Deeper than primes
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Originally Posted by doronshadmi
From a structural view 0.25 has more than a one location along 0_______1
"From a structural view," those locations are no longer determinable.
From a structural view 0.25 has more than a one location along 0_______1
"From a structural view," those locations are no longer determinable.
The Man
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If it is “not a set and not a member of a set” then cardinality simply does not apply. So stop using cardinality in reference to your “magnitude of existence” and define your “magnitude of existence” independent of cardinality.
As you assert above your “non-local AND non-complex, which is not a set and not a member of a set” it has no cardinality.
Nope, I asked a question. Can you answer it or not?
Once again who ever claimed that a segment could be reduced to a zero dimensional point? Stop simply trying to posit asspects your own failed reasoning onto others. Once again that it is “smaller than any strict or non-strict value > 0 that exists under [base 10] AND it is also > 0” is your assertion and it simply asserts that it is smaller than it self. Now you could assert that it is “smaller than any strict or” any other “non-strict value > 0 that exists under [base 10] AND it is also > 0”, which would not be claiming that it is smaller than itself, but still claiming that it is your smallest segment.
Again show what member of that infinite collection is not a member of that infinite collection, otherwise it is complete.
As explained to you before Doron cardinality specifically refers to the size sets, as such your use of cardinality in reference to your “magnitude of existence” and your “non-complex thing like the non-locality of 1-dimansional space” remains simply meaningless.
Once again you can’t distinguish between cardinality and your fantasy “magnitude of existence”
Then it has no cardinality. Once again please define your “magnitude of existence” independent of cardinality.
As already explained to you the “1” in “1-dimensional” specifically refers to the number of coordinates needed to define a point in that space. If you are going to try to define your own “dimensionality” without points then your are going to need something to quantify that dimensionality. So please define your “dimensionality” without points and how you quantify that “dimensionality”
Doron your continuing self-inconsistent assertions simply demonstrate that you have no clue what you are trying to claim with you “OM”.
As already explained to your Doron what “enables Quantity” in the case of cardinality is the members (or more specifically the size) of the set being consider
No Doron it is defined by the set, specifically what does and does not constitute a member of that set, but of course that has been explained to you before as well.
Once again Doron a “1-dimensional space” is defined by the coordinates required to locate a point within that space. No points no dimension. So please define your “dimensionality” and specifically how you quantify it without points.
Be aware that your ridiculous, self contradictory and generally contradictory claims do no constitute facts.
Remember Doron.
“The amount of a given collection is defined by the linkage among Non-locality and Locality”
And
“The simplest result (without a loss of generality) of this linkage is done between 1-dimensional space, known as Line, and 0-dimensional spaces, known as Points.”
The empty set has neither points nor a line. So much for your “linkage” “done between 1-dimensional space, known as Line, and 0-dimensional spaces, known as Points.” defining the “amount of a given collection”
No Doron any cardinality, including zero, is dependent only on the size of the defined set.
Oh and “belong XOR not belong” is still always FALSE.
Then your “"{" "}" or ...________..., “ simply do not have that “non-local property” you claim otherwise your “distinct objects and "{" "}" or ...________..., “ would have that property as it includes both your “distinct objects and” your “ "{" "}" or ...________..., “
Are you claiming that you lose your “non-local property” of your “"{" "}" or ...________..., “ in your “linkage between the distinct objects and "{" "}" or ...________..., “?
doronshadmi
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EDIT:
My answer is about the traditional and the non-traditional definition of Cardinality.
------
First let's reply to your last comments, which are related to my non-traditional definition of Cardinality:
The non-traditional definition of Cardinality:
Cardinality is the magnitude of existence of a given thing, which has at least the weakest or strongest magnitude of existence, as follows:
a) Emptiness, which is defined as "that has no predecessor", has cardinality 0, which is the weakest cardinality.
b) Fullness, which is defined as "that has no successor", has cardinality ∞, which is the strongest cardinality.
The strongest cardinality is stronger than any other given cardinality, such that any object that has the strongest cardinality, is at and beyond any existing thing that is compared to it (Emptiness, which has cardinality 0, is excluded because it is not an existing thing).
Non-locality and Locality are the result of the comparison between that has the strongest cardinality, with that does (or do) not has (or have) the strongest cardinality (Emptiness, which has cardinality 0, is excluded because it is not an existing thing).
Under this comparison (or linkage, if you will) among existing things; that has the strongest cardinality is simultaneously at AND beyond any existing thing (or collection of existing things) that does (or do) not have cardinality ∞.
1-dimensional space is the smallest existing thing that has cardinality ∞.
0-dimensional space is the smallest existing thing that has cardinality > 0, such that 0-dimensional space is the immediate successor of Emptiness, if Cardinality is the magnitude of existence of a given thing.
Because 1-dimensional space and 0-dimesional space are both existing things, then (as mentioned above) 1-dimesional space enables to be simultaneously at AND beyond any given 0-dimesional space along it, where any given 0-dimensional space along 1-dimensional space, can't be simultaneously at AND beyond 1-dimentional space.
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Second, let's reply to your last comments, which are related to the traditional definition of Cardinality:
The traditional definition of Cardinality:
Cardinality is the number of distinct objects that belongs to an existing thing, called a set.
a) Emptiness, is the absence of any distinct objects of an existing set.
b) Fullness, is an undefined concept under the traditional notion of Set.
