Cantor used the expression 2^aleph0 in order to represent the magnitude of R set.

Since base 2 can be represented as a tree diagram, we can use it in order to research a collection of infinitely many elements.

For example, let us look at the infinitely long Top_to_Bottom blue tree, which is also represented as
{1, 2, 4, 8, 16, ...}.

It is obvious that we always find finitely many leafs in any arbitrary level of this tree, so this tree cannot have the magnitude of 2^aleph0.

Furthermore, since in any arbitrary level we are still in N set, we can never define aleph0 as a transfinite number.

Now let us say that we start by a collection of infinitely many R members, which are represented by infinitely many brown points.

In this case, we know that we can never start to use base 2 in order to construct a Bottom_to_Top tree, if our collection of points can construct a solid line, and if we do that, we discover that we get infinitely many identical trees that cannot have |R| (if, again, R set is like a solid line).

So my question is: How can we write 2^aleph0, if base 2 cannot exist when we deal with |R|?

(This question is good for any base n)



1 |
2^0 _______._______ |
/ \ |
2^1 ___.___ ___.___ |
/ \ / \ |
2^2 _._ _._ _._ _._ |
/ \ / \ / \ / \ |
2^3 . . . . . . . . |
/ \ / \ / \ / \ / \ / \ / \ / \ V
2^4 . . . . . . . . . . . . . . . . oo


8 8
2^3 ___.___ ___.___
/ \ / \
2^2 _._ _._ _._ _._
/ \ / \ / \ / \
2^1 . . . . . . . .
/ \ / \ / \ / \ / \ / \ / \ / \ ---> oo
2^0 . . . . . . . . . . . . . . . . . . . . ...


4 4 4 4
2^2 _._ _._ _._ _._
/ \ / \ / \ / \
2^1 . . . . . . . .
/ \ / \ / \ / \ / \ / \ / \ / \ ---> oo
2^0 . . . . . . . . . . . . . . . . . . . . ...


2 2 2 2 2 2 2 2
2^1 . . . . . . . .
/ \ / \ / \ / \ / \ / \ / \ / \ ---> oo
2^0 . . . . . . . . . . . . . . . . . . . . ...


1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ---> oo
2^0 . . . . . . . . . . . . . . . . . . . . ...

If we look at http://mathworld.wolfram.com/PowerSet.html then we can see that the expression 2^S (where S is any number) is a general notation of a power set.


It means that |{0,1}| standing in the base of any power set.

And it is easy to show that, for example, 2^3 is equivalent to the power set of 3:


8
2^3 ___.___
/ \
2^2 _._ _._
/ \ / \
2^1 . . . .
/ \ / \ / \ / \
2^0 . . . . . . . .
1 2 3 4 5 6 7 8

is equivalent to |{{}, {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, {0,1,2}}| = 2^3
1 2 3 4 5 6 7 8


And because base |{0,1}| cannot be found, then the whole idea of the transfinite cardinals does not hold, if the R set elements have no room to construct a Binary tree.

And if they have a room to construct a Binary tree, then R set is enumerable, and Cantor's second diagonal method does not hold.

It has been along time since I read about Cantor and his transfinite numbers, so there might be some errors in the following.

I do not think that Cantor used expressions like 2^Aleph0, although such an expression is implied by his logic. .

He defined Aleph₀ as the transfinite number associated with the set of all integers. He proved that a subset of Aleph₀ could be put into a one-to-one correspondence with the entire set. He used that property as a definition of a transfinite set.

He proved that the set of all subsets of any set could not be put into a one-to-one correspondence with the original set, while a set with twice as many members could be put into one-to-one correspondence with the original set if the original set was transfinite. . .

He then defined Aleph₁ as the transfinite number associated with the set of all subsets of Aleph₀. He defined Aleph₂ as the set of all subsets of Aleph₁, et cetera.

From the above, you can say that 2*Aleph₀ is the same transfinite number as Aleph₀, but that 2^Aleph0 is a larger transfinite number.

I think that in formal texts, references are always made to the cardinal number associated with a set, avoiding notation like 2*Aleph₀, but I am not sure of this.

By the way, I do not think there has ever been a proof associating the set of all real numbers with any Aleph_n, although it seems to me and many others that Aleph₁ is that set. Such a proof might have been devised since the last time I read about Cantor sets.

I once devised a simple proof that Aleph₁ was the set of all real numbers, but was told by my mathematics professor that the proof was flawed. I never understood why the the proof was invalid, but am willing to believe that there was a flaw. The proof was so simple that I did not believe I was the first to devise it.

I do not remember if there can be a set with more members than Aleph_n and less members than Aleph_n+1. There might be a proof that there is or is not such a set, but I am not sure.

My previous post does not contribute anything worthwhile to this thread.

Sorry about that, Chief! I read the thread and posted while under the afluence of some incohol.

One should not try to deal with Cantor and his transfinite sets in an informal fashion, which is what this thread is attempting.

An informal discussion of the subject cannot be accepted as a valid criticism, much less a proof that any of Cantor’s notions are invalid.

Can you provide an exact quote from a text on the subject and point to an error in the logic shown?

No problem,

Please read all of http://www.geocities.com/complementarytheory/TRANSFINITES.pdf in order to understand why I do not agree with Cantor's approach about the Infinity concept.

Doron Shadmi: I read your citation, but did not study it. There is a lot of notation which I did not understand. There are also sentences which do not seem to make sense. Perhaps it is double talk or perhaps the author's primary language is not English. For example, the following sentence makes no sense to me.More you simple less you depended, therefore more exist.A long time ago, I took a course which went through the development of Cantor's ideas. They were obviously based on sound mathematical logic, although the results were counterintuitive. My memory of the course includes little that seems related to the PDF file you wrote.

Without something more understandable, I can only assume that you have difficulty accepting counterintuitive concepts, in spite of their being supported by rigid mathematical logic.

The concept of pairing members of two sets to determine which set (if either) has more members seems valid. His definition of an infinite set seems valid: If a set can be put into one-to-one correspondence with a proper subset of itself, then it is an infinite set. The set of all integers conforms to this definition. His diagonal proof that the set of all real numbers has more members than the set of all integers looks good to me.

I remember that some of the course was difficult, and I am not sure that I understood it all, but I did pass the course with a very high grade.

Your PDF paper does not convince me that there is anything wrong with Cantor’s concepts.

More you simple less you depended, therefore more exist.

Try to define P(X) without the existence of X, for example:

Try to define P({}) (={{}}) without the existence of {}.

