/ALL
> permutations of the same list without seeing any of the lists.
> i.e I can give you ANY Diagonal from any permutation, only, allowing
> room to move with the data I provide.
[RUPERT] YES.
------------------------------------------------
[CARDINALITY VERY STRONG CLAIM - DIAGONALS OF PERMUTATIONS]
VERY STRONG CLAIM
There is a procedure that given some diagonal(s) from any
permutation(s) of any list L, will output 1 antidiagonal for
each given diagonal, all of which are not in L.
e.g.
DIAG(L) = 0.0555...
DIAG(L') = 0.1555...
DIAG(L'') = 0.2555...
DIAG(L''') = 0.3555...
DIAG(L'''') = 0.4555...
DIAG(L''''') = 0.5555...
DIAG(L'''''') = 0.6555...
DIAG(L''''''') = 0.7555...
DIAG(L'''''''') = 0.8555...
DIAG(L''''''''') = 0.9555...
->
AD(L) = 0.5000...
AD(L') = 0.6000...
AD(L'') = 0.7000...
AD(L''') = 0.8000...
AD(L'''') = 0.9000...
AD(L''''') = 0.0000...
AD(L'''''') = 0.1000...
AD(L''''''') = 0.2000...
AD(L'''''''') = 0.3000...
AD(L''''''''') = 0.4000...
The first digit of the given diagonal is arbitrary!
------------------------------------------------
MUTE SHEEP!
RussellE
> On 2011-03-12, RussellE <reaste...@gmail.com> wrote:
>
> > In ZF without the Axiom of Infinity, we can show any set that is
> > provably an ordinal has a largest element.
>
> Oh, this should be good: go ahead and prove it then.
>
> Hint: you can't. Every theorem of ZF-Infinity is also a theorem of
> ZF.
What is the definition of an ordinal in ZF-I?
Hint - I said a set that is provably an ordinal.
Barb Knox
<5916f49f-1dbb-4c95...@q40g2000prh.googlegroups.com>,
RussellE <reas...@gmail.com> wrote:
> >
> > Russell, I'd like your permission to use the above in a class on set
> > theory. My current thought is to use it as a homework assignment, with
> > 20 marks given for identifying and refuting each provably incorrect item.
>
> Please do.
Thank you.
> Of course, you can't give credit for answers like
> "it works in base 4".
Of course not. Bald assertions are not allowed -- each answer must
include a fairly rigourous proof using the axioms and
theorems-proven-so-far in the set theory.
IMO your lack of rigour is a big part of why you produce nonsense. You
have no way of sorting legitimate arguments out from nonsensical ones.
> And, your students aren't allowed to use induction.
It so happens that they aren't (yet), since we have not yet constructed
the set N which represents the standard model of arithmetic. But once
we do, and prove that this N satisfies induction, they they will be able
to use induction. Regardless, induction is not needed for any of the
refutation proofs; they are all quite straightforward.
[snip]
BTW, does anyone know a Latin translation for "You've got to know when
to hold 'em, know when to fold 'em"?
--
---------------------------
| BBB b \ Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum videtur.
| BBB aa a r bbb |
-----------------------------
Ben Bacarisse
That's very gracious of you. Unfortunately I can't be as gracious in my
response: you fail the proportion of UPPER CASE letters test so I won't
be continuing this discussion.
Most often I post on the off-chance that it might help the discussion by
clarifying a point or explaining an argument. I have no vested interest
in whether you accept what I say. When I am wrong I hope others will
point it out and, if I'm able to follow (and accept) their argument,
I'll be all the better for it. Sometimes I continue in my ignorance,
though of course I never know that this has happened.
<snip>
--
Ben.
Jesse F. Hughes
> On Mar 11, 2:16 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> Transfer Principle <lwal...@lausd.net> writes:
>> > So why do I accept mainstream _science_ but not math? All I have
>> > to do to make nonstream math appear is choose different axioms,
>> > but no axiom can make UFO's suddenly appear.
>> So far, your attempts at making [insult] mathematics appear
>> respectable
You know, I don't really appreciate your stupid attempts at censoring my
posts. I wrote "crank" and I meant it.
> Let's discuss this new UFO analogy in further detail.
>
> So far, your UFO analogy appears to be like this: The mainstream
> doesn't accept the existence of UFO's. Therefore, when someone
> claims to have seen a UFO, the only rational response is to
> assume that the claim is mistaken or a hoax and keep in line with
> the mainstream theory, in which UFO's don't exist.