I claim that Fullness is used by mathematicians as an essential property of Set, even if it is not formally defined by them, as follows:
1) The fact that a set is an existing thing even if it does not have members, is not formalized by the traditional mathematicians, and as a result the cardinality of a set is defined only as the number of the distinct members, that belongs to a given existing set.
2) The existence of a set beyond the existence of its members is exactly the non-local property (which is not formalized by the traditional mathematicians) of the concept of set, which is not influenced by the number of the distinct members that are related to it.
3) Actually without the existence of a set beyond its members (which can be shown right from the existence of the empty set beyond Emptiness), there is no way to formally define any collection.
4) By understanding this fundamental fact, there is a way to formalize the existence of a set, such that it exists beyond any given collection of distinct members, which are related to it.
5) By understanding (1),(2),(3),(4), one defines the existence of a set beyond its members, as the non-local property of a set w.r.t its members, where the collection of any non-emply set is local w.r.t the existence of the set.
6) (5) is logically demonstrated by the inaccessibility of any amount of distinct objects to be beyond "{" and "}" , which is equivalent to the inability of any amount of points or segments to be beyond 1-dimensional space.
1-dimensional space exists beyond of any amount of points or segments along it, exactly as the empty set exists beyond the existence of members.
But traditional Math never formalized this fundamental fact, and as a result it lives under the illusion that there is such a thing like infinite AND complete collection of distinct objects.
Furthermore, according to Traditional Math one has to show some missing member of a given infinite collection of distinct members, in order to prove that such a collection is incomplete.
By doing that, Traditional Math ignores the fact that the concept of Set (and it is shown right from the existence of the empty set beyond its "members") exists beyond the existence of any given collection, and as a result any given collection is incomplete w.r.t the existence of the concept of Set.
Some claims that the name and cardinality of a given set is defined by the property and amount of its members, but it does not change the fundamental fact that a set exists beyond its members, no matter what cardinality or name are given to it.
Place holder X exists no matter what name or cardinality is given to it, and this is a fundamental fact of any generalization of some mathematical framework, which you, The Man, and other traditional thinkers, simply can't comprehend.
In other words, the definition of Set is not formally satisfied, if Non-locaily\Locality linkage is not defined.
My answer is about the traditional and the non-traditional definition of Cardinality.
------
First let's reply to your last comments, which are related to my non-traditional definition of Cardinality:
The non-traditional definition of Cardinality:
Cardinality is the magnitude of existence of a given thing, which has at least the weakest or strongest magnitude of existence, as follows:
a) Emptiness, which is defined as "that has no predecessor", has cardinality 0, which is the weakest cardinality.
b) Fullness, which is defined as "that has no successor", has cardinality ∞, which is the strongest cardinality.
The strongest cardinality is stronger than any other given cardinality, such that any object that has the strongest cardinality, is at and beyond any existing thing that is compared to it (Emptiness, which has cardinality 0, is excluded because it is not an existing thing).
Non-locality and Locality are the result of the comparison between that has the strongest cardinality, with that does (or do) not has (or have) the strongest cardinality (Emptiness, which has cardinality 0, is excluded because it is not an existing thing).
Under this comparison (or linkage, if you will) among existing things; that has the strongest cardinality is simultaneously at AND beyond any existing thing (or collection of existing things) that does (or do) not have cardinality ∞.
1-dimensional space is the smallest existing thing that has cardinality ∞.
0-dimensional space is the smallest existing thing that has cardinality > 0, such that 0-dimensional space is the immediate successor of Emptiness, if Cardinality is the magnitude of existence of a given thing.
Because 1-dimensional space and 0-dimesional space are both existing things, then (as mentioned above) 1-dimesional space enables to be simultaneously at AND beyond any given 0-dimesional space along it, where any given 0-dimensional space along 1-dimensional space, can't be simultaneously at AND beyond 1-dimentional space.
------
Second, let's reply to your last comments, which are related to the traditional definition of Cardinality:
The traditional definition of Cardinality:
Cardinality is the number of distinct objects that belongs to an existing thing, called a set.
a) Emptiness, is the absence of any distinct objects of an existing set.
b) Fullness, is an undefined concept under the traditional notion of Set.
I claim that Fullness is used by mathematicians as an essential property of Set, even if it is not formally defined by them, as follows:
1) The fact that a set is an existing thing even if it does not have members, is not formalized by the traditional mathematicians, and as a result the cardinality of a set is defined only as the number of the distinct members, that belongs to a given existing set.
2) The existence of a set beyond the existence of its members is exactly the non-local property (which is not formalized by the traditional mathematicians) of the concept of set, which is not influenced by the number of the distinct members that are related to it.
3) Actually without the existence of a set beyond its members (which can be shown right from the existence of the empty set beyond Emptiness), there is no way to formally define any collection.
4) By understanding this fundamental fact, there is a way to formalize the existence of a set, such that it exists beyond any given collection of distinct members, which are related to it.
5) By understanding (1),(2),(3),(4), one defines the existence of a set beyond its members, as the non-local property of a set w.r.t its members, where the collection of any non-emply set is local w.r.t the existence of the set.
6) (5) is logically demonstrated by the inaccessibility of any amount of distinct objects to be beyond "{" and "}" , which is equivalent to the inability of any amount of points or segments to be beyond 1-dimensional space.
1-dimensional space exists beyond of any amount of points or segments along it, exactly as the empty set exists beyond the existence of members.
But traditional Math never formalized this fundamental fact, and as a result it lives under the illusion that there is such a thing like infinite AND complete collection of distinct objects.
Furthermore, according to Traditional Math one has to show some missing member of a given infinite collection of distinct members, in order to prove that such a collection is incomplete.