In other words {} can exist without P({}) but not vice versa.

It is trivially understood that Cantor’s approach does not hold, because he ignores the most basic principle of any formal language, which is: SYMMETRY.

In short, if he uses concept like “The Empty Set”, he simply cannot ignore its symmetrical opposite, which is: “The Full Set”.

Cantor ignored it, and by this he missed the deep understanding of the Infinity concept, and the clear and simple fruitful difference between Actual infinity (represented as an infinitely long non-composed solid element) and Potential infinity (represented as a collection of infinitely many elements).

This asymmetrical state creates a non-consistent mathematical framework, which is based on arbitrary and artificial methods, and these forcing methods clearly can be seen when they are compared to a symmetrical method (where both “The Empty Set” and “The Full Set” are used).

Any definition that tries to capture a collection of infinitely many elements in a one definition like the Cantorian aleph0, goes beyond the collection itself, but then this aleph0 is not related anymore to the collection of infinitely many elements, and we find ourselves in the state of Actual infinity (http://www.geocities.com/complementarytheory/RiemannsLimits.pdf), which is too strong to be used as an input, and therefore cannot be manipulated by any Language, Including the Language of Mathematics.

My concept of aleph0 is based on "cloud-like" magnitude of any collection of infinitely many elements.

For example:

aleph0+1 > aleph0

If A = aleph0 and B = aleph0 - 2^aleph0, then A > B by 2^aleph0, where both A and B are collections of infinitely many elements.

Also 3^aleph0 > 2^aleph0 > aleph0 > aleph0 - 1, etc...

So, as you can see my aleph0 is much more flexible and rich concept than the standard Cantorian approach.

Fore more details please look at: http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

Strictly speaking, Actual infinity is too strong to be used as an input.

Potential infinity (which never reaches Actual infinity, and therefore cannot be completed) is the name of the game.

For further information please look at:

http://www.geocities.com/complementarytheory/ed.pdf

http://www.geocities.com/complementarytheory/9999.pdf

http://www.geocities.com/complementarytheory/Anyx.pdf

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

Also Cantor's proof, which is not based on the second diagonal method http://en.wikipedia.org/wiki/Cantor's_first_uncountability_proof (The first is 1-1 and onto between N and Q), is actually failed because of a very simple conceptual mistake, which is:

If A set, c point, and B set are clearly distinguished from each other, then there cannot be a gapless state between them, simple as that!!


In short, Cantor uses simultaneously two different models (3_distingueshed_states_AND_a_solid_line) that are clearly contradicting each other.

Therefore this proof does not hold.

Doron Shadmi: Did you ever try to get your concepts published in some serious mathematical journal?

Yes,

Some of my papers are considered now in several serious mathematical Journals.

http://www.asa3.org/ASA/PSCF/1993/PSCF3-93Hedman.html


Whether Transfinite Numbers Exist

Cantor distinguished three levels of existences: 1) in the mind of God (the Intellectus Divinum); 2) in the mind of man (in abstracto); and, 3) in the physical universe (in concreto.) Cantor believed that Absolute Infinity exists only in the mind of God. But he argued that God instilled the concept of number, both finite and transfinite, into the mind of man. Cantor frequently appealed to their existence as eternal ideas in the mind of God as the basis for the existence of the transfinites in the mind of man.18 I will pursue the implications of this appeal for our understanding of contingent rationality under Epistemology below. Cantor adamantly defended the existence of the transfinites in abstracto, even arguing that God had put them into man's mind to reflect his own perfection.19 Cantor advanced infinite series representations of irrationals to claim that their existence was equivalent to that of the transfinites.

An asymmetric pure framework cannot be consistent, and in the case of set theory, if we use a concept like {} (which is the Empty set) we defiantly cannot ignore its opposite which is {__} the Full set.

The Full set is a non-composed infinitely long element (cannot be described by infinitely many elements) and it is considered as an Actual infinity.

Any model of infinitely many elements cannot be but a Potential infinity, and no Potential infinity can be an Actual infinity.

Because of this reason no collection of infinitely many elements is a complete collection by definition (no such a collection can reach the state of Actual infinity) therefore no universal quantification can be related to any collection of infinitely many elements.

According to the previous post, Cantor avoided Actual infinity because of what he believed, and without any connection to this reason, there is no reason to continue and teach his fatal conceptual mistake about Infinity, as a part of the Modern Language of Mathematics.

In short, his method must get of stage and be replaced by a symmetrical framework, where both {} and {__} are used.

According to the previous post, Cantor avoided Actual infinity because of what he believed, and without any connection to this reason, there is no reason to continue and teach his fatal conceptual mistake about Infinity, as a part of the Modern Language of Mathematics.

In short, his method must get of stage and be replaced by a symmetrical framework, where both {} and {__} are used.


{} and {__} are just two aspects of a more fundamental property.



The universal laws of nature are explained in terms of symmetry. The completed infinities, mathematician Georg Cantor's infinite[transfinite] sets, could be explained as cardinal identities, akin to a type of "qualia" from which, finite subsets, and elements of subsets, can be obtained.

Completed transfinite infinities, posess the property of distributive identity, similarly to the way that a set of "red" objects has the property and "identiy" of redness. Predicates like "red" are numbers in the sense that they interact algebraically according to the laws of Boolean algebra. Take one object away from the set of red objects and the distributive identity "red" still describes[predicates] the set. The definitive - distributive property, "natural number" or "real number" predicates, and gives identity to an entire[infinite] collection of individual subsets - numbers.


Yet, if we have a finite set of individual objects, the number of elements in that set, does not give it an identity. It does not distribute over its individual subsets. Take anything away from the set and the finite number of elements contained in its previous self, ceases to describe it.

Symmetry is analogous to a self evident truth and is distributive via the laws of nature, being distributed over the entire set called universe. A stratification of distributive identities.


[1.] Nothingness is a difficult concept, or rather, a difficult
"non-concept" since it is ...nothing.


[2.] To say that nothingness exists is to create a contradiction,
since nothingness is non-existence. Nothingness "non-exists".



[3.] Nothingness is non-existence; therefore nothingness is
non-relational. Nothingness has no distinctiveness in and of itself,
hence nothingness cannot be recognizable.

[4.] Being-ness means basically "to be", and to exist. Being-ness
exists.


[5.] Take any two distinct quantities-attributes of existence:



A<------P------>B

A and B perceive each other to be different with perception P. That is
to say A is perceived to be different from B and B is perceived to be
different from A.


