Cantor's "diagonal argument". My Objection.
John Jones
I don't make any argument for or against Cantor's methodology. I argue
instead that it does not tackle what it was intended to tackle.
DISCUSSION
Cantor's theorem goes something like "there are infinite sets which
cannot be put into one-to-one correspondence with the infinite set of
natural numbers".
If we helpfully translate that term 'sets' into something more
substantive, Cantor is saying that, for example, the real numbers (those
between 0 and 1, such as 0.2361...) are uncountable while the natural
numbers (any integer, such as 2361...) are countable. Another way of
saying this is that there is a different "cardinality" between the real
numbers (for example, 0.2361...) and the natural numbers (for example,
2361...). If Cantor is right, then these two types of number cannot be
put into a one to one correspondence with each other: there are more
real's than natural's.
Cantor's proof employs a made to measure 'square'. The natural numbers
are employed as an index or list where each number identifies the
position of each real number. Cantor's proof, simply put, amounts to the
idea that when we add or subtract 1 to all the digits of the real
numbers, then that new real number can't be found on the list given to
us by the natural numbers.
MY OBJECTION
My objection is simple: Cantor makes no substantive distinction between
the real numbers and the natural numbers. Either can be used to list the
other.
PROOF
No distinction is made, for example, between "0.2361..." and "2361...".
Either one can be used in a list to identify the position of the other.
The inclusion of the decimal point is a redundant pictorial artifice, a
non-mathematical glyph.
An assumption in Cantor's 'proof' is that the real's can't be used as a
list and that the natural's can be used as a list. We like to think that
decimals can't be used as a list because they start at the infinite end,
whereas the natural numbers start at the finite end and can be used as a
list. Yet there is no practical shortcoming for translating "decimals"
into the elements of an index or list. The decimal point, contributing
nothing to the list, can be dropped.
CONCLUSION
A list erases the distinction between the real's and the natural's.
Either can be used to list the other. Cardinality can be imposed equally
on the real's and the natural's, once we get over losing the empty
psychological significance of what is now the redundant decimal glyph.
Whether or not Cantor's methodology is sound, his proof fails to tackle
what he wants it to tackle and that, consequently, it shows no more than
that what goes for the real's goes for the naturals.
Max
> No distinction is made, for example, between "0.2361..." and "2361...".
> Either one can be used in a list to identify the position of the other.
> The inclusion of the decimal point is a redundant pictorial artifice, a
> non-mathematical glyph.
Except that one is intended to be infinitely long, and the other not.
No natural number can begin with 2361 and then continue _forever_,
whereas 0.23611111 etc goes on.
Lest you object to the idea of an 'infinitely long' sequence, just say
that there is one number in every decimal place corresponding to every
natural number (the 1st, 2nd, 3rd etc decimal places). You and I both
agree that there is no largest natural number, ergo, there is no last
decimal place for a real number. For an example of a number with an
infinite decimal expansion that can't be reduced to a finite one
(e.g., a real number that is not rational) look into the square root
of two.
Best,
Max
Aatu Koskensilta
> I don't make any argument for or against Cantor's methodology.
An apt summary, as incoherent blather is not an argument at all.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
george
> INTRODUCTION
> I don't make any argument for or against Cantor's methodology.
> I argue instead that it does not tackle what it was intended to tackle.
You have completely mis-
characterized the theorem.
The theorem says that EVERY set, IRrespective of size, HAS MORE
subsets than it has elements. Infinity is NOT relevant to the proof
of
the theorem. The theorem and its proof remain UNchanged EVEN if
you DENY the axiom of infinity (if you replace it with an axiom
insisting
that nothing infinite exists, or that all infinite classes are
proper).
In other words, by stating the following...
> DISCUSSION
> Cantor's theorem goes something like "there are infinite sets which
> cannot be put into one-to-one correspondence with the infinite set of
> natural numbers".
...
you are proving that YOU DON'T KNOW "What it was intended to tackle".
> If we helpfully translate that term 'sets' into something more substantive,
NOTHING is "more substantive". In point of fact, it is EVERYthing
ELSE
that can get "translated into" sets. Every natural number, for
example,
is translated into the set of all smaller natural numbers. Since no
natural
number is smaller than 0, 0 is translated into the empty set. Since
the
only natural number smaller than 1 is 0, 1 is translated into { { } }.
Ad inf.
Real numbers are translated into subsets of N via a bit-string
mapping.
> Cantor is saying that, for example, the real numbers (those
> between 0 and 1, such as 0.2361...)
It DOESN'T MATTER whether you restrict these reals to being between 0
and 1,
but if you want to, go ahead. Just don't screw up your grammar so
badly as
to imply that THE real numbers ARE numbers between 0 and 1.
> If Cantor is right, then these two types of number cannot be
> put into a one to one correspondence with each other: there are more
> real's than natural's.
This IS NOT specific to Cantor IN ANY way except that he got there
FIRST.
NOWadays this IS A THEOREM from SOME AXIOMS of set theory. In FIRST-
order logic. So for this to fail, it will not be enough for ANYthing
about CANTOR
to be wrong: LOGIC will have to be wrong.
> Cantor's proof employs a made to measure 'square'.
Well, obviously, IT HAS to be square because YOU (as OPPOSED to
Cantor)
are assuming that there is ONLY ONE size of infinity. So the list
HAS to be
infinity x infinity in size and shape.
> The natural numbers are employed as an index or list
NO, DUMBASS: The natural numbers are employed BY DEFINITION
as THE ONLY POSSIBLE index FOR EVERY list. If a thing is not indexed
by the natural numbers THEN THE THING IS NOT a list -- BY DEFINITION
OF
"list"!
> where each number identifies the position of each real number.
You forgot that IT'S SQUARE.
Each natural number ALSO identifies a COLUMN on the row indexed by
any natural number. It takes an ordered pair of natnum co-ordinates
to
tell whether a particular real-on-the-list (row) does or does not
contain
a particular natural number (i.e. is or is not zero at that particular
digit-
position WITHIN the real number).
> Cantor's proof, simply put, amounts to the
> idea that when we add or subtract 1 to all the digits of the real
> numbers, then that new real number can't be found on the list given to
> us by the natural numbers.
The list IS NOT GIVEN to us by the natural numbers!
The list IS NOT GIVEN *AT ALL*!!
The argument applies TO ANY list!
NO properties of the list (aside from its being square) are used in
the proof!
> MY OBJECTION
> My objection is simple: Cantor makes no substantive distinction between
> the real numbers and the natural numbers.
Dipshit: The inherent difference
IS BETWEEN ELEMENTS AND SUBSETS of a given set.
The powerset axiom IS AN AXIOM. EVERY set has a DIFFERENT set
of its subsets, from its elements. THAT IS an inherent difference.
Thanks for playing. Moron.
> Either can be used to list the other.
BY DEFINITION, THE NATURAL NUMBERS *ARE THE ONLY*
things that can be used to "list" ANY things IN ANY list!
Sorry you were so damn stupid you didn't know what a list was.
John Jones
>> PROOF
>> No distinction is made, for example, between "0.2361..." and "2361...".
>> Either one can be used in a list to identify the position of the other.
>> The inclusion of the decimal point is a redundant pictorial artifice, a
>> non-mathematical glyph.
>
> Except that one is intended to be infinitely long, and the other not.
> No natural number can begin with 2361 and then continue _forever_,
> whereas 0.23611111 etc goes on.
That's not right though. If that's the only distinction then it is
straightaway obvious that the reals are bigger than the naturals.
> Lest you object to the idea of an 'infinitely long' sequence, just say
> that there is one number in every decimal place corresponding to every
> natural number (the 1st, 2nd, 3rd etc decimal places).
Ok, but my point was that such a correspondence, when expressed as a
list, removes the distinction between the reals and the naturals.
> You and I both
> agree that there is no largest natural number, ergo, there is no last
> decimal place for a real number. For an example of a number with an
> infinite decimal expansion that can't be reduced to a finite one
> (e.g., a real number that is not rational) look into the square root
> of two.
But there is an unjustified asymmetry between largest and smallest: the
discrete smallest cannot be used as a list (the reals), whereas the
discrete largest can be used as a list (the naturals). Without an
explanation for this assymetry we would not be advised to favour one
over the other in the contender for an index or list.
John Jones
> On Dec 1, 4:22 pm, John Jones <jonescard...@aol.com> wrote:
>> INTRODUCTION
>> I don't make any argument for or against Cantor's methodology.
>> I argue instead that it does not tackle what it was intended to tackle.
>
> You have completely mis-
> characterized the theorem.
> The theorem says that EVERY set, IRrespective of size, HAS MORE
Yes I know. Let's drop the language of sets. It's not necessary and
tiresome.
> Infinity is NOT relevant to the proof
Good job I didn't use it then.
> of
> the theorem. The theorem and its proof remain UNchanged EVEN if
> you DENY the axiom of infinity (if you replace it with an axiom
> insisting
> that nothing infinite exists, or that all infinite classes are
I never attended to the idea of infinity even though, no doubt, a great
deal of boollocks is mentioned in regard of it elsewhere, here even.
> In other words, by stating the following...
>> DISCUSSION
>> Cantor's theorem goes something like "there are infinite sets which
>> cannot be put into one-to-one correspondence with the infinite set of
>> natural numbers".
> ...
> you are proving that YOU DON'T KNOW "What it was intended to tackle".
>> If we helpfully translate that term 'sets' into something more substantive,
>
> NOTHING is "more substantive". In point of fact, it is EVERYthing
> ELSE
> that can get "translated into" sets. Every natural number, for
> example,
> is translated into the set of all smaller natural numbers. Since no
> natural
> number is smaller than 0, 0 is translated into the empty set.
That's boring, unnecessary talk. Just drop 'sets'. If everything is a
set of everything, somewhere, somehow, then it just looks pretentious to
keep having to mention it.
> Since
> the
> only natural number smaller than 1 is 0, 1 is translated into { { } }.
> Ad inf.
There's that set thing again. Look, I will advise you. A set IS {}.
That's all it is. A shape that we say of 'oo look that's a set'. Just
drop sets. Stick to the point. Eat your greens. Use toilet paper. Stay
in your HOME!
> Real numbers are translated into subsets of N via a bit-string
> mapping.
Is that 'bit string mapping'? or 'mannered butt scratching'?
>> Cantor is saying that, for example, the real numbers (those
>> between 0 and 1, such as 0.2361...)
>
> It DOESN'T MATTER whether you restrict these reals to being between 0
> and 1,
> but if you want to, go ahead.
I said 'for example'. In your enthusiasm do not cover parts of your
screen in spit.
> Just don't screw up your grammar so
> badly as
> to imply that THE real numbers ARE numbers between 0 and 1.
Let there be no pandemonium among us, let not strange garblings scorch
our ears, inflame our souls and plunge us into committing irredeemeable
errors..
>> If Cantor is right, then these two types of number cannot be
>> put into a one to one correspondence with each other: there are more
>> real's than natural's.
>
> This IS NOT specific to Cantor IN ANY way except that he got there
> FIRST.
> NOWadays this IS A THEOREM from SOME AXIOMS of set theory. In FIRST-
> order logic. So for this to fail, it will not be enough for ANYthing
> about CANTOR
> to be wrong: LOGIC will have to be wrong.
Logic IS wrong. How else could we know that it is WE who are RIGHT!
>> Cantor's proof employs a made to measure 'square'.
>
> Well, obviously, IT HAS to be square because YOU (as OPPOSED to
> Cantor)
> are assuming that there is ONLY ONE size of infinity. So the list
> HAS to be
> infinity x infinity in size and shape.
Kellog's Cocopops are round with a hole. This makes them infinite AND
null. They make the milk go brown and the white go null. But I like my
milk to be white and stay white. How about you, Stu?
>> The natural numbers are employed as an index or list
>
> NO, DUMBASS: The natural numbers are employed BY DEFINITION
> as THE ONLY POSSIBLE index FOR EVERY list.
That makes Cantor's proof a proof by definition.
> If a thing is not indexed
> by the natural numbers THEN THE THING IS NOT a list -- BY DEFINITION
> OF
That makes Cantor's proof a proof by definition.
>> where each number identifies the position of each real number.
That makes Cantor's proof a proof by definition.
> You forgot that IT'S SQUARE.
> Each natural number ALSO identifies a COLUMN on the row indexed by
> any natural number. It takes an ordered pair of natnum co-ordinates
> to
> tell whether a particular real-on-the-list (row) does or does not
> contain
> a particular natural number (i.e. is or is not zero at that particular
> digit-
> position WITHIN the real number).