By doing that, Traditional Math ignores the fact that the concept of Set (and it is shown right from the existence of the empty set beyond its "members") exists beyond the existence of any given collection, and as a result any given collection is incomplete w.r.t the existence of the concept of Set.
Some claims that the name and cardinality of a given set is defined by the property and amount of its members, but it does not change the fundamental fact that a set exists beyond its members, no matter what cardinality or name are given to it.
Place holder X exists no matter what name or cardinality is given to it, and this is a fundamental fact of any generalization of some mathematical framework, which you, The Man, and other traditional thinkers, simply can't comprehend.
In other words, the definition of Set is not formally satisfied, if Non-locaily\Locality linkage is not defined.
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doronshadmi
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EDIT:
The Man, since your definition of Cardinality refers to the amount of objects, without the understanding of how multiplicity (the "s" at the end of a word like "objects") is possible in the first place, you simply have nothing to say about Cardinality, or what enables this concept.
The traditional definition about "the size of a set" is misleading because it ignores the fundamental fact of the existence of the concept of Set beyond the number of its members, exactly as we see right in the case of the Empty set, which is an existing thing that has no members.
As long as this misleading notion stands at the basis of Set Theory, this theory is invalid.
This narrow reasoning has no chance to be developed into the level that enables to distinguish between the complex and the non-complex, where in the case of "1-dimensional space", the "1" defines the the smallest existing thing that has cardinality ∞ , which is the cardinality of Non-locality, such that anything that has this cardinality can be simultaneously at AND beyond any limited domain.
Moreover, if one claims that 1-dimensioanel space is completely covered by a collection of 0-dimensional spaces, then one actually claims that 1-dimensional space is actually a collection 0-dimensional spaces. But by using this claim (which is definitely your claim, The Man) a line segment is reducible into a single 0-dimensional space, because 1-dimensional space actually does not exist.
In other words, The Man, you claim things al along this thread, without the results that are derived from them.
I explicitly say that the cardinality of the non-local is greater than the cardinality of any given collection.
The Man said:
The Man, since your definition of Cardinality refers to the amount of objects, without the understanding of how multiplicity (the "s" at the end of a word like "objects") is possible in the first place, you simply have nothing to say about Cardinality, or what enables this concept.
The traditional definition about "the size of a set" is misleading because it ignores the fundamental fact of the existence of the concept of Set beyond the number of its members, exactly as we see right in the case of the Empty set, which is an existing thing that has no members.
As long as this misleading notion stands at the basis of Set Theory, this theory is invalid.
Again you demonstrate your limited reasoning, which gets things only in terms of points, and the number of names that are related to them.The Man said:
This narrow reasoning has no chance to be developed into the level that enables to distinguish between the complex and the non-complex, where in the case of "1-dimensional space", the "1" defines the the smallest existing thing that has cardinality ∞ , which is the cardinality of Non-locality, such that anything that has this cardinality can be simultaneously at AND beyond any limited domain.
It is already done, but since you are unable to grasp Emptiness, Fullness Locality, Non-locality etc…, you simply unable to distinguish between the non-complex ( for example: ∞( 2(),0() ) and the complex ( for example: ∞( 2(0(x,y)) ) ) no matter how many times it is explained to you.The Man said:
Yet, it is an existing thing, even it does not have any members (which is a fact that you do not understand its influence on Set Theory and the Mathmatical Science).The Man said:
If one claims that the sum of converges lengths has the exact value of a given limit, then one actually claims that a given line segment is reducible into a point.The Man said:
Moreover, if one claims that 1-dimensioanel space is completely covered by a collection of 0-dimensional spaces, then one actually claims that 1-dimensional space is actually a collection 0-dimensional spaces. But by using this claim (which is definitely your claim, The Man) a line segment is reducible into a single 0-dimensional space, because 1-dimensional space actually does not exist.
In other words, The Man, you claim things al along this thread, without the results that are derived from them.
Not at all.The Man said:
I explicitly say that the cardinality of the non-local is greater than the cardinality of any given collection.
Let me correct it for you. If Cardinality is the number of distinct objects of a given set, then the cardinality of a non-empty set is defined by the linkage among Non-locality and Locality.The Man said:
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The Man
Unbanned zombie poster
Once again Doron cardinality specifically deals with the size of sets and if your not referring to sets then your not referring to cardinality. So once again please define your “magnitude of existence” independent of cardinality. You can define your “magnitude of existence” any way you want, but if you are going to refer to the concept of cardinality then you are referring to the size of sets.
What exactly is infinite about your “1-dimensional space” space?
Based on your so called “definition” above your “magnitude of existence” isn’t related to anything quantitative so the “magnitude” of your “magnitude of existence” is evidently something you just pull out of your arse.
Neither of those assertions are part of the definition of cardinality. Stop simply trying to posit aspects of your own failed reasoning onto others.
Your claim is false.
The empty set is an axiom of some set theories.
What “The existence of a set beyond the existence of its members” “is not influenced by the number of the distinct members that are related to it”? You really do need to start checking the self-consistency of your assertions Doron.
“the empty set beyond Emptiness”? So now your empty set isn’t, well, empty? What do you mean by “beyond Emptiness”?
Doron a set is a collection of it’s defined members, so anything you might fantasies “beyond” that is simply not the same defined set.
See above
Well of course the members of a set are not “beyond” that set, that what makes them members of that set. However the point 5 is “beyond” the one dimentional space of the interval [1,5) in the reals exactly because it is not a member of the set representing that interval.
