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As I said the property of mortality is not chaneged by the number of members that can be found in some collection (finite or infinite). But the Cardinality of non-finite cannot be found, because this is exactly the difference between a finite collection (its Cardinality can be found) and a non finite collection (Its Cardinality cannot be found). In order to see my point of view, please read: http://www.geocities.com/complementarytheory/EProp.pdf Thank you.

No Bloodhound, it is not ovbious at all, please read post #1 to see it for yourself.

What is "complete Quantity"? For example: Please show us stap by stap how can we define the complete number of elements in {1,1,1,1,1,1,1,...}? Thank you.

In the case of 1-1 between two collections of Natural numbers' date=' then the cardinality the total number (the SUM) of 1-1 mapping that can be found. Yeh, I know The SIZE of a collection is actually its Cardinality. Can you please explain what is the difference between Size of a collection and Sum of a collection? Thank you.

One of the most devastating things in the Language of Mathematics and its logical reasoning is a hidden assumption, and the worst thing is a first-order hidden assumption. Let FOHA be a First-Order Hidden Assumption. Let UQ be Universal-Quantification. Is there a FOHA in the way we use UQ concept? The UQ is based on the term ‘For All’. The meaning of the word ‘All’ is synonym to the word ‘Complete’ and if it is related to a collection of elements, then from a quantitative point of view ‘All’ is actually the SUM of this collection, where in the level of SUM we are no longer in the level of each single element that exists in this collection. The SUM of a collection is actually its Cardinality. If we define some property that can be found in each single element in this collection, then this property cannot be SUMMERIZED, because this definition can exist only in the level of each single element. For example: All men are mortal. The property of being mortal does not changed by the quantity of men. In this case we cannot use the Quantity concept because the existence of this definition cannot be developed beyond the Quantity 1 (each single element). So we see that there can be found some common property to several distinguished elements, but this common property is not related to the Quantity concept, and therefore it cannot be SUMMERIZED, or in other words, this common property cannot be summarized beyond the quantity 1, and the meaning of quantity 1 in this case is synonym to the term ‘There exists’. So we can clearly see that a common property that cannot be summarized is actually based on the term ‘There exists’, where this common property’s existence is in the level of each distinguished element that exists in the collection. Let FA be ‘For All’. Let TE be ‘There Exists’. It is very important not to mix between FA and TE. Now let us say that we have a collection of infinitely many elements. For example, the collection of the Natural numbers. The existence of this collection is based on these definitions: a) 1 is a Natural number b) N is a container of only Natural numbers c) 1 is in N d) n = 1 e) If n is in N then n+1 is in N. By term (e) we know that for each arbitrary single element in N there is another element that is bigger by 1 from this arbitrary single element. In this case the common concept (that is actually based on TE) is no other then the “famous” Successor concept. Since the Successor is a TE product, we cannot conclude anything about the complete Quantity of N collection. Furthermore, we cannot use this definition: For All n in N, If n is in N then n+1 is in N. We can use FA only if we can summarize the elements of N, but since there are infinitely many elements, then their SUM cannot be found and this if a fundamental difference between a collection of finitely many elements (that its SUM can be found) and a collection of infinitely many elements (that its SUM cannot be found). Since the meaning of the word ‘All’ (in the ‘For All’ term) is synonym to the word ‘Complete’ and if it is related to a collection of elements, then from a quantitative point of view ‘All’ is actually the SUM of this collection. But then we can clearly see that by using the prefix ‘For All n in N’ we actually use a hidden assumption that gives us the illusion that we can define the Cardinality of a non-finite collection. Some one can say: But we can use 1-1 mapping between collections of elements, and by this technique we can find if there is or there is no difference in the quantity between the collections, even if we do not know the exact SUM (the Cardinality) of each collection. Let us check this claim: By 1-1 mapping, we take two collections A and B (for example) and try to find a 1-1 map between each element in A and each element in B. If the two collections are finite collections, then we can summarize the Quantity of the 1-1 mapping, and then we compare this result with the cardinality of each examined collection, and only by these 3 Cardinals we can conclude if both collections are Equal to the Cardinal of the 1-1 mapping, or not. If the collections are not equal, then it is obvious that the cardinal (the SUM) of the 1-1 mapping is equal to the smaller collection. But if we use the 1-1 mapping technique between two collections where each one of them has infinitely many elements, then: In this case no cardinality can be found, and our conclusions are depend on TE. But also in this case we want to find out if the two collections have the same quantity of elements, or not. By TE (There Exist) the only information that we can get is only in the level of each single 1-1 mapping, where the SUM of these 1-1 mapping cannot be summarized, because TE cannot go beyond quantity 1. So we clearly can see that if the cardinality of the two examined collections cannot be found, then we cannot use FA (For All) on these collections but only TE (There Exists). Conclusion: We can get a result out of 1-1 mapping technique only if the cardinality of the 1-1 mapping can be found, and it cannot be found if both examined collections are non-finite. Now we can clearly see the fundamental conceptual mistake of using FA on a collection of infinitely many elements, which gives us the illusion that we can conclude some meaningful conclusions about the quantitative property of the examined collections. FA (‘For All’) is the Universal Quantifier (which is based on the Quantity concept) and there is a First Order Hidden Assumption in the way we use the Universal Quantifier concept on collections of infinitely many elements.

