Why the attack on Cantor ?
VERDIGRIS
are many people who find something to object to about
Cantor's Theory. I believe that the attack on Cantor's
theory will not end until someone has found a way to
drive a stake right through the heart of the theory.
*****
Verdigris: I think Cantor was right to 'invent' infinite
categories but was wrong in the way he chose to
do it. I think the criterion of counting embodied in the
concept of bijection is wrongly applied to infinite sets.
This is because they cannot be enumerated as a temporal
operation in the way that finite sets can be enumerated.
Although one can always begin to map one-one from
n to f(n) the operation can never be completed, even
in principle. The assertion that there is a complete
mapping from n to f(n) is therefore unjustified.
The idea that the cardinality of such sets exists is
suspect as a result of their necessay incompleteness.
In other words, an incomplete set can have no total
so it is meaningless to assert that the cardinality of f(n)
is the same as the cardinality of n.
*****
Our whole culture has changed radically in the last twenty
years. Things that might have seemed reasonable to
the last generation will seem absurd to the new generation.
One of the things that must seem absurd to the new
generation is Cantor's Theory.
What has changed in the last twenty years is the ubiquity
of computers. Today's children are growing up surrounded
by computers, and they're growing up in families where
the parents are also comfortable with computers. They're
living in a computer culture.
So what does that have to do with Cantor's Theory?
To someone who has grown up with computers, the world
of computation must seem like a very real world. It's not
a physical world, to be sure, but it is concrete and has an
objective existence nonetheless. It's a world revealing a
rich set of phenomena. And in fact, it's a maximally rich
world, in the sense that any phenomena in any world can
be modeled in the world of computation.
So what's the point?
To someone who is comfortable with thinking about the
world of computation as a real world, "mathematics" is
most usefully defined as the study of phenomena that are
observable in the world of computation. That is, we can
think of the computer as a microscope that helps us peer
into the world of computation, and "mathematics" as the
science that studies the phenomena observed through
that microscope.
*****
Verdigris: Mathematics can not be confined to the
computable or demonstrable. Consider the totality of
all functions of the form y = f(x); there is no possible
computer program that could enumerate each individual
function, or even a class of funtions, particularly if they
have real parameters. The infinite nature of number
extends to geometric forms. In any case, if the universe
is finite, there will always be sufficiently large finite
numbers (parameters) which will render computations
based on them impossible. Computation must always
fail at some point to match definite large finite numbers.
*****
So where does Cantor's theory fit in to this scenario?
It doesn't. And that's the problem.
To someone who is comfortable with the notion of a world
of computation, it is extremely plausible to define the world
of mathematical objects that "exist" to be the universe of
objects that can be seen through the microscope. Cantor's
theory formally implies the "existence" of objects that cannot
be seen through the microscope, and hence Cantor's theory
must be viewed as a mythology.
When the classical mathematicians hear about all this, they
generally go into a tizzy and refuse to think straight about it.
They go into a defense mode. They seem to believe that
they are about to lose everything they've worked for. But in
fact, very little is lost.
For one thing, nothing of applied mathematics is lost. The
idea of applying mathematics means to create computational
models of real world phenomena in which the phenomena
observed in the model correspond to the phenomena observed
in the real world. So this notion of mathematics being the
study of phenomena observable through computation fits
applied mathematics like a glove, so to speak.
*****
Verdigris: Mathematics is about generalisations rather than
computable objects.
*****
But we also don't lose most of the abstractions that the pure
mathematicians are fond of, though we are forced to understand
them from a different perspective.
For starters, we can build up a theory of the infinite, so long as
we don't allow anything into the universe of the infinite that does
not have corresponding approximations that can be seen
through our microscope. Thus, for example, we can say that
perfect circles "exist" in the sense that we can see arbitrarily
close approximations to circles. Likewise, pi "exists" in that
we can compute arbitrarily close approximations. Functions
like exp(x) "exist" likewise. Fourier Transforms "exist".
Manifolds "exist". Solutions to differential equations "exist".
What's more, infinite sets "exist", as long as we keep in mind
what it means to say that they "exist". It means that we can
see arbitrarily close approximations to them. For example, for
the set of positive integers, we can "see" sets of the form {1..N}
for arbitrarily large N, and such sets can be taken to be
approximations of the set of all positive integers. For the set of
real numbers (a continuum), an approximation must be a finite
set of finite approximations of real numbers.
*****
Verdigris: This cannot work because no finite operation can
discriminate between two different infinite entities because
the available objects are either too large or too small. The
analogy is the inability of a conventional microscope to see
the small objects exposed by an electron microscope.
*****
So if infinite sets "exist", and power sets "exist", what's wrong
with Cantor's theory?
What I'm proposing is that mathematics should be built upon
a notion of "observability". We can say that a mathematical
object is observable if we can observe arbitrarily close
approximations to it in our microscope. We can say that
the object has certain properties, if and only if we can see that
approximations to the property hold for approximations to
the object. We can say that mathematical assertions have
observable content if they make predictions that can be
observed in our microscope.
This notion of observability leads to a unique and minimalist
theory of the infinite. And that theory is not Cantor's Theory.
Cantor's Theory implies that there "exist" things that are not
observable.
So what was Cantor's mistake?
Cantor started off asserting that infinite sets exist, without
consideration of what it means to observe those infinite sets.