Yes, yes,
>> Cantor's proof, simply put, amounts to the
>> idea that when we add or subtract 1 to all the digits of the real
>> numbers, then that new real number can't be found on the list given to
>> us by the natural numbers.
>
> The list IS NOT GIVEN to us by the natural numbers!
> The list IS NOT GIVEN *AT ALL*!!
Then the natural numbers are employed BY DEFINITION as THE ONLY POSSIBLE
index FOR EVERY list.
> The argument applies TO ANY list!
You mean lists that are not natural numbers employed BY DEFINITION as
THE ONLY POSSIBLE index FOR EVERY list.
> NO properties of the list (aside from its being square) are used in
> the proof!
I think you were hauled up before a despotic tribal leader at some point
and asked to give an account of yourself or suffer the death meted out
to unbelievers.
>> MY OBJECTION
>> My objection is simple: Cantor makes no substantive distinction between
>> the real numbers and the natural numbers.
>
> Dipshit: The inherent difference
> IS BETWEEN ELEMENTS AND SUBSETS of a given set.
There's that sets fixation again. It makes no difference how Cantor
words it. The methodology is the same.
> The powerset axiom IS AN AXIOM. EVERY set has a DIFFERENT set
> of its subsets, from its elements. THAT IS an inherent difference.
> Thanks for playing. Moron.
Yes, yes, yes,
>> Either can be used to list the other.
>
> BY DEFINITION, THE NATURAL NUMBERS *ARE THE ONLY*
> things that can be used to "list" ANY things IN ANY list!
That makes Cantor's proof a proof by definition.
> Sorry you were so damn stupid you didn't know what a list was.
The train's gone. You missed it.
Jan Burse
>> Except that one is intended to be infinitely long, and the other not.
>> No natural number can begin with 2361 and then continue _forever_,
>> whereas 0.23611111 etc goes on.
>
> That's not right though. If that's the only distinction then it is
> straightaway obvious that the reals are bigger than the naturals.
I wouldn't say they are bigger.
They are much many more.
Look see:
pi is bigger than 3
4 is bigger than pi
There is no difference in magnitude
that can be reached for the reals
and the naturals.
But the reals are "more numerous"
that the natural numbers.
Bye
Jan Burse
Anyway JJ, you might become the next famous Obscurant,
if only you would be well educated...
http://en.wikipedia.org/wiki/Obscurantism
Bye
John Jones
You are right of course. I shouldn't say 'bigger' I should say 'more
numerous'. Or I could say 'bigger numerous'.
MoeBlee
> Cantor's proof, simply put, amounts to the
> idea that when we add or subtract 1 to all the digits of the real
> numbers, then that new real number can't be found on the list given to
> us by the natural numbers.
I don't know what that is supposed to mean.
The basis of the proof is that given a list of denumerable binary
sequences, we can flip the digits of the diagonal sequence to make an
anti-diagonal sequence that is a denumerable binary sequence not on
the given list.
MoeBlee
Ross A. Finlayson
The antidiagonal argument doesn't apply to binary (base two) expansions
(because of dual representation). Also it doesn't apply to base three,
or base one (tally marks), or a representation with an infinite alphabet.
Regards,
Ross F.
David C. Ullrich
wrote:
There is no such assumption.
You should really learn some basic English grammar, by the
way.
>and that the natural's can be used as a list. We like to think that
>decimals can't be used as a list because they start at the infinite end,
>whereas the natural numbers start at the finite end and can be used as a
>list. Yet there is no practical shortcoming for translating "decimals"
>into the elements of an index or list. The decimal point, contributing
>nothing to the list, can be dropped.
>
>CONCLUSION
>A list erases the distinction between the real's and the natural's.
>Either can be used to list the other. Cardinality can be imposed equally
>on the real's and the natural's, once we get over losing the empty
>psychological significance of what is now the redundant decimal glyph.
>Whether or not Cantor's methodology is sound, his proof fails to tackle
>what he wants it to tackle and that, consequently, it shows no more than
>that what goes for the real's goes for the naturals.
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
jesko
You are really wrong!
Natural numbers can be used to list decimal places.
And so cantor did.
There's a distinction between real and natural numbers in Cantorian
system: reals numbers are infinite objects and they are countable!
The main proof infact starts with the definition of which reals are
simply
irrational and which are trascendental.
Diagonal proof is just the popolar way of proof that different kind of
infinity
should exist! Another proof is the official one and your arguments are
not sufficiently
sound to deal with.
Thanks
Daryl McCullough
>The antidiagonal argument doesn't apply to binary (base two) expansions
>(because of dual representation).
Of course it does. You just have to take the multiple representations
into account. For example:
If r_0, r_1, ... is an infinite sequence of reals, define a new
real d between 0 and 1 as follows: (letting d[n] = the nth bit of the
binary representation of d)
1. d[2n] = 1 - r_n[2n]
2. d[4n+1] = 0
3. d[4n+3] = 1
It is clear that d does not end with all 0s or all 1s. It is also
clear that for any n, d is unequal to r_n.
>Also it doesn't apply to base three, or base one (tally marks),
>or a representation with an infinite alphabet.
Of course it does. This is one of those shibboleths to distinguish
someone minimally competent in mathematics from a complete crackpot:
Ask him: Do you think there is something fishy with Cantor's theorem?
(Similarly, Godel's Theorem).
Out of those who are minimally competent in mathematics, we can
distinguish the well-adjusted ones from the immature, impulsive
ones by saying: "Cantor's proof is nonsense" and seeing if the
person tries to convince you otherwise.
--
Daryl McCullough
Ithaca, NY
MoeBlee
wrote:
> MoeBlee wrote:
> > On Dec 1, 1:22 pm, John Jones <jonescard...@aol.com> wrote:
>
> >> Cantor's proof, simply put, amounts to the
> >> idea that when we add or subtract 1 to all the digits of the real
> >> numbers, then that new real number can't be found on the list given to
> >> us by the natural numbers.
>
> > I don't know what that is supposed to mean.
>
> > The basis of the proof is that given a list of denumerable binary
> > sequences, we can flip the digits of the diagonal sequence to make an
> > anti-diagonal sequence that is a denumerable binary sequence not on
> > the given list.
> The antidiagonal argument doesn't apply to binary (base two) expansions
> (because of dual representation).
Any denumerable sequence of denumerable binary sequences is anti-
diagonalized by exchanging 0 for 1 and 1 for 0.
> Also it doesn't apply to base three,
I'm not familiar with that.
> or base one (tally marks),
I don't know how one would represent an interval of reals in base 1.
> or a representation with an infinite alphabet.
We prove that the set of denumerable sequences of natural numbers is
great4er than the set of natural numbers.
MoeBlee
Daryl McCullough
>Any denumerable sequence of denumerable binary sequences is anti-
>diagonalized by exchanging 0 for 1 and 1 for 0.
I think Ross is worried about the fact that
0.1000...
and
0.0111...
represent the same real number in base 2. So diagonalization
of a list containing the first representation might produce
the second representation.
John Jones
>>
>> An assumption in Cantor's 'proof' is that the real's can't be used as a
>> list
>
> There is no such assumption.
Call it an index then, if you like. The reals can be used as the index
just as much as the naturals. Used as an index, there is no distinction
between the reals and the naturals.
John Jones
I never questioned Cantor's method. I said that he did not distinguish
between the naturals and the reals because both can be used as an index.
But it looks like the response I get is, once more, more arm-flapping
than tackling.
MoeBlee
wrote:
I agree that is what he's talking about. Then, as you and I agree, it
does not refute the ordinary diagonal arguments.
MoeBlee
MoeBlee
ANY set is an index set of some functions or another into any other
set. So what?
The set of real numbers is NOT the index set of a function of CERTAIN
KIND, that certain kind being a sequence (or, as it is called, "a
list"). This is simply from the DEFINITION of 'sequence' (or 'list').
MoeBlee
Arturo Magidin
>Daryl McCullough wrote:
[...]
>I never questioned Cantor's method.
That's a rather lot of empty rhetoric and arm-flapping. Of ->course<-
you are questioning his method. You are questioning his basic
assumptions that underpin the method.
>I said that he did not distinguish
>between the naturals and the reals because both can be used as an
>index.
Nonsense. The natural numbers are used as an index in their natural
order, which is a well order. There is a first natural number, and
given any collection of naturals that is not all the naturals, there
is a "smallest natural not in the collection." That makes them
particularly good as indices.
In what manner is it that you propose to use the reals as an index set
that will satisfy those conditions, which ->are<- part of the
methodology that Cantor later utilizes?
>But it looks like the response I get is, once more, more arm-flapping
>than tackling.
It looks like the response you get is not what you like, so you simply
claim they do not address your point. In point of fact, you don't
address the proof, and whenever anyone tries to steer you in its
direction, you say things like "let's drop 'sets', because that only
complicates matters." In other words, you flap your arms and you AVOID
the proof instead of "tackling it."
In other words, you seem to be the very black thing calling the
non-black thing black.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin
magidin-at-member-ams-org
jesko
Is a index finite or infinite?
Well if one can use reals as indexes then how many finite signs one
have to add to theory?
Thanks
LudovicoVan
> We prove that the set of denumerable sequences of natural numbers is
> great4er than the set of natural numbers.
We who? You cannot prove that without Cantor, can you?
-LV
MoeBlee
Anyone who wishes to apply first order logic to the axioms of Z set
theory.
MoeBlee
Daryl McCullough
>I never questioned Cantor's method. I said that he did not distinguish
>between the naturals and the reals because both can be used as an index.
What you said made no sense, whatsoever. It showed that you have no
understanding of mathematics.
0.1278...
is a perfectly legitimate real number, while
1278...
is *NOT* a natural number. A natural number has a finite number
of digits, while the decimal representation of a real number may
have an infinite number of digits.
Aatu Koskensilta
What notion of proving "with" or "without Cantor" do you have in mind?
LudovicoVan
> LudovicoVan <ju...@diegidio.name> writes:
> > On 2 Dec, 18:45, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > > We prove that the set of denumerable sequences of natural numbers
> > > is great4er than the set of natural numbers.
>
> > We who? You cannot prove that without Cantor, can you?
>
> What notion of proving "with" or "without Cantor" do you have in mind?
The OP (whichever the specific merit) is an objection to the diagonal
argument. Now, the proof or theorem (along with any specific axioms it
leverages) MoeBlee mentions above, isn't it *fundamentally* relying on
assuming Cantor's results (the diagonal argument and the theorems) to
begin with? Can such a "proof" be really used as logical evidence pro
or con the soundness and validity of Cantor's results, along with the
very existence of the uncountabile reals, and so on? I mean, if one
makes an axiom of it, I'll have to think one is just begging the
question...
Am I missing something?
-LV
John Jones
respond in kind is that it might attract people with an agenda. Don't
you think?
Reals and naturals can be regarded as sets, but my objection in this
instance is that talk of sets adds another level of problematic
interpretation and possibility for diversion, and that this is not
helpful in directly tackling the problem in hand.
> The natural numbers are used as an index in their natural
> order, which is a well order. There is a first natural number, and
> given any collection of naturals that is not all the naturals, there
> is a "smallest natural not in the collection." That makes them
> particularly good as indices.
"Being first" in an index can be represented equally by 1 and .1.
The dot is a non-mathematical glyph. That's the weak argument, which is
good enough for your purposes. We can take it further: any sign whatever
can be "first". My aunty was first in the queue at the post office. She
did not look like a 1, or a .1.
Further, just as for Cantor, you are saying (whether you intended to or
not) that the distinction we make between the naturals and the reals,
and between what is countable and what is not, depends on the property
of ordering. Refer now to 1) again.
John Jones
I don't see why it is significant to look at the diagonal the other way,
from right to left. Besides, I don't think we need a diagonal. Any
random zigzag will do just as well. Or a straight line along a column or
row will do.
John Jones
(Don't use the word set. It is confusing. If I say 'three hundred and
forty six', and then I say 'the set of three hundred and forty six', it
looks as if I am saying something different. It's also ambiguous. The
set of real numbers isn't itself a number. Just use numbers like we are
used to please)
Now, the point is, is that if either the reals or the naturals can be
used as an index for identifying a particular value of the other, then
we can't say that one is countable and the other is not - we cannot
establish a difference between them. As I said at the end of my original
post, what goes for one goes for the other.
Aatu Koskensilta
> The OP (whichever the specific merit) is an objection to the diagonal
> argument. Now, the proof or theorem (along with any specific axioms it
> leverages) MoeBlee mentions above, isn't it *fundamentally* relying on
> assuming Cantor's results (the diagonal argument and the theorems) to
> begin with?