Nope.
Though evidently simple mathematics and geometry simply exist well “beyond” your understanding.
Please define any location in a one dimensional space that is “beyond” a location in that one dimensional space.
Again show what member of such an infinite collection is not a member of that infinite collection such that it is incomplete.
Actually Doron that is just a result meaning of “incomplete”, that something is missing. If nothing is missing then it is in fact complete.
Once again Doron the empty set is defined by exactly its lack of members, but evidently that is still just beyond you.
The fact that the concept of a set or collection is not limited to any one set or collection (including the empty set) still does not make any set or collection incomplete. In fact the very concept of set or collection makes any abstract set or collection complete as it specifically has and is limited to all of its members.
Again Doron what member of a set is not a member of that set?
Oh waiter, can I get some dressing on this word salad?
As you can’t define your “Non-locaily\Locality linkage” formally, that claim is demonstrably false.
doronshadmi
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The Man, please answer only by yes or no to the following question:The Man said:
The Man, do you claim that the empty set and emptiness is the same?
He actually did and was right about it. If the member of the infinite collection is a number expressed as 0.999..., for example, then the traditional math showed with "varying degrees of rigor" that numbers whose fractional part comprises a single infinitely self-repeating digit don't exist.
http://en.wikipedia.org/wiki/0.999...
doronshadmi
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0.999...[base 10] is a numral that represents number 1, according to Traditional Math.
Let us take again the example of 1/2+1/4+1/8+1/16+1/32+1/64+…
________________________________________________________________ 1/2 +
________________________________________________________________ 1/4 +
________________________________________________________________ 1/8 +
________________________________________________________________ 1/16 +
________________________________________________________________ 1/32 +
________________________________________________________________ 1/64 + ...
Traditional Math does not understand that if the sum of infinitely many added and convergent segments = 1, then there are only finitely many Blue\Red Red\Blue cases.
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jsfisher
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So, you are again confirming that, according to you, 2 is not a member of the set {2}.
So, you reconfirm you claim while demonstrating you blunder. Neither {{a}} nor {{b}} are members of the set {{a},{b}}.
So, you are again confirming that, according to you, 1/4 is different from 0.25.
So, you are again confirming that, according to you, sets, maps, and functions are all the same thing.
What has this to do with Geometry? Are you claiming (now) that Geometry does not have a concept of 'infinite'?
Certainly for any post by you, independent of the forum in which you place it, mathematical concepts are divorced from their formal meaning.
That's great, but are you confirming or retracting your claim?
You have also claimed the Cantor Set contains only one point. Do you stand by that statement?
doronshadmi
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EDIT:
Let me be clearer about my argument against 1/2+1/4+1/8+1/16+1/32+1/64+… = 1
________________________________.________________________________ 1/2 +
________________________________________________.________________ 1/4 +
________________________________________________________.________ 1/8 +
____________________________________________________________.____ 1/16 +
______________________________________________________________.__ 1/32 +
_______________________________________________________________._ 1/64 + ...
If 1/2+1/4+1/8+1/16+1/32+1/64+… = 1, as Traditional Math claims, then the most right (Red or Blue) segment
must be reduced into a single 0-dimensional space.
In this case we get these results:
1) 1-dimensional space is reducible into 0-dimesional space, or in other words, 1-dimensional space does not exist, and all we have is an infinite collection of 0-dimensional spaces.
2) The number of Blue\Red Red\Blue collection of segments, is a finite collection, because the transition form a collection of a single 0-dimensional space into a collection of more than a single 0-dimensional space and vice versa, has no intermediate degrees of transitions (the transition from cardinality 1 to cardinality 2 and vice versa, has no intermediate degrees of transitions).
In general, Traditional Math's reasoning fails to deal with the one-to-many many-to-one transition, as shown by this case without a loss of generality.
Let me be clearer about my argument against 1/2+1/4+1/8+1/16+1/32+1/64+… = 1
________________________________.________________________________ 1/2 +
________________________________________________.________________ 1/4 +
________________________________________________________.________ 1/8 +
____________________________________________________________.____ 1/16 +
______________________________________________________________.__ 1/32 +
_______________________________________________________________._ 1/64 + ...
If 1/2+1/4+1/8+1/16+1/32+1/64+… = 1, as Traditional Math claims, then the most right (Red or Blue) segment
must be reduced into a single 0-dimensional space.
In this case we get these results:
1) 1-dimensional space is reducible into 0-dimesional space, or in other words, 1-dimensional space does not exist, and all we have is an infinite collection of 0-dimensional spaces.
2) The number of Blue\Red Red\Blue collection of segments, is a finite collection, because the transition form a collection of a single 0-dimensional space into a collection of more than a single 0-dimensional space and vice versa, has no intermediate degrees of transitions (the transition from cardinality 1 to cardinality 2 and vice versa, has no intermediate degrees of transitions).
In general, Traditional Math's reasoning fails to deal with the one-to-many many-to-one transition, as shown by this case without a loss of generality.
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jsfisher
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You continue to misrepresent what is meant by such a summation. You have been corrected many, many times, yet you persist. Are you convinced that by the sheer power of repetition you will somehow morph your bogus statement into a true one?
At any rate, since your premise is false, your conclusions have no basis. The fact your inferences are also faulty only embellishes your spectacular failing.
The Man
Unbanned zombie poster
If your “magnitude of existence” does not specifically involve the size of set it simply does not involve cardinality, regardless of how desperately you apparently want to associative your “magnitude of existence” with cardinality.