He told us that power sets exist, without telling us what it
means to say that they exist. He went on talking about
properties of the objects, without telling us what it means
to say the objects have those properties (other than the
superficial notion that what it means for an object to have a
property is that a sentence in the formal language can be
interpreted to say that the object has a property). And then he
came to the conclusion that there must "exist" more objects
than can be observed. Cantor chose axioms which would
formally lead to the conclusion he wanted to reach, without
regard to the observable universe.
*****
Verdigris: the use of axioms and definitions is in no way
resricted to what can be observed (whatever that might
mean) but relates only to what can be definitely conceived.
*****
Emphatically, Cantor has not shown that there is any logical
inconsistency in assuming that the universe of observable
objects is the whole mathematical universe. He has not
given us a compelling reason to accept his fantasy world,
which goes well beyond the observable universe.
*****
Verdigris: The compelling reason to accept the infinite is
that it can be defined in a comprehensible way. Infinite
entities exist in the sense that we know what we mean
by them. As definite ideas they can be discussed and
understood but do not correspond to real models. As
you have pointed out, there are no real models of the
circle or any other perfect mathematical forms.
*****
I should point out that by accepting this notion of observability,
we do not even lose Cantor's theory (as a formal theory). That
is, formal theories themselves are objects that exist in the
observable universe. They are objects which mathematicians
might want to study. It's just that Cantor's theory has no right
to the claim of being part of the foundation of mathematics.
What I am suggesting is that we must seek a theory in which
every object in our theory corresponds to an object in the
universe of objects observable through computation, and
likewise, every object in the observable universe should
correspond to an object in our theory. That's the goal. If we
don't set that as our goal, then the parts of the theory which
don't correspond to observation will grow into entrenched
dogma, stifling growth and creativity, and providing a
springboard from which oppressive dominance hierarchies
can grow. The most vicious battles in human affairs are the
battles that derive from such entrenched dogmas. Cantor's
Theory is not heading us in the right direction.
Those who have been on the net for a long time may recall that
I presented these ideas in sci.math many years ago. They weren't
well received. I gave up believing that there was any reason for
me to press the issue, as the likely result would be a tremendous
effort without reward.
Well, just recently we've seen a rather precocious eleven year
old boy take up the challenge of debunking Cantor's Theory. And
not surprisingly, the defenders of the faith have called him stupid,
insane and worse. I can't just ignore that.
Granted, the situation is somewhat like the story of the little boy who
said "the king has no clothes". Maybe he can't yet explain to true
believers what is wrong with their theories purporting to prove the
king is dressed in fine splendor, but he sure knows what his eyes
tell him. I'd like to encourage him to continue (while warning him of
the dangers; he'll be battling with some insanely devious minds).
I think he just might have enough spunk to win the battle. And the
rewards for winning could be quite substantial.
Cantor was a mad genius. He created a "logically consistent"
(assuming reality checks are not an essential criterion of logic)
and seductive fantasy world. His dreams of unifying theology and
mathematics were never realized, but he did succeed in seducing
the mathematicians for a hundred years, and thereby turning
mathematics into a pseudo-theology.
Cantor's Theory is a dogma. It's a mythology. It's an intellectual
fraud. It's destructive. It deserves to die.
*****
Verdigris: One can see that Cantor would have liked to create an
ultimate mathematical entity that shows the nature of God. This
Platonic ultimate exists as an idea but cannot be expressed in a
definite way. The crux of the matter is that an ABSOLUTE infinity
is either the only infinity or a contradiction in terms. If it is
the only infinite entity, then nothing transcends or exceeds it, otherwise
it
is not the ultimate infinity. The alternative is that an infinite entity
can be exceeded by another infinite entity. The question then
becomes: how many such entities are there and can there be a
greatest one? If there are many such entities then there can be
no greatest definable entity, even though we can well understand
the notion of an infinity which cannot be transcended. The problem
is that we must choose between the principle of absolute
transcendence, whereby any entity can be transcended
and the the possibility of an entity that cannot be transcended.
Cantor decided that there were many infinite entities but said that
the cardinality of the reals could not be exceeded. This conclusion
clearly contradicts the idea that there is no ultimate entity that can't
be transcended. Cantor's arithmetic purports to achieve closure
by forbidding entities that transcend the reals on the number line.
However, this is an arbitrary decision to forbid the occurrence of
readily definable infinitesimals on the number line. This restriction
makes sense by banishing all infinite entities from arithmetic. This
fits in well with your desire to see only finite entities in arithmetic
even though they may never be computable even in principle.
The idea of computability is a useless one because it arbitrarily
discriminates between the computable finite and the incomputable
finite. It is worse than useless when it comes to the infinite, which it
cannot describe in any way.
My preference is to retain the principle of transcendence but restrict
its use to 'constructible' infinities. The result is that absolute infinity
can never be defined as a constructible entity. The resulting realm
of the infinite is much greater than that envisaged by Cantor because
it begins where he left off.
VERDIGRIS
*****
Nico Benschop
[...]
absolute transcendence, whereby any entity can be transcended -
and the possibility of an entity that cannot be transcended.
Cantor decided that there were many infinite entities but said that
the cardinality of the reals could not be exceeded. This conclusion
clearly contradicts the idea that there is no ultimate entity that
can't be transcended. ...(#)
Cantor's arithmetic purports to achieve closure by forbidding
entities that transcend the reals on the number line. However, this is
an arbitrary decision to forbid the occurrence of readily definable
infinitesimals on the number line. This restriction makes sense by
banishing all infinite entities from arithmetic. This fits in well
with your desire to see only finite entities in arithmetic
even though they may never be computable even in principle.
The idea of computability is a useless one because it arbitrarily
discriminates between the computable finite and the incomputable
finite. It is worse than useless when it comes to the infinite, which
it cannot describe in any way.