A and B are different elements of a larger picture, but they
also[must] share certain common attributes of the larger reality
including them. From that perspective A and B are the same, because
aspects of A transform into B and aspects of B transform into A. At a
higher level of generality and symmetry, A cannot be distinguished
from B, and B cannot be distinguished from A. For example, dogs and
cats are distinctively different, yet dogs and cats are the same from
the perspective that both are "mammals".


[6.] Following this premise, it stands to reason that all attributes &
aspects of reality can be transformed into each other, and hence they
have no distinctions from each other[at continually higher and higher
levels of symmetry].



[7.] Therefore, it follows, that at a top[infinite?] level of
symmetry, everything that exists is related by a singular consciousness; a fundamental unity.

QED.




The Universal Mind creates itself - self referentially -
retroactively. Retroactive teleological causation can be described
by the transactional interpretation of quantum mechanics, where
wave distributions of probability originate in the past, present, and
future - giving a super Copernicanative juxtapositional -
teleological basis FOR our perceptual reality... There is no
preferred frame of reference. This is emerges from the fundamenmtal
essence that is the unspoken - undefinable TAO.

Events become actualized when waves from the past and the future
interfere with each other, converging to the ever- present NOW on the
Euclidean plane of a holographic n-1 dimensional hypersurface. This is
manefested by the time invariance of energy conservation, such, that
time itself is actually a perceptual illusion. The wave
patterns[probability distributions] also create what we perceive as
spacetime[gravitons], mass, and "energy". Gravity is also virtual.

Since the Universe has no exterior reference frame, and it must refer
to itself, such, that its overlapping Lorentz invariant geometry is a
computational self configuration. The evolution of spacetime, as
dictated by GR and QM, means that the past history of the Universe is
carried along with the present, FOR the Universe. A densification of
the Universal space-like Euclidean hypersurface. The Universe is a
function of itself. Space becomes relativistically "contracted",
"virtual" time becomes relativistically dilated.

As the time evolution proceeds in the thermodynamic direction of
"virtual" time, the n-dimensional Euclidean hyper-plane continually
increases in information density. The information storage of
perceptual space-time.
A metric field can be defined by the primary substratum of events.
Thus the intrinsic geometrical structure of spacetime is predicated on
the pseudo-Riemannian spaces via the affine relationships — all
physical events are fully reducible to manifestations of the
substratum i. e. the event density generating a metric field.


Stochastically speaking, gravity is must be taken beyond the limits of
classical reality, where the mean value of the stress energy tensor of
quantum fields also has fluctuations as a source of stochastic
Einsteinian vacuum equations. Such is the necessary foundation for
neo-classical gravity and the viability of inflationary cosmology
based on the vacuum energy dominated phase. Metric fluctuations and
spacetime gravitons form an elementary substrate.


Yes, the shortest distance between two points is a straight line;
energy is conserved.


Form a space-time triangle with vertices A,B,C, of three lines - two
null and one spacelike line, such that the triangle "spans" the
timelike plane.


Because of the one-to-oneness of the mapping, the image of the
triangle with the vertices A,B,C , "spans" the transformed plane as
different points are mapped onto different points. Therefore, the
three lines forming the transformed triangle must be coplanar. In
general, the images of all lines lying in a time-light plane must be
coplanar. Thus, timelike planes map into planes.


Any time light-cone line, is the intersection of two time-light
planes. Since timelike planes are mapped onto planes, they intersect
into a line. Thus, any timelike line is mapped into a line.


Basically, all three types of lines - lightlike(null), spacelike, and
timelike, are mapped onto lines.


The uncertainty principle and gravity are related to the same
mathematical properties. The proof of the uncertainty relation
involves the Cauchy Schwartz inequality. The triangle inequality
follows from the Cauchy Schwartz.


Each event in space-time has its own intrinsic measure of time, its
own "present moment", which is a point in the the Euclidean separation
plane of past-present-future, with the future as an uncertainty.

Represent the present moment of an arbitrary observer as the
inward collapse of the "past" light-cone/circular cross section to the
point of the "present moment" and the outward expanse of the future
light-cone/circular cross section into the uncertain future.

A 2-dimensional planar "cross-section" of the present moment, which is
the overlapping of past history, present moment, and future
uncertainty. p is the observational center of the overlapping cross
sections. The "proper time".

[<-[->[<-[p]->]<-]->]

Now it appears that the "past" moments are cumulative and are
increasing in information density = Shannon entropy as a cumulative
overlapping Lorentz invariant circular cross sections on a
holographic substratum.

Curvature and Riemannian geomety can be expressed as light-cone
deformations - generating an equivalent set of non-linear partial
differential equations equivalent to the Einstein field equations.

The two dimensional hyper-Universe gives us an illusion of 3
dimensions plus time. The other dimensions are "virtual dimensions"
analogously to a virtual image in a plane mirror. 2 real dimensions +
2 virtual dimensions = 4 dimensional Universe. Other "hypothesized"
physical dimensions, including those higher dimensions postulated by
string theory, are compactified into the two dimensional substratum.
Thus, The present moment is created and recreated constantly -
analogous to continually opposing-juxtaposing-evolving-reflective
mirror images… Heisenberg uncertainty provides a resolution boundary
and the invariant relational fabric for a translation between Planck
scale space and macroscopic-experiential reality. Unstable dimensional
states at a given level are always "compactified" - stabilized and
bounded by eigenstates - into virtual & symmetric "higher" dimensions.
In the semiclassical approximation giving 2 real + 2 virtual = 4
dimensional Universe. Also, gravity[and gravitons] becomes a
"virtual" force, along with inertial resistance to acceleration, hence
the equivalence principle. Mathematics has the complex numbers at
right angles to the real numbers. These are "virtual" numbers, a
reflection of the reals, that has an infinitely complex structure.

The Universe emerges from the rippling effects of immense[n-->
infinite] numbers of criss-crossing interference
fields/waves/worldines on an equally emergent substrate of
interconnected polarized graviton networks, which themselves originate
from the taowian "ultimate" reality.

The geometric invariance of fields becomes more fundamental than the
fields themselves. Our percptual infrastructure axiomatically
constructs "concrete" reality via a quantum phase filtration, via the
overlapping of Lorentz - invariant - wave - distributed boundaries,
such that the realm of meaningful, patterned, primary reality
transcends time and space. Therefore, the mind/brain is an embedded
hologram, holographically interpreting a holographic universe. All
existence consists of embedded holograms within holograms, and their
interrelatedness somehow gives rise to our existence via perceptual
sensory images.