Once again Doron the ignorance remains completely yours.
Doron the only “misleading” is your own and as such it is simply your notion that is invalid
I asked you to define your dimensionality without points, specifically how it is quantified, so far you have not even attempted to do so.
So the “1” of your "1-dimensional space" is just another quantity you pull out of your arse, how surprising.
Your nonsense notations and labels do nothing to quantify your notion of “dimensionality”, as such your “"1-dimensional space" might as well be a ‘blue-dimensional space’ or ‘apple-dimensional space’ it is just a name you use to evidently mislead people into considering that there is actually something quantifiable about your concept of “dimensionality”.
Once again the fact that it “does not have any members” is exactly what makes it the, well, empty set and as I said before it is an axiom of some set theories.
Nope, we have been over this before Doron, in order for the current segment being summed to be zero length (a point) the previous sum must already have reached the limit
Nope those are once again only your deliberately misleading claims that you try to ascribe to others. Again please show any location in a 1-dimensional space that can not be covered by points. It is the fact there a point is zero dimensional that insures it can cover every and any location in a one dimensional space. Once again it only takes two points to define a one dimensional space and one can always fit an infinite number of other points between those two.
No Doron you just like to make ridiculous claims and then simply try to ascribe them to others, in fact you seem rather compulsive about it. Have you ever consider getting professional help as you seem unable to resist the compulsion?
So then your ‘linked’ complex still has that “non-local property” of your “"{" "}" or ...________..., “, in spite of what you claimed before?
And you are explicitly wrong since cardinality is specifically related to the size of a collection. If your “non-local is” not a collection then cardinality simply does not apply.
Nope still wrong, let me correct it for you. “If Cardinality is the number of distinct objects of a given set” then any cardinality (including that of the empty set) “is the number of distinct objects of a given set”.
Are you claiming that your “linkage among Non-locality and Locality” fails for the empty set?
No.
doronshadmi
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EDIT:
Moreover, the traditional notion of Cardinality can't deal directly with Emptiness as the level of existence that has Cardinality 0, and it does it indirectly by using an existing thing like The Empty set in order to define cardinality 0.
This is a fundamental conceptual mistake that Organic Mathematics fixes, and this correction is important for further development of The Mathematical Science.
Only the linkage between that has cardinality ∞ and that has cardinality 1 enables to define cardinalities which are > 1 AND < ∞, where that has cardinality 1 is local w.r.t that has cardinality ∞ (where Locality is the property of being at most simultaneously at one and only one location w.r.t to that has cardinality ∞), and that has cardinality ∞ is non-local w.r.t that has cardinality 1 (where Non-locality is the property of being at least simultaneously at and beyond the location w.r.t to that has cardinality 1).
Yet, as written above, this distinction is wrongly interpreted and used by Traditional Mathematics, because the traditional notion of Cardinality can't deal directly with Emptiness as the level of existence that has Cardinality 0, and it does it indirectly by using an existing thing like The Empty set in order to define cardinality 0, which is a fundamental conceptual mistake.
You are using a concept like collection, without understanding its existence.The Man said:
No, I am claiming that the traditional notion of Cardinality as the number of objects of a given set, can't distinguish between the weakest existing thing (a point), which has cardinality 1, and the minimal form of the strongest existing thing (a line), which its existence is stronger than any collection of objects along it, and its cardinality is ∞.
Moreover, the traditional notion of Cardinality can't deal directly with Emptiness as the level of existence that has Cardinality 0, and it does it indirectly by using an existing thing like The Empty set in order to define cardinality 0.
This is a fundamental conceptual mistake that Organic Mathematics fixes, and this correction is important for further development of The Mathematical Science.
Only the linkage between that has cardinality ∞ and that has cardinality 1 enables to define cardinalities which are > 1 AND < ∞, where that has cardinality 1 is local w.r.t that has cardinality ∞ (where Locality is the property of being at most simultaneously at one and only one location w.r.t to that has cardinality ∞), and that has cardinality ∞ is non-local w.r.t that has cardinality 1 (where Non-locality is the property of being at least simultaneously at and beyond the location w.r.t to that has cardinality 1).
In that case you have no problem to distinguish between the definition of The Empty set, which is based on the fact that an existing thing has no sub-existing things, and the fact that The Empty set is an existing thing, and therefore its existence is beyond Emptiness.
Yet, as written above, this distinction is wrongly interpreted and used by Traditional Mathematics, because the traditional notion of Cardinality can't deal directly with Emptiness as the level of existence that has Cardinality 0, and it does it indirectly by using an existing thing like The Empty set in order to define cardinality 0, which is a fundamental conceptual mistake.
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doronshadmi
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The Man, do you really don't get that if the previous sum already have reached the limit, then there is no such a thing like "the current segment" (where a segment is defined as an object that its length > 0)?The Man said:
"the current segment being summed to be zero length" is an utter nonsense.
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jsfisher
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I see you continue to struggle to disprove a definition.
I have noticed that cardinality doesn't distinguish between blue and green. Does that add to you disproof of the definition? (Oh! Dear! OM doesn't either! We are all doomed!!!!)
You persist on this fool's errand, Doron. Give it up. You want something different? Great! Define something different. Knock yourself out. But please stop all this verbal idiocy you spew just because you don't understand basic Mathematics.
The Man
Unbanned zombie poster
Once again you are simply wrong. A set containing a point has a cardinality of one a set containing a line has a cardinality of one, while a set containing all the points of that line has an infinite cardinality. Again if you are not referring to sets then cardinality simply is not applicable.