My preference is to retain the principle of transcendence but restrict
its use to 'constructible' infinities. The result is that absolute
infinity can never be defined as a constructible entity. The resulting
realm of the infinite is much greater than that envisaged by Cantor
because it begins where he left off. -- VERDIGRIS
------------------------------------------------------------------**]
Re(#): The whole idea of some Ultimate Infinity, not to be
transcended, is quite immature since it considers only quantities,
and not qualities. Mapping Everything onto a "real" line is quite
absurd: don't there exist Graphs, (Semi-)Groups, Languages, etc..etc?
Even in Cantor's case: comparing 2^N with N is comparing two distinct
TYPES of objects -- Flattening all onto a single line, with 'density'
or some other measure supposedly capturing the lost essence is short
of ridiculous, or to put it mildly: ill advised. The powerset 2^N of
Peano's sequentially generated naturals N = {+1}* has a totally
different structure, namely that of a Boolean Lattice (intersection
and union) - given some axioma that these finite concepts do make
sense for infinite objects. So what?
Why count & compare dissimilar things? Are you sensibly comparing the
Alhambra with the sandgrains it consists of eventually...? Whole
generations of mathematicians are raised on Cantor's diagonal, and
still new generations revive it in the P?=NP arena -- All this about
some form of infinity, *without* considering differences in 'quality'
cq 'type-of-structure'..
The limit IMHO is this: For a realistic person, there are a^n distinct
strings of length n over an alphabet A of size |A|=a. And if it makes
you happy: let this also hold for infinite strings over some infinite
alphabet ;-)
There is an algebraic structure S (semigroup of all transformations
of a set T), defined over a set T of n symbols (states or letters)
that contains maximally n^n objects (functions of T into T), which
even for finite n has a general structure/ decompostion that is not
understood fully. Groups [n!] and Lattices [2^n], Languages [a^n] and
Arithmetic and Logic, they all fall under this denominator (the
associative algebra of strings under concatenation, including Graphs
and Matrices and StateMachines, etc... ;-).
Can't we decide to waste a bit less intellectual energy on pressing
these all onto a single "real-line" - And rather spend some effort in
distinguishing the various types of objects, with corresponding
composition operations, and build some integrated understanding of all
these things and their structure - Finite first... and Infinite later
- after we al least understand them in-the-finite? Otherwise (doing
the infinite before the finite: would suggest some urge to escape,
would'nt it ?-) --------- Voer voor psychologen;-) --------
Just a suggestion...
Ciao, Nico Benschop (email\X) -- http://www.iae.nl/users/benschop
Jonathan W. Hoyle
after the fact. It seems as anachronistic to me as someone arguing in
favor of a state's right to seceed from the Union as in the Civil War.
In the same way, Cantor's diagonal proof is part of mainstream
Mathematics.
You will not find Mathematicians today debating this point. I'm sure
there are a few exceptions out within the crank literature, just as you
will find people who do not believe in relativity or quantum mechanics,
but professionals in the field (by an overwhelming majority) accept the
reality of each of these. To argue against Cantor, you're going to
sound to Mathematicians an awful lot like people from the Flat Earth
Society [their amusing site at
http://www.calpoly.edu/~lgranill/flatearth.html].
Thousands of years ago, there were people who refused to accept that the
square root of 2 is irrational. This just went violently against their
preconceived philosophies. Hundreds of years ago, non-Euclidean
Geometry was rejected outright by many people. Less than 100 years ago,
Weierstrass' everywhere-continuous nowhere-differentiable function was
considered a monstrosity to some. And still today, there are cranks who
are dead-set against Goedel's theorems. But such people are not
considered seriously in the Mathematics community today.
Mathematics is constantly improving its precision. An inconsistency in
Cantor's original Set Theory was fixed by Zermelo and Fraenkel. Cantor
was unable to prove his Continuum Hypothesis, which was later proven to
be undecidable. But these refinements don't undo his work, they only
make them better. But mathematical theorems don't become "unproven"
later with time, by some whim of prevailing philosophy. No one is going
to later "unprove" the irrationality of the square root of 2. And
Cantor's results aren't going to "die" because it may offend someone's
philosophical preference.
If our philosophies conflict with mathematics, it's our philosophies
that must change, not our mathematics.
Torkel Franzen
>I am amazed that this discussion on Cantor is even debatable 100 years
>after the fact.
The anti-Cantor crackpot brigade will always be with us, as will be
people like David Petry, who dislike set theory and go on about it
in portentous droning tirades, and people like "Nathan" who yank
other people's chains by asking about infinite natural numbers and
such. It all goes into the big melting pot!
David Petry
Torkel Franzen wrote in message ...
Torkel, you forgot to give us your age in this post.
John Savard
>Cantor's Theory is a dogma. It's a mythology. It's an intellectual
>fraud. It's destructive. It deserves to die.
There are difficulties with the transfinite - but that simply means
that, unlike some of the more pedestrian parts of algebra and
calculus, there is still the opportunity to resolve the contradictions
and achieve new understanding.
The discomfort does not mean it deserves to die, it means that it is
very much alive.
John Savard (teneerf is spelled backwards)
http://members.xoom.com/quadibloc/index.html
Paul Hammond
Verdigris wrote:
>
> Cantor decided that there were many infinite entities but said that
> the cardinality of the reals could not be exceeded. This conclusion
> clearly contradicts the idea that there is no ultimate entity that can't
>
No he didn't! He proved that for _any_ set, it's powerset has a greater
cardinality. The diagonal argument just proves that |R|>|N|, _not_ that
for any set X, |R|>|X|.