Unless you are willing and able to investigate further, I suppose this
is a reasonable response -- but not because the mainstream doesn't
accept UFOs. Rather because a rational consideration of the existing
claims and evidence has determined that UFOs are unlikely.
I know this will shock you, but some of us don't determine beliefs by
counting the heads of believers.
> Similarly, the mainstream accepts Cantor. Therefore, when someone
> claims to have disproved Cantor, the only rational response is to
> assume that the claim is mistaken and keep in line with the
> mainstream theory, ZFC, which proves Cantor.
Why must you do that? If you have the time and inclination, you can
evaluate his argument and see whether or not it is a theorem of ZFC. If
not, you can ask him what theory he had in mind? (You should try to
determine whether he really understands the question. Does he
understand the role of axiomatic theories?)
If he can't tell you what axioms he had in mind, that's a sure sign that
he doesn't know what he's talking about and a pretty damned good bet
that there's nothing much to his argument.
> If this is an accurate depiction of the UFO analogy (if not, then
> please correct me), here are my comments about it. First of all,
> there are several counterexamples to Cantor. We have theories
> such as NFU (in which non-Cantorian sets exist), zuhair's theory
> ND, as well as Lowenheim-Skolem (which isn't a theory, rather it
> proves the existence of countable models).
No one has denied that Cantor's theorem is provable in every theory.
Nor is L-S relevant, of course. You know enough mathematics to know
damned well that L-S is not a "counterexample" to Cantor's theorem.
> These aren't "UFO's" -- i.e., objects (theories) that some claim
> to have seen, but the mainstream doesn't accept as existing. NFU
> is an _actual_ theory. One can see the _actual_ post in which
> zuhair states his theory ND. L-S proves the existence of _actual_
> countable models of ZFC -- this is being discussed by others
> elsewhere in the thread.
If someone shows up and says here on sci.math, "I have a mathematical
theory and Cantor's theorem is not provable in that theory," and if they
can give a clear presentation of the axioms and argument, then *of
course* they are not treated as cranks.
None of this has anything to do with your pet project.
>> have been no more successful than the UFO parties trying to flag down
>
> At this point, it appears that you are referring to my attempts
> to rigorize not the "Cantor deniers" in this thread, but those
> posters like AP, TO, tommy1729, and possibly even MR. Each of
> these posters have not just one, but several desiderata that he
> demands be satisfied in his theory. Trying to find a rigorous
> theory in which _all_ of the desiderata are satisfied proves to
> be rather difficult. And so, returning to the analogy, one says
> that it's more likely that one will see a real live UFO than a
> rigorous theory in which all of the desiderata are satisfied.
>
> To emphasize this analogy, let's call a rigorous theory where
> all of a single such posters's desiderata are satisfied an
> "Unidentified Flying _Theory_" or UFT.
>
> Now, what should a poster do if he simply does not like a
> certain result of ZFC or mainstream theory? It seems as if
> there should be an alternative besides just simply accepting
> ZFC/mainstream theory and calling out several desiderata that
> amounts to a UFT.
Oh no! Two possibilities again! I feel a dilemma coming on.
> How about this -- a theory in which just _one_ of a poster's
> desiderata is satisfied (and isn't satisfied by ZFC or the
> mainstream theory)? And the theory should be a theory that is
> already known -- and I include zuhair's theories here since
> he's better at making theories than I am. This means that for
> the opponent of ZFC, the theory would be preferable to ZFC as
> the former proves one desideratum which fails in ZFC. But for
> those on the standard side, the theory would be preferable to
> a UFT, since this is an _actual_ theory that can be confirmed
> by looking up the theory in a book (posters on this side
> usually have books) or one of zuhair's posts. So this would
> end up being a compromise theory.
>
> Examples of the above:
>
> For the Cantor deniers in this thread, the three examples
> mentioned earlier (NFU, ND, L-S) should suffice.
I sincerely doubt that many of the cranks that you're referring to will
understand the question, but put it to them like this:
There is no doubt that ZFC proves Cantor's theorem. There is no point
in continuing to argue that the perfectly valid argument is invalid.
But there are other set theories in which the theorem is unprovable.
Here are a few. Are any of these consistent with your views?
> For posters who want infinitesimals, Robinson's hyperreals
> are a compromise theory. For the infinitesimalist, even
> though Robinson's infinitesimals aren't adjacent, surely
> having any nonzero infinitesimals at all is preferable to
> classical analysis with _no_ positive infinitesimals. For
> the mainstream, Robinson's hyperreals are known, while
> adjacent infinitesimals are a UFT.