No. The only non-logical ingredient in the argument is (predicative)
comprehension.
> Can such a "proof" be really used as logical evidence pro or con
> the soundness and validity of Cantor's results, along with the very
> existence of the uncountabile reals, and so on?
What is an "uncountable real"?
> Am I missing something?
Apparently.
John Jones
> Is a index finite or infinite?
> Well if one can use reals as indexes then how many finite signs one
> have to add to theory?
You can use any sign you like as an index, and in any combination. You
can use shapes, colours, sounds, people, naturals, reals, in any mix
whatever.
Are the naturals and the reals infinite? Are sounds, shapes and colours
infinite? Well, what's the limit we reach for making different colours?
Is there a shape-based system, apart from numbers, for making as big
an index as we like?
And can we take enough finitely populated systems to make as big an
index as we like?
John Jones
>
> You are really wrong!
> Natural numbers can be used to list decimal places.
> And so cantor did.
> There's a distinction between real and natural numbers in Cantorian
> system: reals numbers are infinite objects and they are countable!
> The main proof infact starts with the definition of which reals are
> simply
> irrational and which are trascendental.
> Diagonal proof is just the popolar way of proof that different kind of
> infinity
> should exist! Another proof is the official one and your arguments are
> not sufficiently
> sound to deal with.
>
> Thanks
I am saying that reals can list naturals and naturals can list reals.
The only difference between them is a dot.
John Jones
I hope you saw my point. I can use either 0.1278... or 1278... in an
index. For 0.1278... the decimal point is superfluous. I can keep it,
but there is no reason to keep it. For my index will be just as useful
whether I keep it or not.
John Jones
> John Jones <jonesc...@aol.com> writes:
>
>> I don't make any argument for or against Cantor's methodology.
>
>
Stick it up your arse then.
Aatu Koskensilta
I see you wish to continue not making any arguments for or against
anything -- a most commendable strategy.
Daryl McCullough
>
>Daryl McCullough wrote:
>> John Jones says...
>>
>>> I never questioned Cantor's method. I said that he did not distinguish
>>> between the naturals and the reals because both can be used as an index.
>>
>> What you said made no sense, whatsoever. It showed that you have no
>> understanding of mathematics.
>>
>> 0.1278...
>>
>> is a perfectly legitimate real number, while
>>
>> 1278...
>>
>> is *NOT* a natural number. A natural number has a finite number
>> of digits, while the decimal representation of a real number may
>> have an infinite number of digits.
>
>I hope you saw my point.
Not at all.
>I can use either 0.1278... or 1278... in an
>index.
I don't see what that has to do with Cantor's theorem.
Cantor proved that *if* you have a set of reals indexed by
natural numbers, then there is some real number that is not
in that set. You cannot index the set of all reals using the
naturals.
In contrast, if you want to index the set of all naturals using
the reals, you can certainly do that.
Daryl McCullough
>I am saying that reals can list naturals and naturals can list reals.
>The only difference between them is a dot.
No, that is not the only difference. It's been explained to you
that that is *NOT* true. A real number is represented by an *infinite*
sequence of digits, while a natural number is represented by a *finite*
sequence of digits. The dot isn't what makes the difference, it is the
infinite versus finite distinction that makes the difference.
Arturo Magidin
>A problem a person may come across in exercising their natural right to
>respond in kind is that it might attract people with an agenda.
Huh?
>Don't you think?
I certainly think. About lots of things.
>Reals and naturals can be regarded as sets, but my objection in this
>instance is that talk of sets adds another level of problematic
>interpretation and possibility for diversion, and that this is not
>helpful in directly tackling the problem in hand.
In short, you'd rather keep things mushy so you can push through your
miscomprehensions. Gotcha.
> > The natural numbers are used as an index in their natural
> > order, which is a well order. There is a first natural number, and
> > given any collection of naturals that is not all the naturals, there
> > is a "smallest natural not in the collection." That makes them
> > particularly good as indices.
>
>"Being first" in an index can be represented equally by 1 and .1.
No, the point is that there is a natural well ordering to the
naturals. This defines an ordering among ALL the naturals, that allows
you to say which natural goes in which position, which natural follows
which natural, etc. It is also a natural order that satisfies
induction and recursion.
The same is not true of the real numbers. They do not have a natural
ordering that covers all the real numbers and is a well ordering.
If you wish the come up with one, go ahead. Be sure it satisfies ALL
the required properties, including satisfying induction and
recursion. Just telling me what goes "first" does not do it. And if
you didn't know it, perhaps you should start ->there<- before alleging
that "you can just as well use the reals" as an index set.
>The dot is a non-mathematical glyph. That's the weak argument, which is
>good enough for your purposes.
Actually, you've presented no argument, weak or otherwise. You failed
to address the point. Lots of flailing of arms instead of tackling, to
use your wording.
> We can take it further: any sign whatever
>can be "first".
Goody! And how does that establish a complete well ordering of the
reals that satisfies the same properties as the one for the natural
number,s in particular recursion/induction, which are required for the
proof? On what grounds could you possibly allege that your gyrations
and arm flailing here in any way whatsoever could possibly be
interpreted by a thinking person as providing an ordering of the real
numbers that satisfies these necessary properties that the ordering of
the natural numbers have?
>My aunty was first in the queue at the post office. She
>did not look like a 1, or a .1.
>Further, just as for Cantor, you are saying (whether you intended to or
>not) that the distinction we make between the naturals and the reals,
>and between what is countable and what is not, depends on the property
>of ordering. Refer now to 1) again.
No, sillypants. What I pointed out is that the natural numbers come
"equipped" with an order, this order being part and parcel of the
set. This order is part of WHY we use them as indices. YOU,
sillyputty, claim that you could use the reals "just as well" for
indices. But in order for that claim to hold water, there should be
some inherent or equippable order to the reals that allow us to use
them for indices "just as well" as the natural numbers. Refering me to
(1) just proves that you have no justification for the assertion that
you can do "just as well" with the real as indices, and that your
probably know it.
MoeBlee
> MoeBlee wrote:
> > On Dec 1, 1:22 pm, John Jones <jonescard...@aol.com> wrote:
> > The basis of the proof is that given a list of denumerable binary
> > sequences, we can flip the digits of the diagonal sequence to make an
> > anti-diagonal sequence that is a denumerable binary sequence not on
> > the given list.
> I don't see why it is significant to look at the diagonal the other way,
> from right to left. Besides, I don't think we need a diagonal. Any
> random zigzag will do just as well. Or a straight line along a column or
> row will do.
The word 'diagonal' is merely for convenience of informal
visualization. No part of the actual proof requires mention of a
'diagonal'. Wherever 'the diagonal function' occurs in such
discussions, you can substitute '{<n f(n)(n)> | new} where f is the
given enumeration of a set of denumerable binary sequences.
MoeBlee
MoeBlee
> MoeBlee wrote:
> > On Dec 2, 11:37 am, John Jones <jonescard...@aol.com> wrote:
> >> David C. Ullrich wrote:
>
> >>>> An assumption in Cantor's 'proof' is that the real's can't be used as a
> >>>> list
> >>> There is no such assumption.
> >> Call it an index then, if you like. The reals can be used as the index
> >> just as much as the naturals. Used as an index, there is no distinction
> >> between the reals and the naturals.
>
> > ANY set is an index set of some functions or another into any other
> > set. So what?
>
> > The set of real numbers is NOT the index set of a function of CERTAIN
> > KIND, that certain kind being a sequence (or, as it is called, "a
> > list"). This is simply from the DEFINITION of 'sequence' (or 'list').
> (Don't use the word set.
If I'm speaking of assertions of set theory, then it matters not
whether I refer to objects as sets or merely as objects.
> It is confusing. If I say 'three hundred and
> forty six', and then I say 'the set of three hundred and forty six',
And I didn't say that. Nothing I said was confusing in my post.
> it
> looks as if I am saying something different. It's also ambiguous. The
> set of real numbers isn't itself a number. Just use numbers like we are
> used to please)
>
> Now, the point is, is that if either the reals or the naturals can be
> used as an index for identifying a particular value of the other,
I don't know what you mean by "for identifying a particular value of
the other".
An index set is a domain of a function. The indices are the elements
of the domain. The indexed set is the range of the function. The
members of the indexed set are indexed by the indices.
> then
> we can't say that one is countable and the other is not - we cannot
> establish a difference between them. As I said at the end of my original
> post, what goes for one goes for the other.
Why don't you start with the definition of 'is countable'?
MoeBlee
Aatu Koskensilta
> No part of the actual proof requires mention of a
> 'diagonal'. Wherever 'the diagonal function' occurs in such
> discussions, you can substitute '{<n f(n)(n)> | new} where f is the
> given enumeration of a set of denumerable binary sequences.
But how is this 'new' given to us? Is it a part of the framework, a
constituent of the matrix, or does it emerge from what informs the
whole as a linguistic expression of an index, something represented to
us?
MoeBlee
> MoeBlee <jazzm...@hotmail.com> writes:
> > No part of the actual proof requires mention of a
> > 'diagonal'. Wherever 'the diagonal function' occurs in such
> > discussions, you can substitute '{<n f(n)(n)> | new} where f is the
> > given enumeration of a set of denumerable binary sequences.
>
> But how is this 'new' given to us? Is it a part of the framework, a
> constituent of the matrix, or does it emerge from what informs the
> whole as a linguistic expression of an index, something represented to
> us?
Always the given is transient in the pattern of understanding that is
inseparable from immutability as expression that is both coursed in
the sign and witnessed by prior context. Isn't that obvious?
MoeBlee
george
> > characterized the theorem.
> > The theorem says that EVERY set, IRrespective of size, HAS MORE
> > subsets than it has elements. (sets)
>
JJ> Yes I know. Let's drop the language of sets.
>It's not necessary and
> tiresome.
IT IS SO TOO necessary, DUMBASS.
The theorem IS ABOUT SETS!
The theorem compares the number of ELEMENTS of a SET
to the number of SUBSETS of THE SAME SET!
You CAN'T even STATE this theorem IN ANY other context!
I suppose that if you put order back into it and talked about lists
instead of sets, then you would get n! "sublists" as opposed to 2^n
subSETS,
of a length-n list, but at a BARE minimum, you would HAVE TO KEEP
ENOUGH
machinery to distinguish between "individual" elements of an
"aggregate"
and "smaller sub"- aggregates of "an aggregate" -- THESE ARE JUST SETS
BY ANOTHER name!! Since we ALREADY HAVE a name, you REALLY DO
have to USE it INSTEAD of trying to re-invent the wheel. There is
perhaps
a more general concept called "monad" that you can define like this:
(map f) m ≡ m >>= (\x -> return (f x))
join m ≡ m >>= (\x -> x)
m >>= f ≡ join ((map f) m)
Here, "join" is like set theory's unary Union operator and map is sort
of like the axiom of Replacement.
But YOU canNOT "throw away" sets here because the relationship
between these two modes or part-hood or combining/collecting
IS WHAT THE THEOREM IS ABOUT! It is amazing that you started by
alleging that the theorem didn't tackle what it was "aiming at" when
YOU DIDN'T KNOW SHIT about WHAT it was aiming at!
>
> > Infinity is NOT relevant to the proof
>
> Good job I didn't use it then.
You DID SO TOO, dumbass! YOU characterized the theorem this way:
> >> Cantor's theorem goes something like "there are infinite sets which
> >> cannot be put into one-to-one correspondence with the infinite set of
> >> natural numbers".
That is just 1 instance.
The proof is not about that; the proof is about the
GENERAL case!
> >> If we helpfully translate that term 'sets' into something more substantive,
YOU HAVE NO IDEA WHAT you might "translate sets" into!
The truth in any case IS ALWAYS IN THE OPPOSITE direction!
It is RATHER than everything ELSE gets translated INTO SETS!
Sets are "the assembly language" of math-as-we-know-it!
>
> > NOTHING is "more substantive". In point of fact, it is EVERYthing
> > ELSE
> > that can get "translated into" sets. Every natural number, for
> > example,
> > is translated into the set of all smaller natural numbers. Since no
> > natural
> > number is smaller than 0, 0 is translated into the empty set.
>
> That's boring, unnecessary talk. Just drop 'sets'.
You CAN'T just drop sets. The theorem SAYS that the number of
ELEMENTS
of the set is LESS than the number of SUBSETS of the set. It's ALL
AND ONLY
ABOUT sets!!
> If everything is a
> set of everything, somewhere, somehow, then it just looks pretentious to
> keep having to mention it.