If you want a bigger set than R, just take P(R), the set of all subsets of
R.
The continuum hypothesis is the theorem that |R|=|P(N)| (alternatively
written C=\aleph_one).
----
p...@maths.nott.ac.uk
Paul Hammond | "Only teenagers could be
Maths Dept | that incoherent"
University of Nottingham |
Nottingham NG7 2RD | - Lisa Simpson
Paul Hammond
> Anyone who follows sci.math must be aware that there
> are many people who find something to object to about
> Cantor's Theory. I believe that the attack on Cantor's
> theory will not end until someone has found a way to
> drive a stake right through the heart of the theory.
> *****
> Verdigris: I think Cantor was right to 'invent' infinite
> categories but was wrong in the way he chose to
> do it. I think the criterion of counting embodied in the
> concept of bijection is wrongly applied to infinite sets.
> This is because they cannot be enumerated as a temporal
> operation in the way that finite sets can be enumerated.
> Although one can always begin to map one-one from
> n to f(n) the operation can never be completed, even
> in principle. The assertion that there is a complete
> mapping from n to f(n) is therefore unjustified.
> The idea that the cardinality of such sets exists is
> suspect as a result of their necessay incompleteness.
> In other words, an incomplete set can have no total
> so it is meaningless to assert that the cardinality of f(n)
> is the same as the cardinality of n.
> *****
[Deleted: chunk of Petry's post with no comments]
> To someone who is comfortable with thinking about the
> world of computation as a real world, "mathematics" is
> most usefully defined as the study of phenomena that are
> observable in the world of computation. That is, we can
> think of the computer as a microscope that helps us peer
> into the world of computation, and "mathematics" as the
> science that studies the phenomena observed through
> that microscope.
>
> *****
> Verdigris: Mathematics can not be confined to the
> computable or demonstrable. Consider the totality of
> all functions of the form y = f(x); there is no possible
> computer program that could enumerate each individual
> function, or even a class of funtions, particularly if they
> have real parameters. The infinite nature of number
> extends to geometric forms. In any case, if the universe
> is finite, there will always be sufficiently large finite
> numbers (parameters) which will render computations
> based on them impossible. Computation must always
> fail at some point to match definite large finite numbers.
> *****
[rest deleted]
You have just said above that you do not like the idea of a bijection
between infinite sets, because the mapping can not be completed, and yet
you are happy to consider the totality of all functions of the form
y=f(x).
Do you consider that the function R -> R defined by f(x)=x^2 (for example)
to exist? Or is the very idea of such a function uncompletable because of
the infinite sets involved?
It seems to me the idea of a bijection f:N -> N\{0} defined by f(n)=n+1 is
much simpler, yet it causes people to pause for thought.
On the other hand, people are very much at home with the concept of y=x^2,
because they have drawn a portion of it on graph paper at some stage in
their lives.
Maybe the idea that something is 'natural' just depends on how much
experience people have in thinking about it.
This, of course, is before we get to the much more complicated 'set' (or
proper class?) of ALL functions R -> R, which you have introduced above
with seemingly no awareness of what bearing such a concept has for your
first observation about uncompleted infinities.
Nico Benschop
>
> Verdigris wrote:
>>
>> Cantor decided that there were many infinite entities but said that
>> the cardinality of the reals could not be exceeded. This conclusion
>> clearly contradicts the idea that there is no ultimate entity that
>> can't be transcended. Cantor's arithmetic purports to achieve closure
>> by forbidding entities that transcend the reals on the number line.
>>
>
> No he didn't! He proved that for _any_ set, it's powerset has a
> greater cardinality. The diagonal argument just proves that |R|>|N|,
> _not_ that for any set X, |R|>|X|.
>
> If you want a bigger set than R, just take P(R), the set of all
> subsets of R.
>
> (alternatively written C=\aleph_one). ---- Paul Hammond
Re(#): I tought that CH was the conjecture that:
No type of infinity exists between that of N and its powerset 2^N.
(for instance no polynomial function N^k, finite k,
yields an cardinality beyond that of |N| : countable.
To me this suggests: no function_type between polynomes
and exponentials ..?)
Or does that boil down to the same?
--
Ciao, Nico Benschop. | AHA: One is Always Halfway Anyway
http://www.iae.nl/users/benschop | xxxxxxxxxxxxxxx1.1xxxxxxxxxxxxxxx
http://www.iae.nl/users/benschop/cantor.htm
Ulrich Weigand
>Paul Hammond wrote:
>> If you want a bigger set than R, just take P(R), the set of all
>> subsets of R.
>>
>> The continuum hypothesis is the theorem that |R|=|P(N)| ...(#)
>> (alternatively written C=\aleph_one). ---- Paul Hammond
Not quite ... That C = |R| = |P(N)| is a simple fact. The continuum
hypothesis is indeed that C = \aleph_one (or, equivalently, |P(N)| =
\aleph_one), but this is *not* simply an alternative formulation of
|R| = |P(N)|. (Why would you think so?)
>Re(#): I tought that CH was the conjecture that:
> No type of infinity exists between that of N and its powerset 2^N.
Yes, that's equivalent. (\aleph_one is per definition the smallest
uncountable cardinal; hence if \alpeh_one equals C, there cannot be
any cardinal in between |N| = \aleph_0 and |2^N| = C.)
> (for instance no polynomial function N^k, finite k,
> yields an cardinality beyond that of |N| : countable.