>
> For posters like tommy1729, a mereology theory as given
> at the Stanford site. For tommy1729, surely any mereology
> at all is better than no mereology. For the mainstream,
> Stanford mereology is known since anyone can look up that
> website, while tommy1729's nonstandard three-valued logic
> makes his theory a UFT.
>
> Then, once a known theory has been established, one can
> attempt to satisfy another desideratum, one at a time,
> until the task becomes impossible (i.e., a UFT), and see
> what develops. I'd much rather do this then just simply
> accept only mainstream mathematical theories and insult
> anyone who tries to challenge them.
Yes, you must do either one or the other.
But see how it goes. Let's see if you can get a single crank to respond
intelligently to your suggestion that he "adopt" an alternative rigorous
theory.
Good luck.
--
"Yup, you guessed it. If worse comes to worse, I *will* turn to the
Army to help me with mathematicians. And then mathematicians don't
think the NSA or CIA can save your asses, as generals LIKE me."
-- James Harris's latest foray into mathematical logic.
Brian Chandler
> Transfer Principle <lwa...@lausd.net> writes:
>
> > The return of RE means that there are now several opponents of
> > Cantor here in a single thread. But unfortunately, those who
> > support Cantor (the ones you call "Cantor believers") are still
> > in the majority, since about three or so of them who haven't
> > posted in a while have also jumped into this thread.
>
> Unfortunately?
>
> If that weren't the case, then you would (by your own silly criterion)
> have to tell Russell that he's wrong.
Right. Well, TP, here's a suggestion: if you can gather together at
least four cranks, I'm sure we on the side of reason^H^H^H^H^H^H
standard analysis can agree to let no more than two (2) of us
participate, and we'll even grant you a sock puppet, a sort of
wheedling character who keeps trying to ensure that the cranks put
together some common claim. Then what will you say to those of us in
the minority?
Brian Chandler
Jesse F. Hughes
> Assume we have a list of all natural
> numbers in the unary numbering system.
>
> x = one
> xx = two
> xxx = three
> ...
>
> Create an "anti-diagonal" as follows:
>
> If the i-th natural number has n non-blank
> positions, make the first n+1 positions of
> the anti-diagonal non-blank.
>
> By construction, the diagonal is one
> position longer than some unary number
> in the list. Since the successor of a
> natural number is also a natural number,
> the anti-diagonal is a natural number
> not in the list.
You know damned well that the "since" does not follow and that the
resulting infinite list of x's is not the successor of any finite string
of x's.
I honestly believe that you're peddling arguments you *know* are
fallacious.
--
Jesse F. Hughes
"Social castigation. Their pictures in the papers. Reporters hounding
them with hard questions. And it won't end during their lifetimes."
-- Oppose James S. Harris and you get post-mortem hardball interviews
Richard Harter
<BLOCKSPA...@your-mailbox.com> wrote:
That's the essence of the matter. Godel's L is a minimal universe
satisfying ZF. L isn't the only countable model for ZF; there are
many. The trouble with the "entire conceptual power set" is that you
can't guarantee it with any finite set of axioms.
Mike Terry
news:501c03f7-b476-4e49...@a21g2000prj.googlegroups.com...
> On Mar 11, 8:18 pm, JimmyJohn <N...@Jtheist.net> wrote:
>
> > The fact that your particular antidiagonal fails does not mean that
> > anyone else's has to.
>
> I never claimed that.
>
> I claim my list contains its antidiagonal.
> I use a standard algorithm.to compute
> the anti-diagonal. In fact, it is pretty much
> the only algorithm that can be used on
> individual bits of a binary number.
>
> > If one adds a rule that the list of numbers be represented in base n,
> > with n > 3 and each antidiagonal digit excludes 0, n-1 and the digit
> > being replaced, then the antidiagonal number produced from a list will
> > NEVER be among those listed.
> >
>
> OK. I have no argument with that.
> Why do we need to make all of these additional assumptions?
> What do you have against binary numbers?
If you really want to know the answer to this question, why not just apply
the proof that the antidiagonal is different from all reals in the list to
your "false antidiagonal", and observe where it breaks down?
Regards,
Mike.
Aatu Koskensilta
> That's the essence of the matter. Godel's L is a minimal universe
> satisfying ZF. L isn't the only countable model for ZF; there are
> many.
As usually understood L, G�del's constructible universe, is not
countable -- it is a proper class. L is minimal in the sense that it
is contained in any inner model of set theory. (There is also a
countable ordinal alpha such that L_alpha is the minimal standard
model of ZF.)