Everything is NOT necessarily a set -- just everything THAT YOU CAN
APPLY
CANTOR'S THEOREM TO.
Cantor's theorem applies to sets of ur-elements even though those
can't
exist in ZFC.
> There's that set thing again. Look, I will advise you.
No. You can't., because YOU'RE STUPID.
> A set IS
Shut the FUCK up! You DON'T know!
A set between US is whatever OUR agreed axioms SAY it is!
If we have NOT yet agreed on some axioms THEN YOU HAVE
NOTHING TO SAY!
> >> If Cantor is right, then these two types of number cannot be
> >> put into a one to one correspondence with each other: there are more
> >> real's than natural's.
> Logic IS wrong.
How would YOU know???
> How else could we know that it is WE who are RIGHT!
We WHO, dumbass?? YOU *don't* know!
ocopops are round with a hole. This makes them infinite AND
> > The natural numbers are employed BY DEFINITION
> > as THE ONLY POSSIBLE index FOR EVERY list.
>
> That makes Cantor's proof a proof by definition.
ALL PROOFS are proofs by definition, DUMBASS!
The axioms FROM which every proof proves its conclusion
ARE THE DEFINITIONS of the terms used in the proof!
YOU'RE A PHILOSOPHY STUDENT AND YOU DIDN'T KNOW THIS???
>
> > The argument applies TO ANY list!
>
> You mean lists that are not natural numbers employed BY DEFINITION as
> THE ONLY POSSIBLE index FOR EVERY list.
That is ungrammatical and incoherent. I mean exactly what I said:
ALL LISTS. PERIOD. BY DEFINITION.
> >> MY OBJECTION
> >> My objection is simple: Cantor makes no substantive distinction between
> >> the real numbers and the natural numbers.
>
> > Dipshit: The inherent difference
> > IS BETWEEN ELEMENTS AND SUBSETS of a given set.
>
> There's that sets fixation again. It makes no difference how Cantor
> words it. The methodology is the same.
THERE IS a substantive difference between elements and subsets
of the same set. YOU DO *NOT* have any sort of "alternative
methodology".
The methodology "is the same" because EVERYBODY IS ALWAYS using sets,
WHETHER THEY LIKE IT OR NOT!
Ross A. Finlayson
> Ross A. Finlayson says...
>
>> The antidiagonal argument doesn't apply to binary (base two) expansions
>> (because of dual representation).
>
> Of course it does. You just have to take the multiple representations
> into account. For example:
>
> If r_0, r_1, ... is an infinite sequence of reals, define a new
> real d between 0 and 1 as follows: (letting d[n] = the nth bit of the
> binary representation of d)
>
> 1. d[2n] = 1 - r_n[2n]
> 2. d[4n+1] = 0
> 3. d[4n+3] = 1
>
That's not "the" "antidiagonal". You can call it "Daryl's Cantor's" but
it's not "antidiagonal".
To avoid the dual representation, the argument could be phrased in terms
of "natural integers to binary-coded powerset of natural integers".
Then arguing against that would involve arguing against the fundamental
axiomatization of omega as regular. That would be in a theory
sufficient to prove metatheoretical statements about ZF, which would
obviously be a supertheory of ZF.
"The" "antidiagonal" of a list of binary expansions representing reals
is _not_ necessarily distinct from each, similarly the case for base
three doesn't hold, in terms of what should be the most simple argument.
In the cases of a single or infinite alphabet, the numbers aren't
represented as expansions (re the sufficiency of Eudoxus/Dedekind/Cauchy).
> It is clear that d does not end with all 0s or all 1s. It is also
> clear that for any n, d is unequal to r_n.
>
>> Also it doesn't apply to base three, or base one (tally marks),
>> or a representation with an infinite alphabet.
>
> Of course it does. This is one of those shibboleths to distinguish
> someone minimally competent in mathematics from a complete crackpot:
> Ask him: Do you think there is something fishy with Cantor's theorem?
> (Similarly, Godel's Theorem).
>
Do you think there's anything wrong with Euclidean geometry? How about
non-Euclidean?
> Out of those who are minimally competent in mathematics, we can
> distinguish the well-adjusted ones from the immature, impulsive
> ones by saying: "Cantor's proof is nonsense" and seeing if the
> person tries to convince you otherwise.
>
> --
> Daryl McCullough
> Ithaca, NY
>
The antidiagonal argument isn't nonsense, per se, even the layman can
grasp its presentation easily.
Actually constructing a counterexample, even beyond the standard, a
mathematical construct, particularly when it's usable a mathematical
primitive, of course the non-mathematician would not do, except to the
extent that it would be implicit in the actual foundations.
Then, where there actually _is_ that object, understanding it in light
of the standard takes a little more consideration than most would
proffer, particularly where it interferes with their established belief
system. (Irrationals were _apeiron_.)
(ZF is incomplete else it's inconsistent, ZF's universe contains itself.)
Many or most users of mathematics really don't care how countable
additivity (particularly and always countable additivity) in standard
measure theory is used to justify their results. There aren't
applications of transfinite cardinals, where concrete mathematics is
replete with them.
EF is a CDF. Well-order the reals, draw a straight line.
Daryl, that's not "the antidiagonal." Mathematicians generally
appreciate specificity and consistency in definition.
Regards,
Ross F.
herbzet
John Jones wrote:
> MoeBlee wrote:
You're right that a diagonal pattern is not of the essence -- it's just
convenient. Not any random zig-zag will do, but some will. A straight
line along a a column "or row" (a-heh-heh-heh) is (like the count of five
for the holy hand-grenade) right out.
--
hz
John Jones
>>> The natural numbers are used as an index in their natural
>>> order, which is a well order. There is a first natural number, and
>>> given any collection of naturals that is not all the naturals, there
>>> is a "smallest natural not in the collection." That makes them
>>> particularly good as indices.
>> "Being first" in an index can be represented equally by 1 and .1.
>
> No,
You mean 'yes'. I stipulate that the representation of all naturals be
preceded by a dot for each natural. I then use this as an index. This
index is as ordered and as useful as the index made without a dot, and
both indexes are the same size.
> the point is that there is a natural well ordering to the
> naturals. This defines an ordering among ALL the naturals, that allows
> you to say which natural goes in which position,
I can do the same for the reals. 0.23242... can come before 0.23241 if
you are using size as a criterion of ordering.
I ignore your idea that a correct position can be defined without a
number ("an ordering among ALL the naturals, that allows
you to say which natural goes in which position").
> which natural follows
> which natural, etc. It is also a natural order that satisfies
> induction and recursion.
The criterion of an order can be anything we stipulate. It could be
larger to smaller, left to right, it could be genetic ie. from act to
act - where the creation of one number depends on another, etc. But
there is no order given as 'natural', and certainly no order coming from
a correct 'position'.
> The same is not true of the real numbers. They do not have a natural
> ordering that covers all the real numbers and is a well ordering.
You are missing the point. I can use the reals just as well as the
naturals for an index. All I have to do is put a dot in front of the
reals. Your argument is also circular. I have to assume that there is a
natural ordering that covers all the reals if I want to present a proof
that shows it.
> If you wish the come up with one, go ahead. Be sure it satisfies ALL
> the required properties, including satisfying induction and
> recursion. Just telling me what goes "first" does not do it.
That's a strange thing to say. Are you saying that there is an absolute,
or correct, idea of 'first' and that only the naturals have it as a
property? And where are these correct positions you referred to, that
are filled by the natural numbers? Why not just use these correct
positions, and forget about filling them with the natural numbers?
>> We can take it further: any sign whatever
>> can be "first".
>
> Goody! And how does that establish a complete well ordering
In order to present a proof or demonstration that the naturals are
well-ordered, it doesn't help if the proof depends on the fact that the
naturals are so ordered.
> there should be
> some inherent or equippable order to the reals that allow us to use
> them for indices "just as well" as the natural numbers.
For any sequence identified as a natural I can present the same sequence
and identify it as a real, and vice versa. All I need do is dispose
of, or insert, the decimal glyph ".".
David C. Ullrich
wrote:
>David C. Ullrich wrote:
>
>>>
>>> An assumption in Cantor's 'proof' is that the real's can't be used as a
>>> list
>>
>> There is no such assumption.
>
>Call it an index then, if you like. The reals can be used as the index
>just as much as the naturals. Used as an index, there is no distinction
>between the reals and the naturals.
Whatever the heck that means, it has no bearing whatever on the
valididty of the proof that there is no function mapping the natural
numbers onto the reals.
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
John Jones
>
> Cantor proved that *if* you have a set of reals indexed by
> natural numbers, then there is some real number that is not
> in that set. You cannot index the set of all reals using the
> naturals.
>
> In contrast, if you want to index the set of all naturals using
> the reals, you can certainly do that.
Two things then. The first is that the properties of well-ordering and
countability are not interdependent. The second is that using the reals
to index the naturals should, by Cantor's methodology, show that there
is some natural number that is not indexed by the reals.
Daryl McCullough
>
>Daryl McCullough wrote:
>> Ross A. Finlayson says...
>>
>>> The antidiagonal argument doesn't apply to binary (base two) expansions
>>> (because of dual representation).
>>
>> Of course it does. You just have to take the multiple representations
>> into account. For example:
>>
>> If r_0, r_1, ... is an infinite sequence of reals, define a new
>> real d between 0 and 1 as follows: (letting d[n] = the nth bit of the
>> binary representation of d)
>>
>> 1. d[2n] = 1 - r_n[2n]
>> 2. d[4n+1] = 0
>> 3. d[4n+3] = 1
>>
>
>That's not "the" "antidiagonal". You can call it "Daryl's Cantor's" but
>it's not "antidiagonal".
It's the antidiagonal that one would get if one took the original
list
r_0
r_1
r_2
...
and interspersed some extra representations of reals:
r_0
.0111...
.1000...
r_1
.0111...
.1000...
r_2
.0111...
.1000...
r_3
...
Cantor's diagonal argument is not directly about real numbers, it
is about bitstrings (functions from N to {0,1}). If you have an
infinite sequence of bitstrings, then there is a bitstring that
is not on the list. Thus, the set of bitstrings is uncountable.
This can be used to prove that the set of reals is uncountable.
John Jones
> The theorem IS ABOUT SETS!
That just mudies up the concepts. Let's just use the words real numbers
and natural numbers, sunshine.
> The theorem compares the number of ELEMENTS of a SET
> to the number of SUBSETS of THE SAME SET!
YEs, yes,
> You CAN'T even STATE this theorem IN ANY other context!
Cantor showed that for every given infinite sequence of real numbers it
is possible to construct a real number that is not on that list.
Consequently, it is impossible to enumerate the real numbers; they are
uncountable. No generality is lost if we suppose that all the numbers on
the list are between 0 and 1. Certainly, if this subset of the real
numbers in uncountable, then the full set is uncountable as well.
Let us write our sequence as a table of decimal expansions:
etcetera
etcetera etcetera
> I suppose that if you put order back into it and talked about lists
> instead of sets, then you would get n! "sublists" as opposed to 2^n
> subSETS,
How are you going make an index with a jumbled collection or set of signs?
> But YOU canNOT "throw away" sets here because the relationship
> between these two modes or part-hood or combining/collecting
> IS WHAT THE THEOREM IS ABOUT! It is amazing that you started by
> alleging that the theorem didn't tackle what it was "aiming at" when
> YOU DIDN'T KNOW SHIT about WHAT it was aiming at!
I don't need to say anything about sets to understand Cantor. Cantor's
methodology has nothing to do with sets. Call a number a set if you
like, but its pointless and pedantic.
>
>>> Infinity is NOT relevant to the proof
>> Good job I didn't use it then.
>
> You DID SO TOO, dumbass! YOU characterized the theorem this way:
>
>>>> Cantor's theorem goes something like "there are infinite sets which
>>>> cannot be put into one-to-one correspondence with the infinite set of
>>>> natural numbers".
I never used the idea of infinity. I don't care whether it's right or
wrong.
>
>>>> If we helpfully translate that term 'sets' into something more substantive,
>
> YOU HAVE NO IDEA WHAT you might "translate sets" into!
> The truth in any case IS ALWAYS IN THE OPPOSITE direction!
> It is RATHER than everything ELSE gets translated INTO SETS!
> Sets are "the assembly language" of math-as-we-know-it!
Sets is just doofing a cap to convention. There is no mathematical
statements about sets that cannot be said better without it.
>>> NOTHING is "more substantive". In point of fact, it is EVERYthing
>>> ELSE
>>> that can get "translated into" sets. Every natural number, for
>>> example,
>>> is translated into the set of all smaller natural numbers.