> To me this suggests: no function_type between polynomes
> and exponentials ..?)
This does seem a somewhat strange way of stating it ... N^k and
2^N are not *functions* here. N^k is simply the set of all k-tuples
of natural numbers, while 2^N is the set of all subsets of N.
--
Ulrich Weigand,
IMMD 1, Universitaet Erlangen-Nuernberg,
Martensstr. 3, D-91058 Erlangen, Phone: +49 9131 85-7688
Nico Benschop
>
> Nico Benschop <bens...@iae.nl> writes:
> [...]
> >Re(#): I tought that CH was the conjecture that:
> > No type of infinity exists between that of N and its powerset 2^N.
>
> Yes, that's equivalent. (\aleph_one is per definition the smallest
> uncountable cardinal; hence if \alpeh_one equals C, there cannot be
> any cardinal in between |N| = \aleph_0 and |2^N| = C.)
>
> > (for instance no polynomial function N^k, finite k,
> > yields an cardinality beyond that of |N| : countable.
> > To me this suggests:
>
> This does seem a somewhat strange way of stating it ... N^k and
> 2^N are not *functions* here. N^k is simply the set of all k-tuples
> of natural numbers, while 2^N is the set of all subsets of N.
> -- Ulrich Weigand
Re(#): Let me clarify the idea, related to integer functions
counting the cardinality of sets:
For set N^k there is polynomial function f(k) = n^k,
where n = |N| of set N, (countably) infinite.
and set k^N yields exponential function g(n) = k^n.
The idea is then, that g(n) cannot be represented by a product
or sum of polynomials of type f(k), that is: of finite degree upto k.
Only for k --> |N|, hence k=n, we can get 'exponential behaviour' n^n.
(and then also 2^n, which cardinally is equivalent to n^n ;-)
My urge is to get way from the 'infinity' concept, and translate it to
the essential difference between polynomial and exponential, hence into
functional or 'structural' difference.
Re: My recent post on Generation vs Closure.
"Too much is mapped onto the number line" ;-)
Aatu Koskensilta
Nico Benschop wrote in message <36DDBF...@iae.nl>...
>Paul Hammond wrote:
>> The continuum hypothesis is the theorem that |R|=|P(N)| ...(#)
>> (alternatively written C=\aleph_one). ---- Paul Hammond
>
>Re(#): I tought that CH was the conjecture that:
> No type of infinity exists between that of N and its powerset 2^N.
[snip]
> Or does that boil down to the same?
They boil down to the same.
Informal explanation: Consider a real number defined as
a function from w (the set of natural numbers, or the set
of finite ordinals, however you like) into {0,1}. . Obviously,
there are card({0,1})^w = 2^aleph-nought such functions.
(We're not taking into account the countable number of
"degenerate" functions describing improper decimal
expansions).
Now, the continuum hypothesis (or one form of it) states that
2^aleph-nought = aleph-1, aleph-1 being the "next" greatest
cardinal. Obviosuly, since alpeh-1 is the cardinal "following"
aleph-nought, there can be no cardinality between these
two. Saying beth-1 = aleph-1, or saying that 2^aleph-nought =
alpeh-1, or saying that "there is no cardinality are equivalent.
Hope this helps.
---
Aatu Koskensilta (squ...@seaga.org)
"Wovon man nicht reden kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
bensc...@my-dejanews.com
Weak answer, David.
Although I'm not impressed with Tokel's 'crackpot brigade' retort either;-(
I think David has a few points worth taking seriously, re: his critique
under heading: too much abstraction and too little object-oriented algebra &c.
In several recent posts, among others in this Cantor thread, I have given
a contribution to an analysis that might help us to go beyond "mapping
all onto the real number line" - and out towards recognizing the various
structures & algebra's in their own right, but in connection with e.g. Cantor
(restricted to associative, but tackling that mysterious "exponentiation" all
right;-)
Unfortunately, none of the esteemed correspondents in this NG deemed it
necessary to answer these suggestions, or to contribute. Sci.math appears
to be a cosy chat medium for mathematicians/scientists with time to spare,
for funny and high-spirited personal attacks, and for condescending talk that
brings us no further. In other words: a perfect waste of this excellent open
medium;-(
And for all practical purposes, viewed from my side: a "black hole"
-- after > 1500 posts since oct95, of which some 500 on FLT/FST, with too low
efficiency of communication / and with little chance on a better alternative
NG as well ...: Nick Halloway's robot-moderated "sci.math.moderated" in status
nascendi barely breathes, while sci.math.research is "moderated to death" by
some narrow-minded clique. It's a sad story all around, noise abounds ;-(
Ciao, do it your way...
Nico Benschop -- http://www.iae.nl/users/benschop
http://www.iae.nl/users/benschop/cantor.htm
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Nico Benschop
Nico Benschop <bens...@iae.nl> writes:
>Paul Hammond wrote:
>> If you want a bigger set than R, just take P(R), the set of all
>> subsets of R.
>>
>> The continuum hypothesis is the theorem that |R|=|P(N)| ...(#)
>> (alternatively written C=\aleph_one). ---- Paul Hammond
Not quite ... That C = |R| = |P(N)| is a simple fact. The continuum
hypothesis is indeed that C = \aleph_one (or, equivalently, |P(N)| =
\aleph_one), but this is *not* simply an alternative formulation of
|R| = |P(N)|. (Why would you think so?)
>Re(#): I tought that CH was the conjecture that:
> No type of infinity exists between that of N and its powerset 2^N.