Interference patterns of waves can be visualised as interacting like
ripples on a pond. At the a fundamental level they create matter and
energy as we perceive them – lifelike three-dimensional effects.

Consciousness and matter share the same quintessential basis,
differing only by degrees of subtlety. There is a strong correlation
between modulations of the brain's electromagnetic field and consciousness.

A<------P------>B


This model of the above idea is more comprehensive:

^
0 XOR 1 = child |
/ |
/ |
father = ? redundancy
\ |
\ |
0 XOR 1 = child |
v
<--uncertainty-->


By this model, symmetry can be found in any examined level, and then can be broken in the child’s’ level, but it always remains symmetrical in the Father's level, which cannot be manipulated by any form of language.

This model of the above idea is more comprehensive:

^
0 XOR 1 = child |
/ |
/ |
father = ? redundancy
\ |
\ |
0 XOR 1 = child |
v
<--uncertainty-->


By this model, symmetry can be found in any examined level, and then can be broken in the child’s’ level, but it always remains symmetrical in the Father's level, which cannot be manipulated by any form of language.

Thank you for the excellent diagram Doron. Your work appears to be progressing.

Dear Analog57,

The idea of this diagram can be found all along my papers, for example:

Look at http://www.geocities.com/complementarytheory/PTree.pdf

or at page 11 of http://www.geocities.com/complementarytheory/Countable.pdf

or at page 18 of http://www.geocities.com/complementarytheory/CATheory.pdf

Analog57: You are a grandmaster of double talk. I have never seen such a display of expertise. It is a lost art last seen by me in vintage 1940's movies on the TCM network. Perhaps there are examples elsewhere.

Now that I think of it, there was some posted @ SciForums about a year ago. The poster’s name was something like Gaudian. He was also adept at the art.

I wonder if Doron is another practitioner.

Dear Dinosaur,

Please explain youself clearly, because I do not understand what your problem is.

Doron Shadmi: I have become amused by this thread and no longer take it seriously. Perhaps when some serious journal accepts one of your papers I might spend some time analyzing your views. Until then, Cantor's ideas seem reasonable to me.

I think that Analog57 is teasing you. The Father/XOR 1 = child diagram conveys no information out of context. I am not sure it would be meaningful in some context.

Dear Dinosaur,

Before you quit, please read all of http://www.geocities.com/complementarytheory/TRANSFINITES.pdf including all of its links.

If you do that, then you will understand why the Cantorian transfinite universe does not hold.

Thank you,

Yours,

Doron

This is interesting.... I was discussing about the idea of infinity in my philosophy class today. This was the argument.

P1: The universe is either finite or infinite.
P2: The universe is not infinite.
------------------------------------------
C: The universe is finite.

My philosophy teacher said that my idea of infinite was wrong. She cited Cantor (just his name and what he talked about )and I was completely speechless cause I've never read about this guy. Now I read this.... :eek:



But anyways...

Is there a connection between Cantor's "transfinites" and Zeno's Paradox? :confused:

At least there seem to be one, eh...?

Doron Shadmi: Sorry that would require more time and intellectual effort than I am willing to expend on this topic.

The first cited article seems to conclude that Cantor’s ideas are basically unsound. My intuition tells me that a more formal proof would be required for this conclusion. Furthermore, if Cantor’s basic concepts can be disproved so easily, I would expect them to have been discarded long ago.

When I took a course dealing with Cantor’s notions about transfinite numbers, it seemed to be very sound mathematical logic, although many of the conclusions were counterintuitive.

TruthSeeker: I doubt that Cantor’s ideas are applicable to the Zeno paradox, but I am not an expert in such matters. Concepts relating to limits and the summing of infinite series are more applicable to the Zeno ideas.

Cantor’s basic ideas are not difficult to comprehend, although a complete course in the subject is can be overwhelming if you do not apply yourself and work hard on it. His basic ideas are as follows. Two sets can be compared by attempting to match each member of one set with a member of the other set. If the sets can be matched, they have the same number of members. If the matching process exhausts one set first, that set has less members than the other set.
Consider matching members of the set of all even integers with the set of all integers. They can be paired as follows: (1, 2), (2, 4), (3, 6), (4, 8), (5, 10) . . . (n, 2n). Neither set is exhausted first. The matching processing indicates that the two sets have the same number of members. An infinite set is defined as one which has this property. A subset of an infinite set can be matched with the entire set. The number of members in the set of all integers was called Aleph₀
The next concept is that the set of all subsets of a set cannot be matched with the set. For example, the set of all subsets of integers has more members than the set of all integers. The number of members of this set is called Aleph₁
Aleph₂, Aleph₃, Aleph₄, et cetera can also be defined.
From here on it gets increasingly more difficult to follow the proofs.The above is a very informal description of a topic requiring strict adherence to formal mathematical logic. Any deviation from precise logic can lead to erroneous conclusions.

There are some interesting conclusions reached. For example, it is proved that there are the same number of points in a 1D space (Id est: a line) as there are in a 2D space (Id est: a plane). This number is called the power of the continuum. I do not think it has ever been proved that this number is the same as some Aleph_n. As you would expect, Cantor proved that there are more points on a line than there are integers. While this seems intuitively obvious, it needs to be proved rather than asserted as being self evident.

Doron Shadmi claims that there is something fundamentally wrong with Cantor's ideas.

it seemed to be very sound mathematical logic

It cannot be consistent by any form or shape ,because Cantor ignored Actual infinity, as it clearly demonstrated in this model:

http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

Please look at it and tell me what problems you find in it?


Any deviation from precise logic can lead to erroneous conclusions.

Please define precise logic.

The common one is the 0_XOR_1 or Excluded Middle logical reasoning, but if you look at http://www.geocities.com/complementarytheory/ed.pdf, you will find that this Black_XOR_White system cannot deal with collections of infinitely many elements in a non-trivial way.

On the contrary Included-Middle logical reasoning can deal with such collection in a much more accurate way then the standard False_XOR_True way, for example:

http://www.geocities.com/complementarytheory/CompLogic.pdf

So dear Dinosaur, when you use a concept like precise logic, you first have to deeply understand it, thank you.

Interesting stuff, both of you...

I think i'm more inclined to Riemann's limits than Cantor's concepts, altough i should still check them out.....

Still, this is all very similar to Zeno's paradox, specially Riemann's limits...