Why bother? Why not use the number itself?
Just say 'all smaller natural numbers'. Why say 'the set of all smaller
natural numbers'? Do you want me to put the word 'set' in front of
everything that looks mathematical?
>>> Since no
>>> natural
>>> number is smaller than 0, 0 is translated into the empty set.
So what? Irrelevant. Just use 0. It looks like you are trying to justify
the use of sets in awkward circumstances like 0. Shutup about sets.
>> That's boring, unnecessary talk. Just drop 'sets'.
>
> You CAN'T just drop sets. The theorem SAYS that the number of
> ELEMENTS
> of the set is LESS than the number of SUBSETS of the set.
Christmas wrapping. It doesn't even look right. Why not say that the
number of permutations of any number is more than the number permutated.
Which is self-evident without Cantor.
>
>> If everything is a
>> set of everything, somewhere, somehow, then it just looks pretentious to
>> keep having to mention it.
>
> Everything is NOT necessarily a set -- just everything THAT YOU CAN
> APPLY
> CANTOR'S THEOREM TO.
> Cantor's theorem applies to sets of ur-elements even though those
> can't
> exist in ZFC.
Whats a decimal got to do with a set? Look - nought point one. Why say
the set of nought point one? Why say the elements of the set of nought
pont one. Just say the digits of nought point one if you have to, but
why? Or just say the elements of nought point one.
Translating the decimal and the natural into a set makes it look as
though we are talking about the same sort of thing.
>
>>> The natural numbers are employed BY DEFINITION
>>> as THE ONLY POSSIBLE index FOR EVERY list.
>> That makes Cantor's proof a proof by definition.
>
> ALL PROOFS are proofs by definition, DUMBASS!
The proof is right (or it wouldn't be a proof) because Cantor defined
its terms such that it couldn't be wrong.
>> There's that sets fixation again. It makes no difference how Cantor
>> words it. The methodology is the same.
>
> THERE IS a substantive difference between elements and subsets
> of the same set. YOU DO *NOT* have any sort of "alternative
> methodology".
There were no sets in Cantor's methodology. All he used was digits and
the decimal point. Not a precious set in sight, whatever that is.
Daryl McCullough
>
>Daryl McCullough wrote:
>
>>
>> Cantor proved that *if* you have a set of reals indexed by
>> natural numbers, then there is some real number that is not
>> in that set. You cannot index the set of all reals using the
>> naturals.
>>
>> In contrast, if you want to index the set of all naturals using
>> the reals, you can certainly do that.
>
>Two things then. The first is that the properties of well-ordering and
>countability are not interdependent.
Right, any countable set can be well-ordered.
>The second is that using the reals
>to index the naturals should, by Cantor's methodology, show that there
>is some natural number that is not indexed by the reals.
That's completely wrong. Look at the proof
of Cantor's theorem, and you'll see that there is a key step that
does not work if you try to index the naturals using the reals.
1. Let S be a set of reals, indexed by the naturals.
2. Let r_n be that element of S with index n.
3. If r is a real, then let r[m] be digit number m in a
decimal representation of the fractional part of r.
4. Lemma: There is a real number d such that for all
naturals n, if r_n[n] = 1, then d[n] = 2, and
otherwise, d[n] = 1.
5. Claim: There is no real r_n such that r_n = d.
Now, try to do the same thing with naturals indexed by reals.
1. Let S be a set of naturals, indexed by the reals.
2. Let n_r be that element of S with index r.
3. If n is a natural, then let n[r] = digit number r in
the decimal representation of n.
4. FALSE LEMMA: There is a natural number d such that forall
reals r, if n_r[r] = 1, then d[r] = 2, and otherwise,
d[r] = 1.
Lemma number 4 is *FALSE*. There is no natural number
having an infinite number of nonzero digits in its
representation.
John Jones
Yes.
How about using a circular list of numbers in Cantor's square? You see,
it's a bit odd that we have to continue the diagonal, zigzag, whatever,
to a place where we lose sight of it. I say let's keep sight of our
operations and do a big circle that's in full view. But then Cantor's
demonstration wouldn't work.
And the point is, no matter how big we make the circle, Cantor's proof
will never work for it. In which case, we must ask ourselves what we are
doing different for a diagonal.
John Jones
What about a circle of any size? Saying that a certain shape won't do
for Cantor puts his proof under some pressure does it not?
jesko
> jeskowrote:
>
> - Show quoted text -
No! Every natural is finite and never infinite.
Is Euler constant finite?
Thanks
John Jones
Then te ought to be a reason why the function for real numbers can't be
used as an index in the same way that a function for the reals can.
> Why don't you start with the definition of 'is countable'?
>
> MoeBlee
I'm happy with the one that Cantor used. Although it shouldn't matter
what the definition is because I was saying that whatever the naturals
are, so are the reals, in that either can be used just as well for an index.
John Jones
> The dot isn't what makes the difference, it is the
> infinite versus finite distinction that makes the difference.
>
Why did Cantor go to the trouble of the diagonal argument if this was
already known?
LudovicoVan
Geniuses at work.
-LV
LudovicoVan
> LudovicoVan <ju...@diegidio.name> writes:
> > The OP (whichever the specific merit) is an objection to the diagonal
> > argument. Now, the proof or theorem (along with any specific axioms it
> > leverages) MoeBlee mentions above, isn't it *fundamentally* relying on
> > assuming Cantor's results (the diagonal argument and the theorems) to
> > begin with?
>
> No. The only non-logical ingredient in the argument is (predicative)
> comprehension.
Which argument?
> > Can such a "proof" be really used as logical evidence pro or con
> > the soundness and validity of Cantor's results, along with the very
> > existence of the uncountabile reals, and so on?
>
> What is an "uncountable real"?
Uncountable real_s_.
> > Am I missing something?
>
> Apparently.
Apparent is your denial.
-LV
Arturo Magidin
>Arturo Magidin wrote:
>
>>>> The natural numbers are used as an index in their natural
>>>> order, which is a well order. There is a first natural number, and
>>>> given any collection of naturals that is not all the naturals, there
>>>> is a "smallest natural not in the collection." That makes them
>>>> particularly good as indices.
>>> "Being first" in an index can be represented equally by 1 and .1.
>>
>> No,
>
>You mean 'yes'.
Now you are being dishonest.
I said "no", I meant "no", and the fact that this is not the end of
the sentence (marked by a COMMA, not a period), indicates that the
next part should not be ommitted before making a comment.
>> the point is that there is a natural well ordering to the
>> naturals. This defines an ordering among ALL the naturals, that allows
>> you to say which natural goes in which position,
See?
> I stipulate that the representation of all naturals be
>preceded by a dot for each natural. I then use this as an index.
This does NOT suffice for using ALL the reals as indices, so your
attempt fails, which is why I said "no".
Duh.
> This
>index is as ordered and as useful as the index made without a dot, and
>both indexes are the same size.
But this "index" is NOT "the reals". It's not even "the reals between
0 and 1." So you have failed miserably at making your case that you
can just as well use "the reals" as the naturals for an index set.
As such, that's it. You lose.
>I can do the same for the reals. 0.23242... can come before 0.23241 if
>you are using size as a criterion of ordering.
>I ignore your idea that a correct position can be defined without a
>number ("an ordering among ALL the naturals, that allows
>you to say which natural goes in which position").
You ignore anything you don't like, so long as it allows you to
continue pushing your nonsense.
>> which natural follows
>> which natural, etc. It is also a natural order that satisfies
>> induction and recursion.
>
>The criterion of an order can be anything we stipulate.
No, sillypants. YOU claimed you could use the reals just as well as
you could use the naturals. The naturals have a bunch of properties
that are important. The reals do not have them. You cannot now say
"Oh, well, I meant that if we change all the rules then we could use
them just as well". Either we can use them just as well, AS IS, or we
can't.
Apparently, we can't. Thanks for realizing that.
> It could be
>larger to smaller, left to right, it could be genetic ie. from act to
>act - where the creation of one number depends on another, etc. But
>there is no order given as 'natural', and certainly no order coming from
>a correct 'position'.
>
>> The same is not true of the real numbers. They do not have a natural
>> ordering that covers all the real numbers and is a well ordering.
>
>You are missing the point.
No, YOU are missing the point.
> I can use the reals just as well as the
>naturals for an index.
No, you can't. You aren't using the reals, you are using only SOME
reals. As such, your argument cannot even get off the ground.
> All I have to do is put a dot in front of the
>reals.
Huh?
> Your argument is also circular.
No, you have no argument, and you were unable to justify your
claim. There was no circularity anywhere, there was just the fact that
you could not even get started.
>I have to assume that there is a
>natural ordering that covers all the reals if I want to present a proof
>that shows it.
No, you have to SHOW ME that you can use the reals "just as well" as
the naturals, by showing me that you can actually use them just as
well. Just saying "I'll use them" is no good. The natural numbers come
well ordered and satisfying recursion, the reals do not. If you want
to use them "just as well", you better make sure the DO just as
well. They don't.
Your entire argument is essentially: "I get in my car to go to work
every morning. But I could just as well get in my bed; nothing special
about my car other than the fact that I'm getting into it."
>
>> If you wish the come up with one, go ahead. Be sure it satisfies ALL
>> the required properties, including satisfying induction and
>> recursion. Just telling me what goes "first" does not do it.
>
>That's a strange thing to say.
No, it is a perfectly reasonable thing to say.
>Are you saying that there is an absolute,
>or correct, idea of 'first' and that only the naturals have it as a
>property?
Now you are just being an idiot on purpose. I am saying that the
naturals come equipped with a natural order that we are familiar with
and which has a number of properties we are all familiar with. The
reals do not have an order we all agree on that has those same
properties. If you claim that you can use the reals "just as well" as
you can use the naturals, then I want them to be "just as good": I
want the reals to have some order that has all those familiar
properties. I know what that order is for the naturals, I do not know
what it is for the reals. You can certainly define a non-standard
ordering, but you have to DEFINE it. I don't know what it is. And
defining an order for the reals does not mean simply telling me ".1
goes first" (which is ALL you did before). You need to say more.
That was perfectly clear from what I said. The way you twisted it is
yet more dishonesty on your part.
>And where are these correct positions you referred to, that
>are filled by the natural numbers? Why not just use these correct
>positions, and forget about filling them with the natural numbers?
>
>>> We can take it further: any sign whatever
>>> can be "first".
>>
>> Goody! And how does that establish a complete well ordering
>
>In order to present a proof or demonstration that the naturals are
>well-ordered, it doesn't help if the proof depends on the fact that the
>naturals are so ordered.
And again you interrumpt my sentence in the middle in order to be an
idiot.
You know what? Go ahead and continue to be a lying idiot. But don't do
it in a logic newsgroup.
Arturo Magidin
>george wrote:
>
>> The theorem IS ABOUT SETS!
>
>That just mudies up the concepts. Let's just use the words real numbers
>and natural numbers, sunshine.
No, it CLARIFIES them. That's why you have to stay away from
them. Because as soon as everything is clear and solid, you run
straight into a wall. Instead, it is better to keep everything
shrouded in a fog so that mushy arguments can seem solid.
Talking about engines and gears just muddies the concepts. Let's keep
them out. The point is, if I can get to work by getting in my car, I
don't see why I can't get to work by getting in my bed. After all, the
point is getting in, isn't it?
Daryl McCullough
It was known (that the cardinality of the set of infinite sequences
of digits is greater than the cardinality of the set of finite
sequences of digits) because Cantor proved it. It wasn't known
*before* he proved it.
Daryl McCullough
>
>george wrote:
>
>> The theorem IS ABOUT SETS!
>
>That just mudies up the concepts.
The evidence is that you are exactly backwards. Those
who understand sets understand Cantor's theorem perfectly
well, and those (such as you) who don't understand sets
are completely confused about Cantor's theorem. Your
statements about Cantor's theorem are complete nonsense,
and you would realize that if you were a little more
competent at the basic skills of mathematics and logic.
LudovicoVan
>
> >> Cantor's proof, simply put, amounts to the
> >> idea that when we add or subtract 1 to all the digits of the real
> >> numbers, then that new real number can't be found on the list given to
> >> us by the natural numbers.
>
> > I don't know what that is supposed to mean.
>
> > The basis of the proof is that given a list of denumerable binary
> > sequences, we can flip the digits of the diagonal sequence to make an
> > anti-diagonal sequence that is a denumerable binary sequence not on
> > the given list.