Yes, that's equivalent. (\aleph_one is per definition the smallest
uncountable cardinal; hence if \alpeh_one equals C, there cannot be
any cardinal in between |N| = \aleph_0 and |2^N| = C.)
> (for instance no polynomial function N^k, finite k,
> yields an cardinality beyond that of |N| : countable.
> To me this suggests: no function_type between polynomes
> and exponentials ..?)
This does seem a somewhat strange way of stating it ...
N^k and 2^N are not *functions* here.
N^k is simply the set of all k-tuples of natural numbers, ...[#]
while 2^N is the set of all subsets of N. ...[#]
-- Ulrich Weigand.
------------------------------------------------------------------**]
Re[#]: We have been at this before ;-)
And I had a question that never was plied to, AFAIK.
Namely: consider 2^N as the Boolean lattice of ordered subsets of N,
each 'level' k representing all binary strings with k ones.
Then the lattice top is the single element 1* (all ones)
and the lattice bottom is single element 0* (all zero's)
and lattice level k=1 are all 0/1 strings with just one 1,
in an obvious way representing the set of naturals N.
Let BL_k be the k-th level in the Boolean Lattice 2^N, being
countable and of order {|N| choose k}, a binomial coefficient.
Now BL_1 = N,
and BL_2 = quadratic polynomial in N
('set-wise': all binary strings with 2 ones ;-)
and BL_k = degree k polynomial in N. ( same ... with k ones )
It is known that no UC (UnCountable) set can be
a countable union of countable sets. ...[1]
But here we have: 2^N = \union BL_k (countable #levels k=0,..,|N|=w)
where each subset BL_k is countable. ...[2]
My question:
By [2] we have: 2^N is a countable union of countable sets,
yet by Cantor's diagonal argument 2^N is uncountable [1], how come?
Does not [2] contradict [1] ? What is the clue here ?
Thanks for any comment. Nico Benschop (email\X)
-- http://www.iae.nl/users/benschop/cantor.htm
Dave Seaman
Nico Benschop <bens...@iae.nlX> wrote:
>My question:
> By [2] we have: 2^N is a countable union of countable sets,
Not 2^N, but 2^N|finite, the set of all finite subsets of N. That is
indeed a countable set.
> yet by Cantor's diagonal argument 2^N is uncountable [1], how come?
Because the set of all infinite subsets of N is uncountable, but is not
included in your lattice.
--
Dave Seaman dse...@purdue.edu
Pennsylvania Supreme Court Denies Fair Trial for Mumia Abu-Jamal
<http://mojo.calyx.net/~refuse/altindex.html>
Arthur L. Rubin
Someone's quoting style is undecipherable...
> Namely: consider 2^N as the Boolean lattice of ordered subsets of N,
> each 'level' k representing all binary strings with k ones.
> Then the lattice top is the single element 1* (all ones)
> and the lattice bottom is single element 0* (all zero's)
> and lattice level k=1 are all 0/1 strings with just one 1,
> in an obvious way representing the set of naturals N.
>
> Let BL_k be the k-th level in the Boolean Lattice 2^N, being
> countable and of order {|N| choose k}, a binomial coefficient.
>
> Now BL_1 = N,
> and BL_2 = quadratic polynomial in N
> ('set-wise': all binary strings with 2 ones ;-)
> and BL_k = degree k polynomial in N. ( same ... with k ones )
>
> It is known that no UC (UnCountable) set can be
> a countable union of countable sets. ...[1]
>
> But here we have: 2^N = \union BL_k (countable #levels k=0,..,|N|=w)
> where each subset BL_k is countable. ...[2]
BL_w is not countable.
--
Arthur L. Rubin 216-...@mcimail.com
bens...@iae.nl
OK, so we have: |BL_k| = [w choose k] countable.
Then binomial coefficient |BL_w| = [w choose w] is UC,
as only one of the terms of |2^N| = \sum_k [w choose k]. --NB
Of course, I must admit not having the foggiest idea what BL_w is, or
[w choose w]. Stretching my imagination to BL_k exhausted me already;-)
bens...@iae.nl
a...@seaman.cc.purdue.edu (Dave Seaman) wrote:
> In article <v2mg3u...@forum.swarthmore.edu>,
> Nico Benschop <bens...@iae.nlX> wrote:
>>My question:
>> By [2] we have: 2^N is a countable union of countable sets,
>
> Not 2^N, but 2^N|finite, the set of all finite subsets of N. That is
> indeed a countable set.
>
>> yet by Cantor's diagonal argument 2^N is uncountable [1], how come?
>
> Because the set of all infinite subsets of N is uncountable,
> but is not included in your lattice. -- Dave Seaman
>
I can imagine BL_w is uncountable,
but it must be in lattice BL, which contains _all_ binary strings.
I'm just wondering _where_ ...;-(
It must be below the lattice top_element, which is 1*,
and above the bottom element 0*.
And in fact ordered above BL_k for any finite k,
but below complementary BL'_k (all strings of k zero's).
Narrowing it down like this, BL_w must be somewhere in the middle,
wouldn't it? Maybe even "by symmetry": precisely halfway up BL...?
I'm afraid this game will remain guesswork forever ;-(
(I mean: without any sensable semantics.., with w around the corner)
Ciao, Nico Benschop -- http://www.iae.nl/users/benschop/cantor.htm
Dave Seaman
>I can imagine BL_w is uncountable,
> but it must be in lattice BL, which contains _all_ binary strings.
> I'm just wondering _where_ ...;-(
> It must be below the lattice top_element, which is 1*,
> and above the bottom element 0*.