Doron Shadmi: I think the Riemann Ball relates to geometric and/or calculus analysis notions of infinity. I do not think it is relevant to Cantor notions which relate to set theory.

I think that the serious mathematicians who do not accept Cantor’s concept do not accept the Axiom of Choice. The Axiom of Choice is/was controversial and is avoided by some branches of mathematics. Acceptance of it leads to various anomalies like the Banach-Tarski Paradoxical Decomposition. If you wish to refute Cantor’s ideas, why not investigate that angle? I think that the Axiom of Choice is required by Cantor’s development. Perhaps your arguments are equivalent to rejecting the Axiom of Choice.

BTW: I think I have read claims that neither acceptance nor rejection of the Axiom of Choice lead to inconsistent systems, suggesting that Cantor’s concepts are logically consistent. I do not understand some of the implications of this axiom, but do understand how it relates to Cantor’s notions.

I am not sure, but I think that complex analysis and related branches of mathematics try to avoid infinity. Instead they talk about variables which grow without bound and do not explicitly permit the use of infinite quantities. Such branches of mathematics might avoid invoking the axiom of choice. Even complex analysis which mentions a circle at infinity avoids actually allowing the use of values at infinity (at least I think it avoids the use of such vlaues).

If you wish to refute Cantor, why not show where in his development he has erred? Show his development of the concept of transfinite numbers and point out the flaw in his argument. You seem to think that flaw is in his basic concepts. His early development is not difficult. If it is flawed, it should be easy to point out the flaw. The burden of proof is really your responsibility since you are trying to show that an accepted theory is invalid.

As I have previously posted: Cantor’s ideas seemed logically sound when I took a course which dealt with his concepts. Some of the conclusions seemed counterintuitive, but one should not accept intuition as superior to logical analysis.

Doron,

People use set theory because it has given years of fruitful mathematics. It also is just fine and dandy consistent. If you were serious about set theory, you'd learn something developed after 1930. There are plenty of great constructive set theories that fit your limitations on the concept of actual infinity, yet they do not deny the consistency of normal set theory. There are many constructive theorems about the limits of the applicabilty of ordinary set theory and its usefulness in generating concepts for discussion.

Try Bishop and Bridges book on constructive mathematics. They are able to succesfully introduce all the concepts in this thread constructively.


1) I am not talking about any specific version of Set Theory.


2) I am talking about a fatal conceptual mistake that Cantor did when he used the idea of sets.



My argument about the Cantorian transfinite system (where aleph0 is not in N), is based on the most simple things that, in my opinion, we should care about when we define a consistent framework, and these things are:


Simplicity and Symmetry.


In this case, we do not think about quantity (fewest possible elements) but about simplicity.

Only the simplest thing can be considered as a building-block of some pure framework.

In the case of Emptiness, it is the lowest concept that cannot be manipulated by any framework that is based on information, and the language of Mathematics is first of all an information system, like any language, formal or informal.

When we have the lowest concept that cannot be manipulated by any framework, then if we want to save the simplicity of our framework, we cannot ignore anymore its internal symmetry.

So, if we use Emptiness, then in order to save the simplicity of our framework, we use symmetry, and define Fullness as the highest concept that cannot be manipulated by any framework.

By saving the simplicity and symmetry of our framework, we actually define its operational domain, where we can work and do interesting Math.

Cantor missed this important insight and created an asymmetrical framework that do not aware to the highest concept that cannot be manipulated by any framework, which is Fullness

And the result is the transfinite universe that ignore Actual infinity (which is both Fullness and Emptiness concepts).

When we look at this diagram http://www.geocities.com/complementarytheory/RiemannsLimits.pdf we can clearly and simply see that Actual infinity is an inevitable fundamental concept of any set theory, which uses Emptiness as one of its concepts.

Strictly speaking, any set theory that uses Emptiness, cannot ignore Fullness, in order to be consistent.

Since the Cantorian approach uses Emptiness but ignore Fullness, it cannot be a consistent framework, and this conclusion is stronger then any proof which is based on some axiomatic system.

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

About Symmerty:


Some climes that Symmetry is: “The possibility of an object to be divided into two equal parts ..."

I think that this interpretation is deeper: (physics interpretation) "the property of being isotropic; having the same value when measured in different directions" (http://www.cogsci.princeton.edu/cgi-bin/webwn2.0?stage=1&word=symmetry) because the original meaning is:

"The same thing"

And "the same thing" is the identity of a thing to itself (which is the basic form of symmetry) or the "the same thing" can be extended to some common relation between different self-identities.

In all cases, the name of the game is Symmetry, and by this concept we can research how self-symmetries (or self identities) interact with each other, in order to expose their deeper common source of symmetry.

Any other interpretation of the Symmetry concept misses the full fruitfulness of this concept.

but one should not accept intuition as superior to logical analysis

The logical reasoning of my argument is based on Included-Middle analysis, which is deeper then Excluded-Middle (False_XOR_True) analysis ( http://www.geocities.com/complementarytheory/CompLogic.pdf ).

:eek:

I had never been able to explain my thought process and now, this.....!

Since the Cantorian approach uses Emptiness but ignore Fullness, it cannot be a consistent framework, and this conclusion is stronger then any proof which is based on some axiomatic system.
I thought he ignored fullness.... :confused:

The logical reasoning of my argument is based on Included-Middle analysis, which is deeper then Excluded-Middle (False_XOR_True) analysis ( http://www.geocities.com/complement...y/CompLogic.pdf ).
You should use the words "exomedius" and "endomedius"... :D

Exomedius: exo-"out", medius-"middle"
Endomedius: endo-"in", medius-"middle"

Doron Shadmi: Your diagrams and arguments might mean something to you. They do not mean much to me.

Do you have any opinion about the Axiom of Choice?

oops..

Do you have any opinion about the Axiom of Choice?

If we have several non-empty sets and there is no rule how to pick a single element out of each set, then we need the axiom of choice, which picks for us these elements, but there is no rule that can guide us what or how to choose, so the result is an arbitrary collection.

I have no problem with such a result because there are more expressions like "arbitrary close" which are very useful in calculus for example, so if there is some need to define an arbitrary collection, then the axiom of choice is useful.

Now please explain to me how aleph0 is not in N and also not an Actual inifinity?

Doron Shadmi: At best I have a vague notion of what you are asking. Now please explain to me how aleph0 is not in N and also not an Actual inifinity?I do not know what you mean by an actual infinity. I assume that Aleph0 is what Cantor calls Aleph₀ (usually pronounced Aleph-Null). I assume that N is the set of all integers.