3) You are correct' date=' there is a new quantifier notated as '[b']@[/b]' and called "For Each", and we have a fundamental Paradigm-Shift in the Language of Mathematics and its Logical reasoning. As for your answer: What an Emptiness!

And this is the fundamental conceptual mistake of the modern Language of Mathematics' date=' which in its framework 'ALL' = 'EACH'. 'ALL' = 'EACH' is a pure non-sense. In the rest of the word 'ALL' not= 'EACH', and this non-sense is used only by the community of professional mathematicians. Your definition of aleph0 is based on fundamental logical contradiction because: a) In your community 'ALL' = 'EACH' . b) You have no logical proof that clearly shows that: (‘Universal quantifier’ AND ‘Collection of infinitely many finite elements’) AND (‘Successor’) --> True and I am still waiting to your rigorous logical proof. Matt' date=' I think that you have to explain how can it be that In your community 'ALL' = 'EACH'? and how (‘Universal quantifier’ [b']AND[/b] ‘Collection of infinitely many finite elements’) AND (‘Successor’) --> True ? Can you answer to these two simple questions?

In this case you do not use the Universal quantifier' date=' but you speak about some property that cannot be found in [b']each single element [/b] in N that is > 9. But you use the Universal quantifier in order to define the cardinality of N. In order to speak about the properties of |N| you first have to define |N|. So' date=' I am still waiting to your proof, which clearly shows that: (‘Universal quantifier’ [b']AND[/b] ‘Collection of infinitely many finite elements’) AND (‘Successor’) --> True because without this proof, |N| cannot be found and no function (mapping) result can be found between collections of infinitely many elements.

No you cannot' date=' becuse you did not show how the cardinality of [b']N[/b] can be found.

Matt and Dankomed, In the basis (the first-order level) of N set there exists the sequence of infinitely many 1 objects, for example: {1,1,1,1,1,1,1,1,1,1,1,1,1,…} Each Natural number is based on nested finite subsets of this sequence, for example: 1= {1} 2= {{1},1} 3= {{{1},1},1} … and {1, 2, 3, … } = { {1}, {{1},1}, {{{1},1},1}, … } The meaning of the Successor is actually, to add 1 to {1,1,1,1,1,1,1, …} sequence, for example: 1 = {1} --> |{1}| 2 = {1}+1 = {{1},1} --> |{1,1}| 3 = {{1},1}+1 = {{{1},1},1} --> |{1,1,1}| … Now by using a Universal quantifier (the term ALL) on N ( where N is a collection of infinitely many finite objects) then if ALL 1 objects are in N, then it implies that +1 (the Successor) cannot be found. Form this result we can conclude that if N is based on infinitely many 1 objects (where each 1 object is a finite and non-composed element), then the logic connectivity between the ‘Universal quantifier’ concept and the ‘Successor’ concept cannot be but a XOR logical connective, for example: (‘Universal quantifier’ AND ‘Collection of infinitely many finite elements’) XOR (‘Successor’) --> True But: (‘Universal quantifier’ AND ‘Collection of infinitely many finite elements’) AND (‘Successor’) --> False If you disagree with me, then you have to prove that: (‘Universal quantifier’ AND ‘Collection of infinitely many finite elements’) AND (‘Successor’) --> True Can you prove that?

Where are the details of YOUR proof?

No dear Haggy, Please explain to me this Blah, blah blah.