I’ll ask again, what exactly is infinite about your “line”?
Your claim is simply false. The empty set is empty by definition and its cardinality is a direct result of the empty set being, well, empty. The only things indirect are you and your nonsense assertions.
The mistake is entirely yours, and rather than fixing your own mistake (and actually learning some mathematics) you have invented some fantasy world so you can revel in your own, apparently deliberate, mistakes.
No Doron once again the definition of cardinality “enables to define” all, well, cardinalities including 0, 1 and “∞”. However since you don’t actually use cardinality or its definition, it has just become another label or buzz word for you to use so you can revel in your own, apparently deliberate, mistakes.
Could you put that in English please?
Doron the empty set and emptiness are both abstract concepts and only ‘exist’ as such. So I would certainly not claim that the “existence” of the empty set is “beyond Emptiness”. In fact the empty set is a particular concept of Emptiness (one related specifically to a set) so I might stipulate that the concept of emptiness is more general or more broadly applicable than the concept of the empty set. Also the empty set is strictly limited to emptiness, otherwise it wouldn’t be, well, empty.
Perhaps you need to explain exactly what you mean by “beyond” and specifically “beyond Emptiness”. Though I have little hope of your explanation being self-consistent, generally consistent or meaningful in any way other than to just help you so you can revel in your own, apparently deliberate, mistakes.
doronshadmi
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jsfisher, Traditional Math uses Cardinality as the size of a given collection, without understanding its existence.
The Man
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Well look who finally caught up. So are you done with your ridiculous claim that “a given line segment is reducible into a point.”?
It’s your utter nonsense Doron, but you may just be catching on to that fact now.
doronshadmi
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You are using a concept like collection, without the understanding of its existence.The Man said:
The Man said:
The Empty set, which is an existing thing (even if this existence is interpreted by you as an abstraction) has an existence that is greater than Emptiness (even if Emptiness is interpreted by you as an abstraction).The Man said:
Traditional Math ignores this abstract fact, and uses an abstract existing thing, in order to indirectly define the cardinality of Emptiness (which is an abstraction that its cardinality is smaller than the cardinality of an existing abstraction like the empty set).
Since you prefer to distinguish between the abstract and the non-abstract, I'll follow your notion and claim that this indirect use of abstract concepts is a fundamental abstract conceptual mistake.
Organic Mathematics defines Cardinality directly, and by this direct approach, Emptiness has Cardinality 0 and The Empty set has Cardinality 1, which is equivalent to the cardinality of a point, because both of them are the minimal abstract existence.
|| = 0
|{}| = |.| = 1
|{{}}|=|{}{}| = |..| = 2
|{{{}}}|=|{}{}{}| = |…| = 3
etc …
|____| = ∞ where ____ is the minimal abstract existing thing that its cardinality is greater (beyond) than any collection, whether the collection is a set or a multiset.
Moreover, any cardinality > 1 is the result of . (or "{""}") ___ linkage, such that:
|{{}}|=|{}{}| = |..| = 2
|{{{}}}|=|{}{}{}| = |…| = 3
etc …
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doronshadmi
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The utter nonsense is yours, because you are unable to get http://www.internationalskeptics.com/forums/showpost.php?p=6575426&postcount=12371 , which clearly demonstrates this utter nonsense of yours about the sum of convergent lengths, which is = to a given limit.
You are a complete ignorant about this fine subject, because you do not understand the results of your own claims.
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jsfisher
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How does this irrelevant, incorrect, and trivial statement impact the definition of cardinality?
jsfisher
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Referring back to a post where you, Doron, re-demonstrate your lack of understanding of Mathematics doesn't support your claim. The only thing clearly demonstrated in that post is your obstinate insistence on misrepresentation.
That's not the main argument, coz that series approaches it's limit (and according to some even reaches it!) different way than 0.999..., the number that has been expelled from R. You need to look again at
http://www.internationalskeptics.com/forums/showpost.php?p=6563628&postcount=12347
and find the bug there.
There were several UFO sightings years ago that made fun of this particular issue. UFO is 1 acronym divided into 3 letters, so it was predefined as UFO = 1/3. The movement and time of that mysterious triangle in the sky suggested some analogy . . .
1/9 = 0.111...
2/9 = 0.222...
3/9 = 0.333...
Clearly, when the denominator is 9, then the numerator repeats itself infinitely in the result provided by the long division. Since the analogy has formed itself pretty, no need to waste time by dividing -- you just enter the numerator into the result . . .
7/9 = 0.777...
8/9 = 0.888...
When you check once again to make sure that the analogy holds by doing the long division, you find out that the analogy really holds. So let's continue . . .
9/9 = 0.999...
But when you do the long division, then
9/9 = 1
Now it's the analogy vs the long division. Since we've got already enough religious wars, no one needs mathematical wars, and so the elders wisely decided on 0.999... = 1.
So that's the alien contribution to all possible explanation to the miraculous identity. There is another based on the etymology of "long division."
1/3 = 0.33333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333...
Since LOL is 1 acronym divided into 3 letters, LOL = 1/3.