Due to not being sure of what you are asking, I cannot answer you directly. I can describe the Cantor view of the set of all integers, which I think you call N.

The set of all even integers can be paired with the set of all integers as follows.
1,2
2,4
3,5
4,8
5,10

. . . .
n, 2n
. . . .

For every member of one set, there is a corresponding member of the other set. The pairing process does not exhaust either set. Thus the set of all integers can be put into one-to-one correspondence with a subset of itself.

A set with the above property is defined to be an infinite set.

Perhaps one might quarrel with this definition, but it is surely not a property of a finite set. One might argue that there is a problem with the above pairing process. The conclusion is counterintuitive, but I do not see an error in the pairing process.

It seems strange, perhaps invalid, that a set can be put into one-to-one correspondence with a subset, but I can find no way to show that the set of all even integers is exhausted before the set of all integers is exhausted. You name a member of the set of all integers, say n, and the above pairing indicates what even integer corresponds to it, namely 2n. If the set of all even integers is exhausted first, I cannot describe or name the unpaired integers.

Cantor defines Aleph₀ as the cardinal number associated with the set of all integers. It represents the number of members of that set. One can say that the set of all integers has Aleph₀ members.

My description of the Cantor view is not as formal as that provided in a good text book, but I think it conveys the general concept.

Do you have some objection to the Cantor view described above?

I do not know what you mean by an actual infinity.
Huuum.... let me give it a shot on this one. Actual infinity should mean "no boundary". In other words.... it goes on forever. It is not an actual number, but a concept...

Dinosaur:
As a side note, I give you the following question:

According to Cantor, an infinite could be expressed as the distance cetween two fingers, for example. Altough it is true that there may be an infinite amount of points between two fingers or in a line, wouldn't it also be tru the fact that the space btween to fingers is said to be a finite space? Otherwise, we would not be able to see the two fingers together, right? In other words.... I ask you... is the distance between two fingers a finite distance or an infinite distance? ;)


[Or both!? ;) ]

Cantor defines Aleph0 as the cardinal number associated with the set of all integers


Dear Dinosaur, the language of Mathematics can work only if there are elements that can be manipulated, and these elements can be used as inputs in our Mathematical tools.

There are two concepts that cannot be used as inputs, and they are:

Emptiness (too weak to be used as an input) and Fullness (too strong to be used as an input) and both of them are actual infinity.

What's left is point-like or segment-like building-blocks, that can be manipulated by the language of Mathematics.

If you take a collection of infinitely many point-like and/or segment-like building-blocks, and use a Universal-Quantification that is related to them, then you are no longer in any model that can be described by infinitely many elements, but only by Fullness, which is an unavailable information form that it is too strong to be manipulated by the language of Mathematics.

The model of fullness is described by an infinitely long (totally pointless) solid element.

There is no middle state between the infinitely_many_elements model and the infinitely_long_(totally_pointless)_solid_element model.

Since aleph0 is not in the domain of the infinitely_many_elements model, it cannot be but in the domain of the infinitely_long_(totally_pointless)_solid_element model.

In this case the transfinite system cannot be manipulated by the language of Mathematics.

Therefore the Cantorian transfinite universe does not hold.

By using the Reimaenn's circle model http://www.geocities.com/complementarytheory/RiemannsLimits.pdf I clearly and simply demonstrate these two models.

Emptiness (too weak to be used as an input) and Fullness (too strong to be used as an input) and both of them are actual infinity.
Really? Prove it! :D ;)

By using the Reimaenn's circle model http://www.geocities.com/complementarytheory/RiemannsLimits.pdf I clearly and simply demonstrate these two models.
Sounds challenging to explain that "clearly" and "simply"... :D

Doron Shadmi: Most of what you post does not make sense to me.

In particular, the Riemann ball you keep citing is a geometric concept, while Cantor analysis deals with sets. I just do not see how one is related to the other.

As posted earlier, I see nothing wrong with the Cantor development of transfinite numbers, although it relies on the Axiom of Choice, which is controversial.

What is wrong with the Cantor definition of an infinite set? What is wrong with his views on matching members to decide which of two sets has more members? Do you consider the set of all integers to have a finite number of members?

the Riemann ball you keep citing is a geometric concept

Sorry Dinosaur, but this model has nothing to do with geometry, it is a pure analytic model that clearly represents the number system itself, and also clearly shows its highest limitation.

Sets are used (in this case) to represent the same number system, so in both models we are talking about the same system.

But my representation is a better way to understand the number system, because through it we get the very impotent insight of the symmetry concept, which is totally ignored by the Standard set representation.

Standard Set system has a bottom information limitation, which is Emptiness, and I add to the same system its top information limitation, which is Fullness.

When I do that, the Standard Set system gets its necessary internal symmetry, that without it, it cannot be a consistent pure framework.

By using this symmetry, we clearly show that the Cantorian aleph0 does not hold.


Really? Prove it!

I do not have to prove it, both of them are based on axioms in my framework:

The Axiom of the unreachable weak limit:
No input can be found by {} which stands for Emptiness.

The Axiom of the unreachable strong limit:
No input can be found by {__} which stands for Fullness.

The logical basis of both of them can be found in pages 4,5,6 of http://www.geocities.com/complementarytheory/CompLogic.pdf

Doron Shadmi: Your Riemann ball is a geometric representation of the set of all real numbers. In the absence of a proof of the continuum hypothesis, it is not pertinent to the Aleph_n concepts developed by Cantor.

Your other diagrams do not seem to prove anything to me.

Asserting unproven axioms about emptiness & fullness and then claiming that they refute Cantor's views does not seem valid.

So far, you seem unwilling to attempt to point out a flaw in Cantor’s development of transfinite numbers.

You could at least answer direct questions like the followingWhat is wrong with the Cantor definition of an infinite set? What is wrong with his views on matching members to decide which of two sets has more members? Do you consider the set of all integers to have a finite number of members?Answers to the above questions would help me understand what you are trying to say.

Asserting unproven axioms ...

Dear Dinosaur, there is some very fundamental thing about the language of mathematics that you miss, which is:
An Axiom is a proposition regarded as self-evidently true without proof (http://mathworld.wolfram.com/Axiom.html)

Your Riemann ball is a geometric representation of the set of all real numbers

No, it can be used to represent N, Z*, Q, R or C sets, and this representation is better then any standard linear string of symbols, because by the standard linear representation, you cannot see the difference between Actual infinity and Potential infinity.

What is wrong with the Cantor definition of an infinite set?