>
> > MoeBlee
>
> I don't see why it is significant to look at the diagonal the other way,
> from right to left. Besides, I don't think we need a diagonal. Any
> random zigzag will do just as well. Or a straight line along a column or
>
> - Show quoted text -
On 2 Dec, 21:44, John Jones <jonescard...@aol.com> wrote:
> MoeBlee wrote:
> > On Dec 1, 1:22 pm, John Jones <jonescard...@aol.com> wrote:
>
> >> Cantor's proof, simply put, amounts to the
> >> idea that when we add or subtract 1 to all the digits of the real
> >> numbers, then that new real number can't be found on the list given to
> >> us by the natural numbers.
>
> > I don't know what that is supposed to mean.
>
> > The basis of the proof is that given a list of denumerable binary
> > sequences, we can flip the digits of the diagonal sequence to make an
> > anti-diagonal sequence that is a denumerable binary sequence not on
> > the given list.
>
> I don't see why it is significant to look at the diagonal the other way,
> from right to left. Besides, I don't think we need a diagonal. Any
> random zigzag will do just as well. Or a straight line along a column or
> row will do.
Not along a row, but yes along a column. The whole thing might indeed
be reduced to the incommensurability of one side to the other of the
phantomatic "square": that you cannot have a correspondence between
columns and rows, so that the square is not a square and the whole
argument is just unsound and then invalid.
Standard math is made of gosts. Good luck.
-LV
LudovicoVan
Indeed, it's not a matter of wording: you have substituted a chimera
with a "new" one.
-LV
Daryl McCullough
LudovicoVan says...
>> > But how is this 'new' given to us? Is it a part of the framework, a
>> > constituent of the matrix, or does it emerge from what informs the
>> > whole as a linguistic expression of an index, something represented to
>> > us?
>>
>> Always the given is transient in the pattern of understanding that is
>> inseparable from immutability as expression that is both coursed in
>> the sign and witnessed by prior context. Isn't that obvious?
>
>Geniuses at work.
Are you familiar with the concept of "parody"? How about "irony"?
LudovicoVan
> In article <30e006e9-ed28-4814-8e5e-7840807c6...@f3g2000yqf.googlegroups.com>,
And how genious are you?
-LV
John Jones
> In article <gh5tva$ra6$1...@aioe.org>, John Jones <jonesc...@aol.com> wrote:
>> george wrote:
>>
>>> The theorem IS ABOUT SETS!
>> That just mudies up the concepts. Let's just use the words real numbers
>> and natural numbers, sunshine.
>
> No, it CLARIFIES them.
No way. And I know that you don't know what a set is supposed to be.
Neither do the people that use the term. Let's get along without it,
please. Why are we talking about sets? Are we fond of them?
> That's why you have to stay away from
> them. Because as soon as everything is clear and solid, you run
> straight into a wall. Instead, it is better to keep everything
> shrouded in a fog so that mushy arguments can seem solid.
Too late. I wrote the above before I read that.
> Talking about engines and gears just muddies the concepts. Let's keep
> them out. The point is, if I can get to work by getting in my car, I
> don't see why I can't get to work by getting in my bed. After all, the
> point is getting in, isn't it?
Excuse me, do you have any sets of bananas left?
Yes, we have a null set.
I'll have one please.
Shove off.
John Jones
Cantor's diagonal isn't a set of numbers. If you think otherwise just
ask yourself 'a set of what?' Cantor's diagonal is a number.
Arturo Magidin
>Arturo Magidin wrote:
>> In article <gh5tva$ra6$1...@aioe.org>, John Jones <jonesc...@aol.com> wrote:
>>> george wrote:
>>>
>>>> The theorem IS ABOUT SETS!
>>> That just mudies up the concepts. Let's just use the words real numbers
>>> and natural numbers, sunshine.
>>
>> No, it CLARIFIES them.
>
>No way.
Simply put, your wrong. And therein lies all your problems. You'd
rather be mushy and wrong and correct. Good luck.
>Shove off.
Yeah. Better to continue being an ignoramus pretending to be a wise
man, than to learn something. It's worked real well for you so far.
John Jones
> In article <gh5oe0$trf$1...@aioe.org>, John Jones <jonesc...@aol.com> wrote:
>> Arturo Magidin wrote:
>>
>>>>> The natural numbers are used as an index in their natural
>>>>> order, which is a well order. There is a first natural number, and
>>>>> given any collection of naturals that is not all the naturals, there
>>>>> is a "smallest natural not in the collection." That makes them
>>>>> particularly good as indices.
>>>> "Being first" in an index can be represented equally by 1 and .1.
>>> No,
>> You mean 'yes'.
>
> Now you are being dishonest.
Oh God no.
> I said "no", I meant "no",
You didn't know what you meant. You just typed n o comma.
> and the fact that this is not the end of
> the sentence (marked by a COMMA, not a period), indicates that the
> next part should not be ommitted before making a comment.
There wasn't a next part. You changed the subject to -
>>> the point is that there is a natural well ordering to the
>>> naturals. This defines an ordering among ALL the naturals, that allows
>>> you to say which natural goes in which position,
>
> See?
I hope you do.
>> This
>> index is as ordered and as useful as the index made without a dot, and
>> both indexes are the same size.
>
> But this "index" is NOT "the reals".
Why not? It was ok for the naturals.
> It's not even "the reals between
> 0 and 1."
OHH yes it is, if I put a dot in front of every natural.
> So you have failed miserably
I am happy in my success.
The only way you can know that is through Cantor's proof!
>
> Now you are just being an idiot on purpose. I am saying that the
> naturals come equipped with a natural order
There's that 'natural' again.
Are you saying that Cantor's argument relies on the fact that there is a
natural ordering genetically prior to the representation of any number?
It's a bit mystical isn't it? or a bit like a mathematical wig?
> that we are familiar with
> and which has a number of properties we are all familiar with. The
> reals do not have an order we all agree on that has those same
> properties.
Neither do the naturals. There is no natural order for the naturals. I
can start from the biggest and work down to the smallest, or vice versa,
Or I can start with the evens, then when I'm done, turn around and work
backwards with the odds.
Or I can put a dot in front of them, and call them the reals, which is
what Cantor did.
> If you claim that you can use the reals "just as well" as
> you can use the naturals, then I want them to be "just as good":
Me to.
> I
> want the reals to have some order that has all those familiar
> properties.
I can stipulate whatever order I like. I don't have to use any
convention regarding what is order and what is not. All I need to have
is enough different signs. I can arrange them how I like for the
purposes of making an index. And whatever decimal is given me, I can
remove the dot and call it a natural.
An index wipes distinctions if those distinctions are based on ordering.
> I know what that order is for the naturals,
I don't.
> I do not know
> what it is for the reals.
In the absence of convention, any order you stipulate.
> You can certainly define a non-standard
> ordering, but you have to DEFINE it.
Yes, by all means.
> I don't know what it is. And
> defining an order for the reals does not mean simply telling me ".1
> goes first" (which is ALL you did before). You need to say more.
You can say start with the largest. After all, we say start with the
smallest with the naturals. No prob, except-
But a better scheme would be to abandon large and small and just say
that 123456789 is the order for each place in the sequence for both
naturals and reals. Cantor can't use concepts of size in his proof if it
is size that is in question.
John Jones
Your step 4 false lemma is risky. Perhaps others will take it up.
But first, why is there no natural having an infinite number of nonzero
digits? That looks as if the meaning of zero has somehow drifted away
from the meaning of the other 9 digits, as if this zero is to be treated
differently to the other digits.
John Jones
You know what I am going to say now, perhaps. Did Cantor have to assume
for the success of his diagonal argument that the cardinality of the set
of infinite sequences of digits is greater than the cardinality of the
set of finite sequences of digits? (Ignoring for the moment this take on
the naturals and reals)
John Jones
> On Tue, 02 Dec 2008 19:37:36 +0000, John Jones <jonesc...@aol.com>
> wrote:
>
>> David C. Ullrich wrote:
>>
>>>> An assumption in Cantor's 'proof' is that the real's can't be used as a
>>>> list
>>> There is no such assumption.
>> between the reals and the naturals.
>
> valididty of the proof that there is no function mapping the natural
> numbers onto the reals.
I can map any natural to any real and vice versa simply by removing the dot.
>
>
> David C. Ullrich
>
> "Understanding Godel isn't about following his formal proof.
> That would make a mockery of everything Godel was up to."
> (John Jones, "My talk about Godel to the post-grads."
> in sci.logic.)
Arturo Magidin
>Arturo Magidin wrote:
>> In article <gh5oe0$trf$1...@aioe.org>, John Jones <jonesc...@aol.com> wrote:
>>> Arturo Magidin wrote:
>>>
>>>>>> The natural numbers are used as an index in their natural
>>>>>> order, which is a well order. There is a first natural number, and
>>>>>> given any collection of naturals that is not all the naturals, there
>>>>>> is a "smallest natural not in the collection." That makes them
>>>>>> particularly good as indices.
>>>>> "Being first" in an index can be represented equally by 1 and .1.
>>>> No,
>>> You mean 'yes'.
>>
>> Now you are being dishonest.
>
>Oh God no.
>
>> I said "no", I meant "no",
>
>You didn't know what you meant.
You go the pronoun wrong: YOU didn't understand what I meant. I knew
exactly what I meant. Just because you are too dumb to figure it out,
kindly do not project your shortcomings on others.
> You just typed n o comma.
And now you are compounding your lie. I typed a bunch of stuff after
it, which you dutifully ignored because that allowed you to claim a
nonexisting confusiong.
>> and the fact that this is not the end of
>> the sentence (marked by a COMMA, not a period), indicates that the
>> next part should not be ommitted before making a comment.
>
>There wasn't a next part. You changed the subject to -
>
>>>> the point is that there is a natural well ordering to the
>>>> naturals. This defines an ordering among ALL the naturals, that allows
>>>> you to say which natural goes in which position,
That ->was<- the next part, you lying sack of ignorance.
>> It's not even "the reals between
>> 0 and 1."
>
>OHH yes it is, if I put a dot in front of every natural.
You don't get all the reals between 0 and 1, you only get SOME of
them.
>> So you have failed miserably
>
>I am happy in my success.
Indeed: you clearly enjoy wallowing in the mud of your ignorance and
your idiocy. Kindly stay there and keep away from us clean folk.
>>> I can use the reals just as well as the
>>> naturals for an index.
>>
>> No, you can't. You aren't using the reals, you are using only SOME
>> reals.
>
>The only way you can know that is through Cantor's proof!
Nonsense. All your "indices" have a finite number of digits after the
decimal point. I know lots of real numbers, independent of Cantor's
proof, that have infinitely many nonzero digits after the decimal
point. Starting with the easiest, 1/3. 1/3 is not among your
indices. Ergo, you are not using all reals. No need to invoke Cantor's
proof (or "Cantor's proof!") to realize that.
Unless, of course, you insist on being an idiot.
>> Now you are just being an idiot on purpose. I am saying that the
>> naturals come equipped with a natural order
>
>There's that 'natural' again.
Replace with "well-understood".
>Are you saying that Cantor's argument relies on the fact that there is a
>natural ordering genetically prior to the representation of any
>number?
No. You should stop putting words into other people's mouth,
especially when those words are just the offal of your ignorance.
>> that we are familiar with
>> and which has a number of properties we are all familiar with. The
>> reals do not have an order we all agree on that has those same
>> properties.
>
>Neither do the naturals. There is no natural order for the naturals.
Yes, there is.
> I
>can start from the biggest and work down to the smallest, or vice versa,
Only one of which is the standard, common, natural order of the
naturals. Even if you stand on your head and eat your own shit, it
still does not change which one is the standard, common, natural order
of the natural numbers.
>Or I can start with the evens, then when I'm done, turn around and work
>backwards with the odds.
Which is not the natural, common, standard order of the naturals.
>Or I can put a dot in front of them, and call them the reals, which is
>what Cantor did.
No, it is not what Cantor did. If you are too ignorant to understand
what Cantor did, don't pretend you do. If you are too steep in your
idiocy to understand, then don't tell us what it is "Cantor did",
since you don't know what he did.
>> If you claim that you can use the reals "just as well" as
>> you can use the naturals, then I want them to be "just as good":
>
>Me to.
Too bad you failed, then.
>> I
>> want the reals to have some order that has all those familiar
>> properties.
>
>I can stipulate whatever order I like.
But unless it is just as good as the one for the naturals, you aren't
fulfilling your goal of getting one that is just as good as the one
for the naturals. Which is why you failed.
>I don't have to use any
>convention regarding what is order and what is not.
You also don't have to use any convention regarding what is a word and
what is not, or what the words mean. But, like Humpty-dumpty, all you
manage when you try to be the Master of Orders that way is to fall
flat on your face and cause a big splat.