> And in fact ordered above BL_k for any finite k,
> but below complementary BL'_k (all strings of k zero's).
I didn't entirely follow your notation the first time. Yes, BL_w would
be above all the finite subsets (the BL_k's) and below all the
co-finite subsets (the BL'_k's). It would consist of all the strings
that have infinitely many 1's and infinitely many 0's.
Ed Hook
bens...@iae.nlX (Nico Benschop) writes:
|>
|> Namely: consider 2^N as the Boolean lattice of ordered subsets of N,
... where N is the set of natural numbers (? -- that seems to
be correct, based upon the context which I mercilessly snipped).
|> each 'level' k representing all binary strings with k ones.
|> Then the lattice top is the single element 1* (all ones)
|> and the lattice bottom is single element 0* (all zero's)
|> and lattice level k=1 are all 0/1 strings with just one 1,
|> in an obvious way representing the set of naturals N.
|>
|> Let BL_k be the k-th level in the Boolean Lattice 2^N, being
|> countable and of order {|N| choose k}, a binomial coefficient.
|>
"|N| choose k" doesn't (strictly speaking) make sense. It is
true, however, that your BL_k is countable _for_each_k_in_N_
(i.e., for each *finite* k).
|> Now BL_1 = N,
|> and BL_2 = quadratic polynomial in N
|> ('set-wise': all binary strings with 2 ones ;-)
|> and BL_k = degree k polynomial in N. ( same ... with k ones )
|>
I don't quite understand how you're introducing polynomials
here, but I don't think that's the fundamental problem ...
|> It is known that no UC (UnCountable) set can be
|> a countable union of countable sets. ...[1]
|>
|> But here we have: 2^N = \union BL_k (countable #levels k=0,..,|N|=w)
|> where each subset BL_k is countable. ...[2]
|>
OK, there you just tried to pull a fast one -- as you indicate
(not very explicitly _at_all_), you have to use BL_w (as well
as all of the BL_k with finite k) in order to build up 2^N.
But BL_w is _not_ countable, it's uncountable.
|> My question:
|> By [2] we have: 2^N is a countable union of countable sets,
|>
|> yet by Cantor's diagonal argument 2^N is uncountable [1], how come?
|>
|> Does not [2] contradict [1] ? What is the clue here ?
|>
Hopefully, you followed the above ...
|> Thanks for any comment. Nico Benschop (email\X)
|> -- http://www.iae.nl/users/benschop/cantor.htm
|>
--
Ed Hook | Copula eam, se non posit
MRJ Technology Solutions, Inc. | acceptera jocularum.
NAS, NASA Ames Research Center | I can barely speak for myself, much
Internet: ho...@nas.nasa.gov | less for my employer
bensc...@my-dejanews.com
a...@seaman.cc.purdue.edu (Dave Seaman) wrote:
> In article <7bmn99$u1i$1...@nnrp1.dejanews.com>, <bens...@iae.nl> wrote:
> >I can imagine BL_w is uncountable,
> > but it must be in lattice BL, which contains _all_ binary strings.
> > I'm just wondering _where_ ...;-(
> > It must be below the lattice top_element, which is 1*,
> > and above the bottom element 0*.
> > And in fact ordered above BL_k for any finite k,
> > but below complementary BL'_k (all strings of k zero's).
That is: BL_w is somewhere in the middle, halfway up the lattice ordering.
>
> I didn't entirely follow your notation the first time. Yes, BL_w would
> be above all the finite subsets (the BL_k's) and below all the co-finite
> subsets (the BL'_k's). It would consist of all the strings
> that have infinitely many 1's and infinitely many 0's. -- Dave Seaman
>
Yes indeed. Isn't it interesting to decompose 2^N by expanding (1+1)^N.
So BL_w is the powerset of N minus the finite_rank strings & their complements
(the rank of a binary string is the #ones in it,
re: the finite subsets of N, and their complements):
BL_w = 2^N \ Sum_finite_k {BL_k v BL'_k}
(union "v", lattice complement BL'_k)
Next question: |BL_w| = |2^N| ?
is Cardinality of BL_w equal to that of 2^N (hence > w)?
(then: removing all finite_rank strings & complements makes no difference)
BTW: Continuing in the detailed structural fashion of Cantor, we have: For
finite and even 2n, the expansion of (1+1)^{2n} has middle binom_coefficient
[2n choose n] = (\prod n+i)/(\prod i), i=1..n, which in the limit n --> inf
might be useful in the above. Denote [n choose k] as n#k: 4#2=6, 6#3=20,
8#4=70, 10#5=252 The limit(n-->oo) of ratio (2n+2)#(n+1) / 2n#n = 4, if I'm
not mistaken, just as 2^{2n+2} / 2^{2n} (re: corresp. Pascal rows) Aim: If
there is a cardinality between |N| and |2^N|, then possibly |BL_w|... And if
not |BL_w|, then none other can exist (?) since no finer partition of 2^N
exists. -- beyond singletons -- [...satisfying some algebraic/axiomatic
obvious condition, like ...]
Comments are welcome.
Ciao, Nico Benschop (bens...@iae.nl)
Dave Seaman
<bensc...@my-dejanews.com> wrote:
>So BL_w is the powerset of N minus the finite_rank strings & their complements
>(the rank of a binary string is the #ones in it,
> re: the finite subsets of N, and their complements):
>
> BL_w = 2^N \ Sum_finite_k {BL_k v BL'_k}
> (union "v", lattice complement BL'_k)
>
>Next question: |BL_w| = |2^N| ?