You don't seem to have answered this, Doron:
jsfisher
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Isn't it sufficient to know that Doron believes 0.25 appears more than once between 0 and 1?
doronshadmi
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From a structural point of view 1/4 (the vertical red line) or 0.01[base 2] have one location along 0________1, where 0.25[base 10] has two locations along 0________1, as follows:
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From a structural point of view 1/4 (the vertical red line) or 0.01[base 2] have one location along 0________1, where 0.25[base 10] has two locations along 0________1, as follows:
What, you mean one 'location' for 0.2 and one for 0.25? You're kidding, right? In what way is that meaningful or useful? So there would be 3 locations for 1/8 (0.125)? A 'location' is just a decimal place, then? And why is there not one at 0.0 for 0.01[base 2]?
doronshadmi
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I clearly show at least two locations for 0.25[base 10] along 0______1
It enables to deal with the structural properties of numbers, in addition to their quantitative properties.
According to the structural properties of numbers, 1/8 has at least 1 location along 0_____1 and 0.125[base 10] has at least 3 locations along 0_____1
No, "decimal" is related only to [base 10], where the locations along different scale levels of a given number, can be found in any base value > 1.
It depends on how you define the structural aspect of the number.
If the initial location (0.0) is considered, then 1/4 has at least two locations, which are the initial location 0 AND 1/4 location along 0______1
In that case 1/4 and 0.01[base 2] have two locations along 0______1,
0.25[base 10] has 3 locations along 0______1,
0.125[base 10] hase 4 locations along 0______1, etc ...
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Go on, then, define it.
According to the drawing, you seem to be moving at a constant speed v away from the point of recovery R. . .
Your "correction"
didn't mention the base 2 nonsensical excuse that you came up with this time.
Theorem: 1 + 1 = 3.
Proof: If we add 1 to 1, we obtain 11, and 11(binary) = 3(decimal).
doronshadmi
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EDIT:
Student: “I have a proof against the assertion that 1/2+1/4+1/8+1/16+1/32+1/64+… = 1, as follows:”
“First, we express 1/2+1/4+1/8+1/16+1/32+1/64+… etc ... by the following diagram:”
________________________________.________________________________ 1/2 +
________________________________________________.________________ 1/4 +
________________________________________________________.________ 1/8 +
____________________________________________________________.____ 1/16 +
______________________________________________________________.__ 1/32 +
_______________________________________________________________._ 1/64 + ...
“If 1/2+1/4+1/8+1/16+1/32+1/64+… = 1, as you assert, then the most right (Red or Blue) segment must be reduced into a single 0-dimensional space.”
“But according to the diagram above, this reduction is impossible, because 1/2+1/4+1/8+1/16+1/32+1/64+… converge as pairs of segments, which have equal lengths , where the values of 1/2+1/4+1/8+1/16+1/32+1/64+… are determined by the most right point of the left segment of any arbitrary convergent pair of Blue\Red Red\blue equal segments, upon infinitely many scale levels”.
“In that case any arbitrary right segment of a given pair of equal segments upon infinitely many scale levels, is exactly the gap between any arbitrary number of 1/2+1/4+1/8+1/16+1/32+1/64+… and the number of the limit.”
Teacher: “Nope, we have been over this before, in order for the current segment being summed to be zero length (a point) the previous sum must already have reached the limit.”
Student: “I agree with you, this reduction is impossible exactly because 1/2+1/4+1/8+1/16+1/32+1/64+… are the most right points of the left segment of any arbitrary convergent pair of Blue\Red Red\blue equal segments, upon infinitely many scale levels”.
“Again, In that case any arbitrary right segment of a given pair of equal segments upon infinitely many scale levels, is exactly the gap between
any arbitrary number of 1/2+1/4+1/8+1/16+1/32+1/64+… and the number of the limit.”
Teacher: “Referring back to your diagram re-demonstrate your lack of understanding of Mathematics doesn't support your claim. The only thing clearly demonstrated in that diagram is your obstinate insistence on misrepresentation.”
Student: “Is this kind of reply is considered by you as a reasonable answer to my proof against
the assertion that 1/2+1/4+1/8+1/16+1/32+1/64+… = 1?”
“Since you assert that 1/2+1/4+1/8+1/16+1/32+1/64+… = 1, then please show it without reducing the most right (Red or Blue) segment into a single 0-dimensional space.”
Student: “I have a proof against the assertion that 1/2+1/4+1/8+1/16+1/32+1/64+… = 1, as follows:”
“First, we express 1/2+1/4+1/8+1/16+1/32+1/64+… etc ... by the following diagram:”
________________________________.________________________________ 1/2 +
________________________________________________.________________ 1/4 +
________________________________________________________.________ 1/8 +
____________________________________________________________.____ 1/16 +
______________________________________________________________.__ 1/32 +
_______________________________________________________________._ 1/64 + ...
“If 1/2+1/4+1/8+1/16+1/32+1/64+… = 1, as you assert, then the most right (Red or Blue) segment must be reduced into a single 0-dimensional space.”
“But according to the diagram above, this reduction is impossible, because 1/2+1/4+1/8+1/16+1/32+1/64+… converge as pairs of segments, which have equal lengths , where the values of 1/2+1/4+1/8+1/16+1/32+1/64+… are determined by the most right point of the left segment of any arbitrary convergent pair of Blue\Red Red\blue equal segments, upon infinitely many scale levels”.
“In that case any arbitrary right segment of a given pair of equal segments upon infinitely many scale levels, is exactly the gap between any arbitrary number of 1/2+1/4+1/8+1/16+1/32+1/64+… and the number of the limit.”
Teacher: “Nope, we have been over this before, in order for the current segment being summed to be zero length (a point) the previous sum must already have reached the limit.”
Student: “I agree with you, this reduction is impossible exactly because 1/2+1/4+1/8+1/16+1/32+1/64+… are the most right points of the left segment of any arbitrary convergent pair of Blue\Red Red\blue equal segments, upon infinitely many scale levels”.