> All I need to have
>is enough different signs. I can arrange them how I like for the
>purposes of making an index. And whatever decimal is given me, I can
>remove the dot and call it a natural.
And you'll be wrong with any decimal that does not terminate. the
decimal expansion for 1/3, when you remove the dot, does NOT yield a
natural, even if you insist on calling it one.
>An index wipes distinctions if those distinctions are based on ordering.
No, it does not. You are not trying to wipe out distinctions based on
ordering, you are trying to wipe out the distinction between
ignorance, idiocy and nonsense on one side (yours), and knowledge,
rationality, and sense on the other (ours).
>> I know what that order is for the naturals,
>
>I don't.
Then go back to school and learn it, instead of showing off your
ignorance and being produ of it.
>> I do not know
>> what it is for the reals.
>
>In the absence of convention, any order you stipulate.
Which you failed to do.
>> You can certainly define a non-standard
>> ordering, but you have to DEFINE it.
>
>Yes, by all means.
Then why did you not do so?
>> I don't know what it is. And
>> defining an order for the reals does not mean simply telling me ".1
>> goes first" (which is ALL you did before). You need to say more.
>
>You can say start with the largest. After all, we say start with the
>smallest with the naturals. No prob, except-
All probs, dumdum. You have not specified an order among the
reals.
All you are doing, SillyPants, is running around in circles with your
fingers in your ears. You are only looking at the real numbers with
terminating decimal expansion. One does not need Cantor to know that
this does not cover all the reals: one only needs to know how to
divide 1 by 3.
Aparently, that is beyond you.
>But a better scheme would be to abandon large and small and just say
>that 123456789 is the order for each place in the sequence for both
>naturals and reals.
Yeah, a better scheme is to say nonsense over and over again until
those arguing with you get tired of talking to a wall of idiocy. That
way, you can pretend you "Won" when all you did was go from idiot to
Idiot. Congrats.
Arturo Magidin
>Daryl McCullough wrote:
>> John Jones says...
[...]
>But first, why is there no natural having an infinite number of nonzero
>digits?
Why is there no circle with four sides?
Why is there no triangle with five internal angles?
> That looks as if the meaning of zero has somehow drifted away
>from the meaning of the other 9 digits, as if this zero is to be treated
>differently to the other digits.
Are leading zeros the same as leading 1s, 2s, 3s, 4s, 5s, 6s, 7s, 8s,
and 9s?
No.
1 is the same as 01 is the same as 001 is the same as 0001 is the same
as 00001, given the meaning of positional notation. However, 1 is not
the same as 11, not the same as 21, not the same as 31,.... not the
same as 91.
So, while the "meaning" is the same (is the number of corresponding
powers of 10 in the expression of the number), adding 0 powers of 10
is not the same as adding 1, 2, 3, 4, 5, 6, 7, 8, or 9 powers of
10. Because adding 0 is not the same as adding any other number. 0
->is<- unique, Dum-dum.
As to the natural numbers: 1 has a finite of nonzero digits in its
expression base 10.
If n has a finite number of nonzero digits, then n+1 has a finite
number of nonzero digits in its expression base 10.
Therefore, inductively/recursively, every natural number has only a
finite number of digits in its expression base 10.
And there, among other places, is where the well-ordering of the
natural numbers comes into play: it satisfies induction. If you can
prove something is true for 1 and you can prove that whenever it is
true for n then it is true for n+1, then that something is true for
each and every natural number.
Arturo Magidin
>David C. Ullrich wrote:
>> On Tue, 02 Dec 2008 19:37:36 +0000, John Jones <jonesc...@aol.com>
>> wrote:
>>
>>> David C. Ullrich wrote:
>>>
>>>>> An assumption in Cantor's 'proof' is that the real's can't be used as a
>>>>> list
>>>> There is no such assumption.
>>> Call it an index then, if you like. The reals can be used as the index
>>> just as much as the naturals. Used as an index, there is no distinction
>>> between the reals and the naturals.
>>
>> Whatever the heck that means, it has no bearing whatever on the
>> valididty of the proof that there is no function mapping the natural
>> numbers onto the reals.
>
>I can map any natural to any real and vice versa simply by removing the dot.
No natural number corresponds to removing the dot from the decimal
expansion of 1/3.
There is no natural number whose expression consists of an infinite
number of 1s.
You don't know what a natural number is. No wonder you can't
understand anything.
David Formosa (aka ? the Platypus)
[...]
> I can do the same for the reals. 0.23242... can come before 0.23241 if
> you are using size as a criterion of ordering.
> I ignore your idea that a correct position can be defined without a
> number ("an ordering among ALL the naturals, that allows
> you to say which natural goes in which position").
What is the second real number?
Herbert Newman
> JJ (psycho) wrote:
>>
>> I can do the same for the reals. 0.23242... can come before 0.23241 if
>> you are using size as a criterion of ordering.
>>
>> I ignore your idea that a correct position can be defined without a
>> number ("an ordering among ALL the naturals, that allows you to say
>> which natural goes in which position").
>>
> What is the second real number?
>
Not sure about the second, but the first is
42.
Herb
John Jones
>
> That ->was<- the next part, you lying sack of ignorance.
>
>>> It's not even "the reals between
>>> 0 and 1."
>>> No, you can't. You aren't using the reals, you are using only SOME
>>> reals.
>> The only way you can know that is through Cantor's proof!
>
> Nonsense. All your "indices" have a finite number of digits after the
> decimal point.
That's taking for granted what Cantor had to prove.
> I know lots of real numbers, independent of Cantor's
> proof, that have infinitely many nonzero digits after the decimal
> point.
Zero is as good a digit as any other. 1.1111... is as informative for an
index as 1.0000...
> Starting with the easiest, 1/3. 1/3 is not among your
> indices.
1/3 can be as much an indice as 3333...
>> There is no natural order for the naturals.
>> Or I can put a dot in front of them, and call them the reals, which is
>> what Cantor did.
>> I don't have to use any
>> convention regarding what is order and what is not.
>
> You also don't have to use any convention regarding what is a word and
> what is not, or what the words mean.
So convention trumps meaning.
>> All I need to have
>> is enough different signs. I can arrange them how I like for the
>> purposes of making an index. And whatever decimal is given me, I can
>> remove the dot and call it a natural.
>
> And you'll be wrong with any decimal that does not terminate. the
> decimal expansion for 1/3, when you remove the dot, does NOT yield a
> natural, even if you insist on calling it one.
Yes it does. 333....
>>> I know what that order is for the naturals,
>> I don't.
>
>
>>> You can certainly define a non-standard
>>> ordering, but you have to DEFINE it.
>> Yes, by all means.
>
> Then why did you not do so?
>
>> You can say start with the largest. After all, we say start with the
>> smallest with the naturals. No prob, except-
>
> All probs, dumdum. You have not specified an order among the
> reals.
I can say any number with most fours is listed first in the index.
>
John Jones
The second real number is .3426.
Daryl McCullough
>
>Daryl McCullough wrote:
>You know what I am going to say now, perhaps. Did Cantor have to assume
>for the success of his diagonal argument that the cardinality of the set
>of infinite sequences of digits is greater than the cardinality of the
>set of finite sequences of digits?
No, he didn't assume that. That is what he *proved*.
Why are you writing about Cantor's argument when you
don't know anything about it? Why don't you find out
what the proof says before you start saying what your
objections are?
Arturo Magidin
>Arturo Magidin wrote:
>>
>> That ->was<- the next part, you lying sack of ignorance.
>>
>>>> It's not even "the reals between
>>>> 0 and 1."
>
>>>> No, you can't. You aren't using the reals, you are using only SOME
>>>> reals.
>>> The only way you can know that is through Cantor's proof!
>>
>> Nonsense. All your "indices" have a finite number of digits after the
>> decimal point.
>
>That's taking for granted what Cantor had to prove.
No. That's knowing what "natural number" means. Since you don't, the
rest of what you have to say is immaterial.
>> I know lots of real numbers, independent of Cantor's
>> proof, that have infinitely many nonzero digits after the decimal
>> point.
>
>Zero is as good a digit as any other. 1.1111... is as informative for an
>index as 1.0000...
No, zero is a unique number. It is the only number that, added to any
other number, will yield the other number as a result. There is
something unique about 0 in the world of positional notation. That's
why it took so long to invent it. Because it is ->not<- the same as
any other.
>> Starting with the easiest, 1/3. 1/3 is not among your
>> indices.
>
>1/3 can be as much an indice as 3333...
The point, SillyPants, is that your scheme for coming up with indices
("add a dot"), never includes 1/3 as an index. Yes, you COULD use 1/3
as an index. But you ARE NOT DOING SO.
>>> I don't have to use any
>>> convention regarding what is order and what is not.
>>
>> You also don't have to use any convention regarding what is a word and
>> what is not, or what the words mean.
>
>So convention trumps meaning.
No, meaning trumps whatIwantittobe. YOU are claiming that things are devoid of
meaning and therefore can be made to mean anything you want.
>>> All I need to have
>>> is enough different signs. I can arrange them how I like for the
>>> purposes of making an index. And whatever decimal is given me, I can
>>> remove the dot and call it a natural.
>>
>> And you'll be wrong with any decimal that does not terminate. the
>> decimal expansion for 1/3, when you remove the dot, does NOT yield a
>> natural, even if you insist on calling it one.
>
>Yes it does. 333....
That is not a natural number. This has nothing to do with Cantor,
before you repeat your ignorance again, it has to do with what the
natural numbers are. That is not a natural number.
>>> You can say start with the largest. After all, we say start with the
>>> smallest with the naturals. No prob, except-
>>
>> All probs, dumdum. You have not specified an order among the
>> reals.
>
>
>I can say any number with most fours is listed first in the index.
You can say whatever you want (within the confines of slander
laws). Alas, most of what you say is nonsense and shows you don't know
what you are talking about. Witness your absurd statements and
ignorance of the natural numbers.
Daryl McCullough
>But first, why is there no natural having an infinite number of nonzero
>digits? That looks as if the meaning of zero has somehow drifted away
>from the meaning of the other 9 digits, as if this zero is to be treated
>differently to the other digits.
All right. Let's look at a particular number, 312, say.
The meaning of this base-ten representation is the following:
312 = 2 + 10*1 + 100*3
So the ones place is 2, the tens place is 1, the hundreds
place is 3. We could keep going, and say that the thousands
place is 0, because
312 = 2 + 10*1 + 100*3 + 1000*0
Then the ten-thousands place will be 0, and so will the
hundred thousands, etc.
In general, for any natural number, only finitely many
decimal places will be nonzero, and the rest will all be
0.
Daryl McCullough
>
>Arturo Magidin wrote:
>>
>> That ->was<- the next part, you lying sack of ignorance.
>>
>>>> It's not even "the reals between
>>>> 0 and 1."
>
>>>> No, you can't. You aren't using the reals, you are using only SOME
>>>> reals.
>>> The only way you can know that is through Cantor's proof!
>>
>> Nonsense. All your "indices" have a finite number of digits after the
>> decimal point.
>
>That's taking for granted what Cantor had to prove.
You clearly don't have any idea what you are talking about.
Herbert Newman
> John Jones says...
>
>> ... why is there no natural having an infinite number of nonzero digits?
>>
Simply put: Because this number would be infinite (infinitely large), but
there is no infinite (infinitely large) natural number.
Herb
David Formosa (aka ? the Platypus)
> David Formosa (aka ? the Platypus) wrote:
>> On Wed, 03 Dec 2008 10:52:49 +0000, John Jones <jonesc...@aol.com> wrote:
>>
>> [...]
>>
>>> I can do the same for the reals. 0.23242... can come before 0.23241 if
>>> you are using size as a criterion of ordering.
>> What is the second real number?
>
> The second real number is .3426.
If size is used for the criterion of ordering, then 0.1713...,
0.08565... and 0 are all before your second real number. However if a
number is 2nd then there can only be 1 position before it. Indeed for
what ever real number you choose for your 2nd element I can find a
real number between it and your first.
John Jones
>
> No, zero is a unique number. It is the only number that, added to any
> other number, will yield the other number as a result. There is
> something unique about 0 in the world of positional notation. That's
> why it took so long to invent it. Because it is ->not<- the same as
> any other.
For the purposes of an index, 0 identifies in the same way as any other
digit.
>>> Starting with the easiest, 1/3. 1/3 is not among your
>>> indices.
>> 1/3 can be as much an indice as 3333...