>
> is Cardinality of BL_w equal to that of 2^N (hence > w)?
Yes. Since 2^N is the union of BL_w and a countable set, it follows that
|BL_w| = |2^N| = c.
> (then: removing all finite_rank strings & complements makes no difference)
None to speak of, since those sets are small.
>BTW: Continuing in the detailed structural fashion of Cantor, we have: For
>finite and even 2n, the expansion of (1+1)^{2n} has middle binom_coefficient
>[2n choose n] = (\prod n+i)/(\prod i), i=1..n, which in the limit n --> inf
>might be useful in the above.
Limits in the traditional epsilon-delta sense don't mix very well with
transfinite ordinals and cardinals. A "limit" in the world of
transfinites is often a set union over some infinite collection of
sets. For example, w is the limit of the natural numbers, because
w = lim k = \union k.
k<w k<w
Aatu Koskensilta
Aatu Koskensilta wrote in message <7bm1p9$idd$1...@tron.sci.fi>...
>Saying beth-1 = aleph-1, or saying that 2^aleph-nought =
>alpeh-1, or saying that "there is no cardinality are equivalent.
This should read "there is no cardinality between the cardinality
of natural numbers and that of real numbers".
"There is no cardinality" is no sort of continuum hypothesis as
far as I know :)
---
Aatu Koskensilta (zap...@sci.fi)
bens...@iae.nl
ho...@nas.nasa.gov wrote:
> In article <v2mg3u...@forum.swarthmore.edu>,
> bens...@iae.nlX (Nico Benschop) writes:
> |>
> |> Namely: consider 2^N as the Boolean lattice of ordered subsets of N,
>
> ... where N is the set of natural numbers (? -- that seems to
> be correct, based upon the context which I mercilessly snipped).
>
> |> each 'level' k representing all binary strings with k ones.
> |> Then the lattice top is the single element 1* (all ones)
> |> and the lattice bottom is single element 0* (all zero's)
> |> and lattice level k=1 are all 0/1 strings with just one 1,
> |> in an obvious way representing the set of naturals N.
> |>
> |> Let BL_k be the k-th level in the Boolean Lattice 2^N, being
> |> countable and of order {|N| choose k}, a binomial coefficient.
> |>
>
> "|N| choose k" doesn't (strictly speaking) make sense. It is
> true, however, that your BL_k is countable _for_each_k_in_N_
> (i.e., for each *finite* k).
>
> |> Now BL_1 = N,
> |> and BL_2 = quadratic polynomial in N
> |> ('set-wise': all binary strings with 2 ones ;-)
> |> and BL_k = degree k polynomial in N. ( same ... with k ones )
> |>
>
> I don't quite understand how you're introducing polynomials
> here, but I don't think that's the fundamental problem ...
>
> |> It is known that no UC (UnCountable) set can be
> |> a countable union of countable sets. ...[1]
> |>
> |> where each subset BL_k is countable. ...[2]
> |>
>
> OK, there you just tried to pull a fast one -- as you indicate
> (not very explicitly _at_all_), you have to use BL_w (as well
> as all of the BL_k with finite k) in order to build up 2^N.
>
> But BL_w is _not_ countable, it's uncountable.
>
> |> My question:
> |> By [2] we have: 2^N is a countable union of countable sets,
> |>
> |> yet by Cantor's diagonal argument 2^N is uncountable [1], how come?
> |>
> |> Does not [2] contradict [1] ? What is the clue here ?
> |>
>
Yes I did, thanks.
See also my answer today to Dave Seaman in this thread.
My idea was to decompose (partition) powerset 2^N, as Boolean
Lattice, into its "rank-sets" (same # of ones). In order to
find a possibly 'smallest' part, under some simple criterion,
that is still UC and of the same cardinality as 2^N:
which is BL_w, the set of all binary strings with w ones
and w zero's ... somewhere "in-the-middle" halfway up the
Boolean Lattice. If this can be shown the finest possible
non-trivial partition of 2^N, and BL_w is UC, than this
might have some consequence for CH (Continuum Hypothesis).
Nico Benschop (email\X)
-- http://www.iae.nl/users/benschop/cantor.htm
Arthur L. Rubin
> "|N| choose k" doesn't (strictly speaking) make sense.
Actually, it does. Let denonte X and Y as arbitrary sets, and
m and n as arbitrary cardinals. We can define
X choose m as the {Y \subset X : |Y| = m)
As, if |X| = |Y|, then |X choose m| = |Y choose M\m|, we can
define "n choose m" as |X choose m" for any X such that |X|=n.
Lee Rudolph
>Ed Hook wrote:
>
>> "|N| choose k" doesn't (strictly speaking) make sense.
>
>Actually, it does. Let denonte X and Y as arbitrary sets, and
>m and n as arbitrary cardinals. We can define
>
>X choose m as the {Y \subset X : |Y| = m)
And if we want to have a categorical/combinatorial
orgy, we might even define "X choose Y" as the quotient
of "injections of Y into X" by the natural action of
Aut(Y) on that object. (It does seem, at second
glance, like one might have to change categories
in the middle of the stream, once we leave Sets
behind. For example, I'd like to say that
"R^n choose R" is real projective (n-1)-space;
similarly, my favorite space in the whole world,
the configuration space of n pairwise distinct
points of C [i.e., the natural model for the
Eilenberg-Maclane space K(B_n,1) of the n-string
braid group], should be "C choose [n]". But how
to make sense of that categorically? I bet
John Baez knows!)
Lee Rudolph