It goes beyond the set itself, because aleph0 is not in N, and the cardinality of any set (finite or infinite) cannot go beyond the domain of its own set.

If we do that, then we are no longer in a model of infinitely many elements (and only in this model we can matching members to decide which of two sets has more members) but in a model of Fullness, which is an infinitely long non-composed and pointless solid element.

Do you consider the set of all integers to have a finite number of members?

Not at all, N set is a collection of infinity many elements that can never reach the state of Fullness (which is the state of an infinitely long non-composed element), no matter how infinitely many elements it has.

The Cantorian transfinite word does not hold.

Its aleph0 is a trivial approach that misses infinitely many information forms, that can be defined by Monadic Mathematics (http://www.geocities.com/complementarytheory/MM.pdf).

Even the most primitive compression, shows the triviality of the Cantorian aleph0, for example:

By Cantor: aleph0+1=aleph0, aleph0-2^aleph0 has no meaning, aleph0 < 2^aleph0, 3^aleph0=2^aleph0, etc...

My solution to Aleph0 concept

My concept of aleph0 is based on "cloud-like" magnitude of any collection of infinitely many elements.

For example:

aleph0+1 > aleph0

If A = aleph0 and B = aleph0 - 2^aleph0, then A > B by 2^aleph0, where both A and B are collections of infinitely many elements.

Also 3^aleph0 > 2^aleph0 > aleph0 > aleph0 - 1, etc...

Fore more details please look at: http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

Strictly speaking, Actual infinity is too strong to be used as an input.

Potential infinity (which never reaches Actual infinity, and therefore cannot be completed) is the name of the game. For further information please look at:

http://www.geocities.com/complementarytheory/ed.pdf

http://www.geocities.com/complementarytheory/9999.pdf

http://www.geocities.com/complementarytheory/Anyx.pdf

Doron Shadmi: You do not seem to understand the implications of your arguments. Consider your statement.It goes beyond the set itself, because aleph0 is not in N, and the cardinality of any set (finite or infinite) cannot go beyond the domain of its own set.Suppose I have five pairs of socks.It is valid for me to say that I have a set containing ten members. Ten is not in the set, since all of its members are socks. Hence my set consisting of 5 pairs of socks cannot have cardinality ten because ten is beyond the domain of the set.If you do not like sets of socks, try the above with a set containing ten complex numbers or ten 3D vectors.

BTW: How can you assert that Aleph₀ is not in the set of all integers? This is equivalent to the claim that either the set of all integers has no cardinality or its cardinality is not an integer. The cardinality (if it exists) is surely an integer. If it is an integer, it should be in the set of all integers.

I am not sure that the logic of my last paragraph is formally valid, but it seems more sound than what you are trying to sell.

The Cantor view seems valid to me if you accept the axiom pf choice.

Suppose I have five pairs of socks.

The idea behind the 10 socks is that you cannot distinguish between the socks and also cannot distinguish between the 5 pairs of socks, but in both cases you know that you have a multiset of 10 elements in the case of {x,x,x,x,x,x,x,x,x,x}, and a multiset of 5 elements in the case of
{{x,x},{x,x},{x,x},{x,x},{x,x}}.

But since a set is not a mutiset, then {x,x,x,x,x,x,x,x,x,x}={x} and {{x,x},{x,x},{x,x},{x,x},{x,x}} = {{x,x}} and if we avoid multisets then
{{x,x}}={{x}}.

So, in the case of pure sets we get: |{x}|=|{{x}}|=1 and we do not need the axiom of choice.

The Cantorian aleph0 cannot be a Natural number if aleph0+n = aleph0, and this is exactly some result of a Cantorian transfinite cardinals arithmetic.

So the Cantorian transfinite system is based on self contradiction, if aleph0 is a Natural number, therefore aleph0 must be beyond any Natural number, and in this case, it does not belong anymore to any model, which is based on infinitely many elements.

The only Cantorian alternative is that aleph0 is Fullness (an infinitely long pointless solid element) and then it cannot be manipulated by the language of Mathematics.

Dear Dinosaur,

Where are you?

Dear Doron,
In your link "Riemann's ball model" (RBM) you have said:
"As we can see from the above example, no infinitely many objects (where an object=an intersection in this model) can reach actual infinity".
Can you explain it to me more detailed, because I can not see it, please...

It is obvious that the line, which is notated by oo does not intersect the two infinitely long lines, which exist below it.

But there are infinitely many diagonal lines between 0 and oo that cross the middle and the bottom infinitely long lines.

Each intersection is some real number that can be found along these two lines.

But when we are in a state of the oo line, no intersection can be found, and we have no information, which can be manipulated by any tool or method of the language of Mathematics.

I call this state Actual infinity, and the collection of the infinitely many sections is called Potential Infinity.

The transition between Actual infinity and Potential infinity and vise versa cannot but a quantum leap.

Furthermore, it is clear that no model which is based on infinity many sections, can be described by a model of a non-composed, sectionless infinitely long solid line, and there is no middle model that can be found between these two models.

Dear Doron,
1. Thanks for explanation. It cleared my suspicion that there is something wrong with Riemann in your posts.
The problem is that Riemann invented his “ball” exactly to illustrate … just contrary to what you are saying. His reflection shows that there is one-to-one correspondence between intersections on line 0 and circumference R=1 and that there is absolutely continuous transit from South pole (point belonging to line 0) to North pole (the point belonging to line 0) through over all members of set (points of mentioned circumference) with the same cardinality as set of points on line 0.).
2. Your statement “It is obvious that the line, which is notated by ∞ does not intersectt the two infinitely long lines, which exist below it” is absolutely strange in your performance, because of fact that you, as I understand, are pretending on creation of a new logically accurate approach to the foundations of Math.
As you should know, nothing will be changed in the Classic Geometry if we will formulate the Axiom of Parallel lines as two lines in the same plate that do intersect only in infinity. So, there is nothing “obvious” in fact do lines you are mentioning intersect in infinity or do not. Only the special hypothesis on that matter could clarify this problem. And Riemann’s ball exactly shows that there is no problem at all: there is a continual transit from line 0 to line ∞ through over set of points of line 0.

Dear Yuriy,

I disagree with you, you cannot define any 1-1 mapping between a model that is based on infinitely many elements, and a model that is based on an infinitely long non-composed and pointless solid element (which is the element that is notated as oo).

Therefore it is a conceptual mistake to clime that two oo elements intersect each other in oo.