> The trouble with the "entire conceptual power set" is that you can't
> guarantee it with any finite set of axioms.
What sort of guarantees do you have in mind?
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Graham Cooper
>
> If he can't tell you what axioms he had in mind, that's a sure sign that
> he doesn't know what he's talking about and a pretty damned good bet
> that there's nothing much to his argument.
IOW, "we've got a formal system and you don't".
How can you in 1 paragraph, dismiss all informal proofs of mathematics
and eliminate any possibility to show an error of formal systems
themselves?
This is standard Dicatator methodology, once in office remove the
elections.
Cantor cranks are NOT trying to prove anything about real numbers!
Cantor cranks are NOT trying to prove anything about real numbers!
Cantor cranks are NOT trying to prove anything about real numbers!
We're trying to tell you your fantastic claims are imaginary.
LISTEN FOR ONCE DUDE!
It's very evident when you consider the following status quo of
Cantor's proof:
[Jim Burns]
All I ask is that you show us such a list.
Pick any list. Tell us what it is.
Use any digit-to-digit conversion. Tell us what it is.
Make your desired anti-diagonal. Tell us what it is.
Then tell us where in your list the anti-diagonal is.
Do that, and I will agree that you have disproven
Cantor's anti-diagonal proof.
If you DON'T do that, WHY don't you? (You don't
even have to answer that question here. Just ask
yourself, "Why am I not doing something that will
give me everything I want?")
Jim Burns
--------------------------------
b e c a u s e . y o u r . a r g u m e n t
r e d u c e s . t o
SHOW ME A DIGIT ON THE DIAGONAL THAT
EQUALS D AND EQUALS D+1
---------------------------------
THAT is your entire argument!
THAT is your proof of BIGGER THAN INFINITY!
Nam Nguyen
>
> How can you in 1 paragraph, dismiss all informal proofs of mathematics
> and eliminate any possibility to show an error of formal systems
> themselves?
What does it mean for a formal system itself to have an error?
Graham Cooper
Garbage In Garbage Out!
The operator just pumping in the same naive function definitions as
informally used in mathematical discourse.
The formal system has no comprehension or representation of a number.
It's nothing more than a notebook.
There are no high level axioms on proper usage of the formal system.
It's bottom up only.
There is no proof of correctness, nor anything close.
Nam Nguyen
But, outside the context of _syntactical_ proofs of formal systems,
what does "correctness" mean?
Graham Cooper
The axioms are consistent with reality? :)
A single partition between accepted fact and not accepted untruths.
It's trivial to disprove 1=2 when Peano Postulates are included in the
inner circle.
MoeBlee
> If the diagonal method fails for a list with only
> two real numbers, why would I assume it works
> for a list with an infinite number of reals?
I addressed your confusion about that.
MoeBlee
Graham Cooper
I get a new sequence of digits with a 2 real list.
Don't see any new sequence of digits with infinite lists!
a slight shift of the argument there!
MoeBlee
>
> The identity function may be a trivial mapping,
> but there is nothing trivial about identifying the
> "set of all natural numbers".
So what? Given that there exists a set of all the natural numbers,
then there is a bijection from that set onto itself. Look, if you deny
there is a set of all natural numbers then it's silly to quibble about
an identity function, as that is not the issue but rather the
existence of the set of natural numbers itself.
> I have already shown no list contains every natural number.
No you haven't. You just continue to dogmatize that there is no such
set and then fail to support with your erroneous and confused
arguments.
> I will give my proof again.
Right, because re-posting your confusions will make them right.
> Prove the set of all natural numbers is uncountable.
>
> Assume we have a list of all natural
> numbers in the unary numbering system.
>
> x = one
> xx = two
> xxx = three
> ...
>
> Create an "anti-diagonal" as follows:
>
> If the i-th natural number has n non-blank
> positions, make the first n+1 positions of
> the anti-diagonal non-blank.
How does n depend on i?
> By construction, the diagonal is one
> position longer than some unary number
> in the list.
You said "anti-diagonal" then you say "diagonal". In any case, you've
not given any coherent construction. You have n depending on i, which
makes no sense.
Anyway, consider your list:
x
xx
xxx
...
To make a diagonal, you can use only 'x's' in it. Now, the diagonal is
an INFINTE sequence of 'x's'. But no natural number is represented by
such an INFINITE sequence. What an "anti-diagonal" here would be is
utterly unclear.