“Again, In that case any arbitrary right segment of a given pair of equal segments upon infinitely many scale levels, is exactly the gap between
any arbitrary number of 1/2+1/4+1/8+1/16+1/32+1/64+… and the number of the limit.”
Teacher: “Referring back to your diagram re-demonstrate your lack of understanding of Mathematics doesn't support your claim. The only thing clearly demonstrated in that diagram is your obstinate insistence on misrepresentation.”
Student: “Is this kind of reply is considered by you as a reasonable answer to my proof against
the assertion that 1/2+1/4+1/8+1/16+1/32+1/64+… = 1?”
“Since you assert that 1/2+1/4+1/8+1/16+1/32+1/64+… = 1, then please show it without reducing the most right (Red or Blue) segment into a single 0-dimensional space.”
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doronshadmi
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With or without the initial point.
doronshadmi
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Wrong theorem, because 0______1 is independent of any base value, as clearly shown by the following diagram:
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doronshadmi
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Sympathic, http://www.schizophrenia.com/meds.html may help you.
jsfisher
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Someday, just maybe, you'll repeat this nonsense, and it will suddenly and magically become true. It will cease to be a gross misrepresentation and misunderstanding on your part.
Today, however, is not that day. You are still blatantly, embarrassingly wrong.
jsfisher
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Doron,
Here, let's give you a leg-up on that "someday." All you need do is disprove the following statement. Then that nasty old summation won't work any more.
Perhaps you can find a simple counter example?
Here, let's give you a leg-up on that "someday." All you need do is disprove the following statement. Then that nasty old summation won't work any more.
[latex]$$$\displaystyle
\forall {\epsilon > 0} ,\: \exists N \; n > N \Rightarrow \epsilon > \left| 1 - \sum_{i=1}^{n} {1 \over {2^i}} \right|
$$$[/latex]
\forall {\epsilon > 0} ,\: \exists N \; n > N \Rightarrow \epsilon > \left| 1 - \sum_{i=1}^{n} {1 \over {2^i}} \right|
$$$[/latex]
Perhaps you can find a simple counter example?
The Man
Unbanned zombie poster
Nope, I have certainly never claimed “a given line segment is reducible into a point.”. That ridiculous and utterly nonsensical claim is just yours Doron. That you would simply like to ascribe it to others does not make it any less your own utter nonsense. You are completely and evidently deliberately ignorant since you just don’t want to comprehend that the results of your claims are still just the results of your claims, regardless of whom you whish to ascribe them to.
The Man
Unbanned zombie poster
"Of Cardinality"
Sung to the tune of
"AT SEVENTEEN" By Janis Ian
"Of Cardinality"
I learned the truth of cardinality
That mathematicians just can’t see
Among existing things with some magnitude
Of values to which I’ll just allude
The concepts that I never knew
The musing of kindergarten youth
Were taken as demonstrable
Of cardinality I learned the truth...
And those of us with 1-D spaces
Just can’t cover all the places
With just points all alone
Imagining math of my own
Replete with inconsistencies
And repeated vague obscurities
It isn't all it seems, of cardinality...
It’s from Fullness that it all abounds
For any non-empty set around
By some vague non-local property
I’ll claim with all sincerity
The set exists beyond itself
and add to it my imagination’s wealth
With even more obscurity
A linkage called complexity...
So remember those who use a set
And haven’t got that linage yet
From claims lacking quality and dubious integrity
I’ll ascribe to them claims of my own design
And unflinchingly claim a line
Exceeds the points it has, in cardinality...
From Emptiness we still must gain
The explanation that never came
Of the linkage from which it all ensues
You don’t get it? Well you lose
That’s too bad, it just a shame
If I can’t explain, it’s just you I’ll blame
Your direct perception just can’t see
This linkage of complexity…
It’s a fantasy that’s just repeated
An infinite set can’t be completed
With it’s members all alone
Imagining math of my own
Replete with inconsistencies
And repeated vague obscurities
It just isn't anything, of cardinality...
Sung to the tune of
"AT SEVENTEEN" By Janis Ian
"Of Cardinality"
I learned the truth of cardinality
That mathematicians just can’t see
Among existing things with some magnitude
Of values to which I’ll just allude
The concepts that I never knew
The musing of kindergarten youth
Were taken as demonstrable
Of cardinality I learned the truth...
And those of us with 1-D spaces
Just can’t cover all the places
With just points all alone
Imagining math of my own
Replete with inconsistencies
And repeated vague obscurities
It isn't all it seems, of cardinality...
It’s from Fullness that it all abounds
For any non-empty set around
By some vague non-local property
I’ll claim with all sincerity
The set exists beyond itself
and add to it my imagination’s wealth
With even more obscurity
A linkage called complexity...
So remember those who use a set
And haven’t got that linage yet
From claims lacking quality and dubious integrity
I’ll ascribe to them claims of my own design
And unflinchingly claim a line
Exceeds the points it has, in cardinality...
From Emptiness we still must gain
The explanation that never came
Of the linkage from which it all ensues
You don’t get it? Well you lose
That’s too bad, it just a shame
If I can’t explain, it’s just you I’ll blame
Your direct perception just can’t see
This linkage of complexity…
It’s a fantasy that’s just repeated
An infinite set can’t be completed
With it’s members all alone
Imagining math of my own
Replete with inconsistencies
And repeated vague obscurities
It just isn't anything, of cardinality...
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