It is exactly as if you say that the sun is shining in the middle of the night, which is a meaningless proposition, and no language (formal or informal) can be based on such propositions and also be considered as a rigorous and logical reasoning framework.

In short, Reimann himself did not understand the infinity concept if he said that there can be any function between a model of an infinitely long non-composed and pointless solid element (which is the element that is notated as oo), and a model that is based on infinitely many elements.

I read parts of it thoroughly, skimmed alot of it.
It seems to me that you're saying that we have assumed there CAN be something greater than infinity in order to prove it. In other words, you're saying that since infinity can never be reached, 2^infinity can never be bigger than infinity because each can never be reached so there is nothing to distinguish between the two sizes. Pretty close?
Aaron

Dear Doron,
Honestly saying, I was counting upon much more serious discussion…
1 I hoped, you will be not against my wish to match usual Euclidian lines with your “non-composed and pointless solid elements” that you noted as “0” and “∞”. I hoped, you will not find any differences in the physical and/or geometrical properties of your “non-composed and pointless solid elements” that you noted as “0” and “∞” and their usual Euclidian “twins” coinciding with them. I hoped, you understand that whole Riemann’s method was invented exactly to show the simplicity of geometric behavior of lines (actually – the functions) in “vicinity” of the infinity. That there is a possibility to see what actually happens “in infinity” instead of doing ambiguous claims like “It is obvious that the line, which is notated by ∞ does not intersectt the two infinitely long lines, which exist below it “ and many others… Are you saying that all these my hopes was wrong?
2. Nevertheless, I do not accept your response due to your own rules.
If lines “0” and “∞” both are “non-composed and pointless solid elements” and if “therefore it is a conceptual mistake to clime that two ∞ elements intersect each other in ∞”, then it is the same conceptual mistake to claim that these two elements do not intersect each other in ∞, as you did in your page “Riemann’s Limit”.
3. And in finally, according to you not only Riemann, but Cantor and all following mathematicians “did not understand the infinity concept” because all them asserted that there is one-to-one correspondence between Euclidean straight line and set of infinitely many elements – the rational numbers.
So, there can be only the one base for productive discussion – acceptance of usual Euclidean nature of Geometry with its relations point-line-square-volume, as far as we want to appeal to the geometrical images, or you will from the beginning redefine each of images you bring up for illustration.

If lines “0” and “∞” both are “non-composed and pointless solid elements”

No thay are totally different models.

Only “∞” is “a non-composed and pointless solid element”.

Number 0 is a single intersection.

The to lines below “∞” are composed by infinitely many elements.

My Math is a non-Euclediann framework (which is not equivalent to a Non-Eucledian geomentry)

In order to understand my work, please look at http://www.sciforums.com/showthread.php?p=720167#post720167

Thank you.

Doron,
1. So, line ∞ is “non-composed and pointless solid elements”; line “0” “is composed by infinitely many elements”. That are your statement, not mine.
2. I hope, we both call “line 0” the straight tangent, the number 0 is lying on in “Riemann’s limit”.
3. You have said: “It is obvious that the line, which is notated by ∞ does not intersect the two infinitely long lines, which exist below it.” So, you operate with lines ∞ and 0 in sense of usual Euclidean geometry (“lines”, “intersect”). But what you just declared as “obvious” is obvious for you, not for anybody, who goes with Riemann’s approach. If you want to invent some new approach, it is your duty to clear all ambiguous statements you make.
4. I read your links very carefully and found them full of unclear, ambiguous statements and definitions like “The minimal concept that cannot be researched is Emptiness, because no information can be found in it.” You are wrong even here. There is a huge information in such thing as Emptiness. Do you know what an information? That there is no information. It means a lot in the sense of Information. Namely this information makes Emptiness recognizable and unique. And you are wrong thinking that there is no way to research Emptiness in the set-theoretical sense. There are a lot of features of Emptiness to be researched. For instance, such universal feature as to be a member of any set that can be defined in the Theory of Sets.
5. So, if you really are interested in serious discussion, please reveal some more attention to my questions, otherwise it becomes a waste of time…

Dear Yuriy,

You mix between Emptiness and the Empty set.

I am talking about the Emptiness concept itself before it is manipulated by using the set concept.

Form this most basic approach, no information can be found in the level of the Emptiness concept itself (before we merge it with the set concept).

The same thing holds for Fullness.

1. I did not mix between Emptiness and the Empty set. Simply I thought that you apply notion Emptiness in the frames of Set Theory as a Empty set. (Because of your statement “By using the set concept we can distinguish between emptiness and fullness, because: |{}| = 0 ….”) If it is not so, and empty set is not a set-theoretical image of Emptiness, then can we keep all further discussion on the set-theoretical features of things like “Riemann’s ball” without using this strange notion Emptiness?
2. Moreover your claims like “… actual infinity (Emptiness and Fullness)…” are not match with classic definition of the actual infinity as given by Hegel. Your claim like “The maximal concept that cannot be researched is Fullness, because it is beyond measurement or manipulation of information” stays absolutely beyond apperception because nobody can understand how you find out about it … if it is beyond any measurement or manipulation of information … Any of that looks like juggling with words…
3. And finally, you have said: “In short, Reimann himself did not understand the infinity concept if he said that there can be any function between a model of an infinitely long non-composed and pointless solid element (which is the element that is notated as oo), and a model that is based on infinitely many elements”. Before anybody will be able to estimate this strong statement can you explain us what is “long non-composed and pointless solid element”? Only some collection of words or it is an image of something that can be long or short, but not composed, but long or short, pointless, but solid and long or short? Why we can not consider it as composed from points, but solid one? Etc.,etc… Why it can have also such a feature as to be … a green? Or it can have it?… It is not a jog. Taking your way of manipulation with words, we can define as actual infinity a round triangle or woodish wood…

Doron Shadmi: I am in Delaware Valley, which is in South Eastern Pennsylvania, which is in North Eastern USA.

Why did you want to know?

Absolute Infinity?

Dear Yuriy,

(If {} then {__}) AND (if {__} then {}) is true.

By this proposition we get the true value, which is based on a symmetry between opposite concepts.

The logical basis of Emptiness is: oo …E nor E nor E … oo = 1

The logical basis of Fullness is: oo …F and F and F … oo = 1

For more details please read pages 4,5,6 of http://www.geocities.com/complementarytheory/CompLogic.pdf

Thank you.

Doron,
You know, I never in my life pressed anyone to accept my expertise.
Take care.

Dear Yuriy,

And so do I.

Take care too.