MoeBlee
MoeBlee
> numbers in decimal:
>
> t(n) = 7 if f(n)(n) not= 7
> t(n) = 8 if f(n)(n) = 7
>
> I am going to change this to:
>
> t(n) = 0 if f(n)(n) not= 0
> t(n) = 9 if f(n)(n) = 0
SO WHAT? You COMPLETELY SKIPPED my explanation about that. You won't
READ.
PLEASE go back to what I wrote about this. I SHOWED that it is
sufficient to find one method that works and that other methods don't
work does NOT refute the argument. And I specifically addressed the
matter of 9's.
READ, you fool.
MoeBlee
MoeBlee
> I am an ultra-finitist. I don't think infinite sets exist.
> Since real numbers are infinite by definition,
> I don't think they exist either.
FINE! But then there's nothing to argue about with you regarding
uncountability. All you have to say is "I don't accept that infinite
sets exist." Then I say, "Okay, suit yourself." And then all your
otherwise incorrect, illogical, confused arguments are irrelevant
anyway. If there is no set of all the naturals and no set of all the
reals, then of course it is nonsense to talk about what bijections
there are between such sets. There are no such sets so there are no
bijections between them. There are no properties of countability or
uncountability to discuss pertaining since there are no such sets of
all the natural numbers and all the real numbers.
> I am demonstrating that the diagonal argument
> creates contradictory results when applied to
> infinite sets.
No, you're demonstrating that you're an ignorant, confused fool.
> I use an algorithm from Wikipedia and produce a list
> of real numbers containing the anti-diagonal.
> No one has given any explaination why a standard
> diagonal algorithm fails to produce a unique real number
> not on my list.
I addressed this matter in DETAIL. I addressed why it is not RELEVANT.
> Saying some other algorithm works is irrelevent.
No, you SKIPPED where I SHOWED that all that is required is one method
that works.
> Why doesn't the algorithm specified by Wikipedia
> work on my set?
We don't need to have it work! How many times can this be explained to
you before a single synapse fires in your brain?
MoeBlee
MoeBlee
> > No. I am an ultra-finitist. I don't think infinite sets exist.
> > Since real numbers are infinite by definition,
> > I don't think they exist either.
>
> So why do you care about the un-countability of the reals? It's a
> theorem in another formal system that you don't accept, so your beef
> should be with the axioms that permit the reals to exit, not with the
> argument about the set's cardinality.
>
> Whether you think a set exists or not has nothing to do with the
> theorems that can be proved from some set of axioms. It may have some
> effect on what axioms you are prepared admit, but the validity of the
> argument is independent of your thoughts about the sets.
RussellE will NEVER understand that. It's been explained to him for
YEARS.
MoeBlee
Aatu Koskensilta
> RussellE will NEVER understand that. It's been explained to him for
> YEARS.
Yes, so naturally it must be explained to him again and again for
years to come, til the end of time itself.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
MoeBlee
>
> In ZF without the Axiom of Infinity, we can show
> any set that is provably an ordinal has
> a largest element.
That's a bit odd since you refer to ZF referring to provability in ZF.
But, okay. But then, so what?
MoeBlee
MoeBlee
> On Fri, 11 Mar 2011 14:40:12 -0800 (PST), MoeBlee
>
> <jazzm...@hotmail.com> wrote:
> >P.S.
>
> >On Mar 11, 4:07=A0pm, c...@tiac.net (Richard Harter) wrote:
>
> >> what has been established
> >> is that a mapping cannot be established within the theory.
>
> >To be more precise: What is established is that the theory has the
> >theorem that there does not exist such a mapping.
>
> >We don't just show that the axioms fail to prove that there is such a
> >mapping, but rather we show that axioms prove that there is no such
> >mapping.
>
> Not quite.
No, we display a proof whose last line is the formula (here rendered
in English, but to be formalized in the language of set theory) "There
is no f such that f is function from the set of natural numbers onto
the set of real numbers".
> We can show that for any model M satisfying the axioms of
> a set theory S there is no such mapping among the sets of M.
Yes. But we also show the exact theorem (aside from models, semantics,
or anything else): "There is no function from the set of natural
numbers onto the set of real numbers")
> In turn,
> for each countable model M it is straightforward to produce such
> mappings; however the mapping is not a model within M.
I don't know what you mean. I don't know what you mean by a model
within a model in this context.
> What is not
> possible is to construct a model M that is actually uncountable.
Construct given what assumptions?
Anyway, what I said originally stands, just as I said it: In Z set
theory, we prove the theorem: "There does not exist a bijection from
the set of natural numbers onto the set of real numbers." That there
are countable models of Z set theory does not contradict what I said.