>
> The point, SillyPants, is that your scheme for coming up with indices
> ("add a dot"), never includes 1/3 as an index. Yes, you COULD use 1/3
> as an index. But you ARE NOT DOING SO.
Alright. Add a 1/ or take away a 1/.
Job done.
>> Yes it does. 333....
>
> That is not a natural number. This has nothing to do with Cantor,
> before you repeat your ignorance again, it has to do with what the
> natural numbers are. That is not a natural number.
Why not? The dots refer to digits.
John Jones
Good. Now we are getting somewhere. The first is 42. The second is 44,
with your permission of course. From there, we go all the way up, then
turn around and do the odd numbers back down to 43. That's a smashing
index. In fact, leave the odd numbers out. There are enough evens.
John Jones
> Are leading zeros the same as leading 1s, 2s, 3s, 4s, 5s, 6s, 7s, 8s,
> and 9s?
00.1, 0.1, 0, 0.0, 00.00, 0.000, 00000000, 0 0 0 00 , , , can
all be used in an index, and they are all different. We don't have to
use size for ordering, or even symbols. But this is off-topic.
>
> 1 is the same as 01
Not in an index it isn't.
Still off-topic.
> As to the natural numbers: 1 has a finite of nonzero digits in its
> expression base 10.
What happened to the rest of them then? Is there a big blank space? like
a void that has sucked up the digit 0?
>
> If n has a finite number of nonzero digits, then n+1
n+1? "plus one"? Plus one what?
> has a finite
> number of nonzero digits in its expression base 10.
Not if "plus one" tells us to add one 0.
> Therefore, inductively/recursively, every natural number has only a
> finite number of digits in its expression base 10.
What happened to the rest of them?
> And there, among other places, is where the well-ordering of the
> natural numbers comes into play: it satisfies induction. If you can
> prove something is true for 1 and you can prove that whenever it is
> true for n then it is true for n+1,
Yes, provided n+1 adds something that makes it true! So it's not an
induction at all. n+1 can add a digit, like 0.
Herbert Newman
Platypus):
>
> ... for what ever real number you choose for your 2nd element I can find
> a real number between it and your first.
>
No, you can't. It's against the law!
Herb
Daryl McCullough
>
>Arturo Magidin wrote:
>
>> Are leading zeros the same as leading 1s, 2s, 3s, 4s, 5s, 6s, 7s, 8s,
>> and 9s?
>
>00.1, 0.1, 0, 0.0, 00.00, 0.000, 00000000, 0 0 0 00 , , , can
>all be used in an index, and they are all different. We don't have to
>use size for ordering, or even symbols. But this is off-topic.
>
>>
>> 1 is the same as 01
>
>Not in an index it isn't.
>Still off-topic.
>
> > As to the natural numbers: 1 has a finite of nonzero digits in its
>> expression base 10.
>
>What happened to the rest of them then? Is there a big blank space? like
>a void that has sucked up the digit 0?
Clearly, you need to brush up on the basics of arithmetic
before philosophizing about Cantor's theorem. You don't seem
to have any idea what you are talking about.
John Jones
> John Jones says...
>
>> But first, why is there no natural having an infinite number of nonzero
>> digits? That looks as if the meaning of zero has somehow drifted away
>>from the meaning of the other 9 digits, as if this zero is to be treated
>> differently to the other digits.
>
> All right. Let's look at a particular number, 312, say.
> The meaning of this base-ten representation is the following:
>
> 312 = 2 + 10*1 + 100*3
Ok.
> So the ones place is 2, the tens place is 1, the hundreds
> place is 3. We could keep going, and say that the thousands
> place is 0, because
>
> 312 = 2 + 10*1 + 100*3 + 1000*0
>
> Then the ten-thousands place will be 0, and so will the
> hundred thousands, etc.
>
> In general, for any natural number, only finitely many
> decimal places will be nonzero, and the rest will all be
> 0.
Ok. Fine.
Let's say that we want to start off with a level playing-field. So, we
initially make the assumption that both reals and naturals have an
infinite number of places reserved for digits.
What would be our justification for making the rule that 'value trumps
place' and getting rid of places when the digits that occupy the places
do not contribute to value?
John Jones
I thought this was what Cantor was trying to show? I don't want to
presume a vital property that is initially in question.
John Jones
> In article <gh6otc$2vl$2...@aioe.org>, John Jones <jonesc...@aol.com> wrote:
>> David C. Ullrich wrote:
>>> On Tue, 02 Dec 2008 19:37:36 +0000, John Jones <jonesc...@aol.com>
>>> wrote:
>>>
>>>> David C. Ullrich wrote:
>>>>
>>>>>> An assumption in Cantor's 'proof' is that the real's can't be used as a
>>>>>> list
>>>>> There is no such assumption.
>>>> Call it an index then, if you like. The reals can be used as the index
>>>> just as much as the naturals. Used as an index, there is no distinction
>>>> between the reals and the naturals.
>>> Whatever the heck that means, it has no bearing whatever on the
>>> valididty of the proof that there is no function mapping the natural
>>> numbers onto the reals.
>> I can map any natural to any real and vice versa simply by removing the dot.
>
> No natural number corresponds to removing the dot from the decimal
> expansion of 1/3.
Get rid of the 1/ then.
>
> There is no natural number whose expression consists of an infinite
> number of 1s.
Yet we are quite prepared to say otherwise for a real.
John Jones
> John Jones says...
>> Daryl McCullough wrote:
>
>> You know what I am going to say now, perhaps. Did Cantor have to assume
>> for the success of his diagonal argument that the cardinality of the set
>> of infinite sequences of digits is greater than the cardinality of the
>> set of finite sequences of digits?
>
> No, he didn't assume that. That is what he *proved*.
Right. So why use what it is that is proved as a necessary premise in
the construction of the proof?
george
> Cantor's diagonal isn't a set of numbers.
That DEPENDS on something. It depends on whether the ORIGINAL
UNDERLYING
SET (The set whose number-of-elements and number-of-subsets ARE BEING
COMPARED) is a set of numbers. The diagonal (or, if you mean
Cantor's, the
ANTI-diagonal -- the COMPLEMENT of the diagonal) is a subset OF THE
ORIGINAL SET. If the original set was a set of numbers, then both the
diagonal
and the anti-diagonal will be sets of numbers as well.
> If you think otherwise just
> ask yourself 'a set of what?'
A set of the SAME things that the ORIGINAL UNDERLYING set was a set
of.
> Cantor's diagonal is a number.
The theorem IS ABOUT SETS, DUMBASS!
It IS NOT about numbers until AFTER You ENCODE THOSE NUMBERS AS
sets!
You are confusing ONE special case -- when Cantor's theorem, WHICH
APPLIES *TO ALL* SETS -- ALL sets have more subsets than they have
elements -- is being applied TO THE SET OF NATURAL NUMBERS, TO THE SET
OF FINITE ORDINALS, TO THE (smallest) SET REQUIRED TO EXIST BY THE
AXIOM OF INFINITY --
WITH the whole theorem!
If the underlying set is the set of natural numbers then the diagonal
IS A SUBSET of the natural numbers and THEREFORE MAPS EASILY
to a real number VIA THE BIT-STRING MAPPING!
John Jones
>>> What is the second real number?
>> The second real number is .3426.
>
> If size is used for the criterion of ordering, then 0.1713...,
> 0.08565... and 0 are all before your second real number.
But they are not on my list of viable possibilities. I checked.
> However if a
> number is 2nd then there can only be 1 position before it. Indeed for
> what ever real number you choose for your 2nd element I can find a
> real number between it and your first.
Ha-Ha. Yes of course! But look at my devilish response..
I can do the same for the naturals!
IF I allow you the possibility of having an infinite placement for the
digits of possible reals, then there is no reason why you should not
allow me an infinite placement for the digits of possible naturals.
After all, for Cantor's sake, we should try and keep conditions the same
for the reals and the naturals before we try to prove anything about them.
So if I have a first real as 0.10, and a second real as 0.20, then you
can come along with a 0.15 to mess up the order.
But I can do the same for the naturals, as long as I keep the number of
places for the digits the same (2 in this case). My first natural is 10,
my second natural is 20, and then someone comes along and mucks it up
with a 15.
The point I would like to make is that, on a level playing-field,
whatever you do with the reals you can do with the naturals. If you
object that I am not treating the naturals like the naturals ought to be
treated, then my response is that for the purposes of creating an index
it does not matter as long as we can map every real to every natural and
vice versa. Good.
george
> You're right that a diagonal pattern is not of the essence --
Of course it is! The fact that it is not "absolutely necessary" does
NOT
STOP it from being ESSENTIAL!
> it's just convenient.
It is NOT MERELY contingently convenient! It is NECESSARILY MAXIMALLY
convenient! The identity bijection is NECESSASRILY THE easiest
injection!
> Not any random zig-zag will do, but some will.
Any injective function on N will do. You have to come up with a
subset
that is guaranteed to disagree with every set (row) on the list about
some
element (column) of the underlying set. WHICH element doesn't matter
as long as every row/set GETS A DIFFERENT element/column -- if 2
elements
got THE SAME column then forcing "disagreement" between BOTH at that
ONE position (in the anti-zig-zag being defined) might be impossible.
> A straight
> line along aa column "or row" (a-heh-heh-heh) is (like the count of five
> for the holy hand-grenade) right out.
Indeed, the column is maximally NON-injective since it maps EVERY row
(domain-element of N) to the SAME column (range-element of N).
And the row (excellent use of heh-heh-heh) is dismissed simply because
in this context; you have to pick SOME position on EVERY row
(otherwise the anti-"zig-zag" might OCCUR on the list, on the row
where NO
DISagreeing column/element/position was picked).
george
> How about using a circular list of numbers in Cantor's square?
THERE IS NO SUCH THING
as a "circular list"!! A *list* HAS to be ordered linearly like the
natural
numbers. But THAT IS NOT EVEN THE POINT! The POINT is,
YOU INHERENTLY HAVE *TWO DIMENSIONS* In this problem!
You INHERENTLY HAVE a COLLECTION OF SUBSETS OF A GIVEN SET,
and you are INHERENTLY COMPARING them for size! SO SQUARE
IS PREDEFINED into the question! "circle" ISN'T EVEN MEANINGFUL
in this context!
> You see,
> it's a bit odd that we have to continue the diagonal, zigzag, whatever,
> to a place where we lose sight of it.
You're being and idiot: WE NEVER lose sight of it! EVERY position in
it
IS FINITE!
> I say let's keep sight of our
> operations and do a big circle that's in full view.
You say that YOU ARE AN IDIOT.
You are CONSTRUCTING A SUBSET of 1 set FROM A LIST
of subsets OF THE SAME SET. THE LIST IS INHERENTLY rectangular
if the set has a natural ordering. The vertical dimension of the list
(along
the rows) IS INHERENTLY linearly and connectedly ordered BECAUSE IT'S
A LIST, and that's PART OF THE DEFINITION of list. The horizontal
dimension
does NOT HAVE to put the elements of the underlying set in any
particular
order, but IF YOU ARE DOING THIS WITH NATNUMS, then the order IS THERE
WHETHER YOU LIKE IT OR NOT. More to the point, you ONLY NEED ONE
element-of-disagreement between the constructed subset and each
subset-
on-the-list, but any line you could draw across a circle cuts it in
TWO places
unless it's tangent -- a circle is just OVERKILL. But more to the
point, since
THE LIST IS INFINITE IN BOTH DIMENSIONS, a circle CAN'T COVER it!
A circle COULD NOT REACH some rows/subsets.
In other words,
YOU'RE STUPID.
> But then Cantor's
> demonstration wouldn't work.
You're also a damn liar. The fact that you chose to draw a circle
COULD NOT STOP the diagonal subset FROM EXISTING, or from getting
complemented. Certain sets HAVE to exist, if certain others do.
> And the point is, no matter how big we make the circle, Cantor's proof
> will never work for it.
NO proof will work for it!
No matter how big you make the circle, IT DOESN'T INTERSECT SOME ROWS!
So it OBVIOUSLY IS NOT RELEVANT to the question!
> In which case, we must ask ourselves what we are
> doing different for a diagonal.
YOU must ask because YOU ARE SO STUPID that you don't already know.
What is different about the diagonal is that it DOES intersect every
row,
in ONE place (and one is all you need). The circle intersects (at
most) 2
rows in 1 place, a finitely few more in 2 places, AND INFINITELY MANY
IN NONE AT ALL. So it is just not relevant to the issue. It doesn't
describe
what happens with the subset regarding elements-corresponding-to-the-
rows
that it (the circle) DOES NOT INTERSECT!