MoeBlee
MoeBlee
> On 2011-03-12, RussellE <reaste...@gmail.com> wrote:
>
> > In ZF without the Axiom of Infinity, we can show any set that is
> > provably an ordinal has a largest element.
>
>
> Hint: you can't. Every theorem of ZF-Infinity is also a theorem of
> ZF.
Notice, though, that he has "provably" in the theorem itself.
MoeBlee
Daryl McCullough
>
>MoeBlee <jazz...@hotmail.com> writes:
>
>> RussellE will NEVER understand that. It's been explained to him for
>> YEARS.
>
> Yes, so naturally it must be explained to him again and again for
>years to come, til the end of time itself.
Isn't that the point of Usenet? Explaining the same things to the same
people, again and again, til the end of time?
At least, I've never seen it used for any other purpose.
--
Daryl McCullough
Ithaca, NY
MoeBlee
>
> What is the definition of an ordinal in ZF-I?
Why should you have to ask? You're bloated with opinions about set
theory but you don't even know simple basics such as the definition of
'ordinal'.
MoeBlee
MoeBlee
> So far, your UFO analogy appears to be like this: The mainstream
> doesn't accept the existence of UFO's. Therefore, when someone
> claims to have seen a UFO, the only rational response is to
> assume that the claim is mistaken or a hoax and keep in line with
> the mainstream theory, in which UFO's don't exist.
No, he didn't say that. You're completely confused about the analogy
and its suppositions.
MoeBlee
Nam Nguyen
> On Mar 13, 2:04 am, Nam Nguyen<namducngu...@shaw.ca> wrote:
>> On 12/03/2011 8:57 AM, Graham Cooper wrote:
>>
>>> On Mar 13, 1:46 am, Nam Nguyen<namducngu...@shaw.ca> wrote:
>>>> On 12/03/2011 8:31 AM, Graham Cooper wrote:
>>
>>>>> How can you in 1 paragraph, dismiss all informal proofs of mathematics
>>>>> and eliminate any possibility to show an error of formal systems
>>>>> themselves?
>>
>>>> What does it mean for a formal system itself to have an error?
>>
>>> Garbage In Garbage Out!
>>
>>> The operator just pumping in the same naive function definitions as
>>> informally used in mathematical discourse.
>>
>>> The formal system has no comprehension or representation of a number.
>>
>>> It's nothing more than a notebook.
>>
>>> There are no high level axioms on proper usage of the formal system.
>>
>>> It's bottom up only.
>>
>>> There is no proof of correctness, nor anything close.
>>
>> But, outside the context of _syntactical_ proofs of formal systems,
>> what does "correctness" mean?
>
>
> The axioms are consistent with reality? :)
But what is "reality" that any axiom has to be consistent with?
Specifically, why are you so sure that, e.g., you'd not have
"Garbage In Garbage Out" in your perception of reality (whatever
it might be)?
Marshall
wrote:
> In article <82ei6ct68o....@A166.veli3.tontut.fi>, Aatu Koskensilta says...
>
>
>
> >MoeBlee <jazzm...@hotmail.com> writes:
>
> >> RussellE will NEVER understand that. It's been explained to him for
> >> YEARS.
>
> > Yes, so naturally it must be explained to him again and again for
> >years to come, til the end of time itself.
>
> Isn't that the point of Usenet? Explaining the same things to the same
> people, again and again, til the end of time?
>
> At least, I've never seen it used for any other purpose.
Certainly it *occasionally* gets used between adults to exchange
information or correct misconceptions. But yeah, it's mostly
cranks vs. anticranks.
Marshall
Nam Nguyen
Yes. But see below.
> But yeah, it's mostly cranks vs. anticranks.
>
Mostly. Yes. But there are times it was (has been) used between
the 2 groups of anticrank: the "orthodox" and the "rebel".
LudovicoVan
wrote:
> In article <82ei6ct68o....@A166.veli3.tontut.fi>, Aatu Koskensilta says...
> >MoeBlee <jazzm...@hotmail.com> writes:
>
> >> RussellE will NEVER understand that. It's been explained to him for
> >> YEARS.
>
> > Yes, so naturally it must be explained to him again and again for
> >years to come, til the end of time itself.
>
> Isn't that the point of Usenet? Explaining the same things to the same
> people, again and again, til the end of time?
>
> At least, I've never seen it used for any other purpose.
Self-fulfilling prophecies tend to be self-fulfilling.
-LV