It depends of how we understand the word 'complete'. A set is complete only if it's cardinality can be found. As I clearly showed, |N|-1 means, that the cardinality of set N cannot be found. This is the exact difference between a collection of finitely many elements (where its cardinality can be found, therefore it is a complete collection) and a collection of infinitely many elements (where its cardinality cannot be found, therefore it is an incomplete collection). Let us check another thing, which is the structure of the Natural numbers. As we know, 1 is the indivisible atom of N members, so let us show only the sequence of atoms of N members: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,…} In order to define an order in such a collection, we use subsets of this infinitely long sequence, for example: {{1}, {{1},1}, {{{1},1},1}, {{{{1},1},1},1}, {{{{{1},1},1},1},1} ,…} = {1, 2, 3, 4, 5, …} But it is clear that basic cardinality of both sets is the same |N|-1 We can create infinitely many variations of subsets, for example: {{{1},1}, {{{{1},1},1},1}, {{{{{{1},1},1},1},1},1} ,…} = {2, 4, 6, …} But still it is clear that the basic cardinality (the cardinality of the atoms) of any collection of infinitely many subsets has the same |N|-1 . In other words, the deep cardinality is based only on the infinitely long and non-complete sequence of infinitely many 1 atoms, where 1 is an atom because it is the non-composed element of the universe of Natural numbers, and no Natural number can be found, if this atom cannot be found. Now, we can clearly and simply understand, that {1,2,3,4,5,…} or {2,4,6,…} are no more then different variations of {1,1,1,1,1,1,…} which is the basis of any one of them, and this is the deep reason why they can be mapped with each other, because in the case of |{1,2,3,…}| and |{2,4,6,…}| we actually compare between {1,1,1,1,1,1,1,…} to itself, where |{1,1,1,1,1,1,1,…}| is no more then |N|-1 . By the way, from this deep understanding |{1,2,3}| NOT= |{2,4,6}| because: |{1, 1,1, 1,1,1}| NOT= |{1,1, 1,1,1,1, 1,1,1,1,1,1}| If we insist that {{1},1} is different from {1} then the reason is first of all based on the difference of the caedinality between |{{1},1}| and |{1}|. So how it can be that (for example) |{ |{1}|, |{{1},1}|, |{{{1},1},1}| }| is equal to |{ |{{1},1}|, |{{{{1},1},1},1}|, |{{{{{{1},1},1},1},1},1}| }| ?

The Cantorian universe, Dedekind's cut (see post #46), Mapping between collections of infinitely many elements, these are the concepts that has no relation to Mathematics, because they are based on fundamental conceptual mistakes of misunderstanding of the Infinity concept, and they are force the impossible on a collection of infinitely many elements, as I clearly show in the case of the Natural numbers and also in post #46. Matt, you did not show to us how 'ALL' and '+1' (the successor) do not contradicting each other. If you cannot give such an example, than you have nothing to say about my argument. Do you understand this?

You are right Dave, but since n+Epsilon is also a Natural number, then if ALL n in N (which means that there is no successor anymore) then n+Epsilon=n+0, which is the wrong, but Epsilon is turned to 0 by forcing the term ALL on a collection of infinitely many elements, and I say that we cannot do it, because Epsilon=1 and not 0. If Epsilon is indeed 1 (and not 0), but we also force ALL term on N (ALL n in N) then we get another impossible result which is: n=n+1. Conclusion: We cannot use the term ALL n in N, or in other words, we cannot force a Universal Quantification on a collection of Infinitely many elements, like set N. It means that the cardinality of N is |N|-1, and the Cantorian transfinite universe does not exist.

Matt, in order to re-search fundamental concept like a Universal Quantification you have to return to the moment before Cantor invented |N| and w. At this stage we have: 4) If n is in N, then n+1 is in N Now we start to analyze this proposition. Here is my analysis: If some Natural number is in N than also its successor is in N Since Natural number+1 is also a natural number, then (Natural number+1)+1 is also in N and so is ((Natural number+1)+1)+1 ... ad infinitum. Let Epsilon = 1 So the general notation to this idea is (n)+Epsilon. If +Epsilon part of the (n)+Epsilon expression is not in N then N is a finite collection, so in order to get an infinite collection +Epsilon exist in N not as (n) but as the next element after (n) (which is written as +Epsilon). If |N| is the cardinality of all n in N and since (n+Epsilon) is also an N member, then n=n+Epsilon, Or in other words n=n+0, which means that if all n in N then there is no successor anymore because N set is completed by an 'all' term. But if Epsilon=0 because of the all term, and N is also infinite, then we are no longer in a model of infinity that is based on infinitely many elements, but we are in a model that is based on an infinitely long non-composed element, which its cardinality cannot be used as an input by any Mathematical tool. So if we want to keep N as a collection of infinitely many elements, then the cardinality of N (where Epsilon=1) cannot be more than |N| - Epsilon. In order to see my two models of infinity, please look at: http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

1) 1 is a Natural number. 2) n = 1 3) N is a set of only Natural numbers. 4) If n is in N' date=' then [i']n[/i]+1 is in N N is at least a well-ordered set, by the above propositions. Now please show us where is the Universal Quantification here?

Dear Matt, I gave a very simple example (the Epsilon>0 argument) of why we cannot use the Universal Quantification with sets like N set. The transfinite universe can be found only if |N| (where N is a well-ordered set) can be found. I clearly show that the best we can get is |N| - Epsilon, which prevents from us to find |N|. So forget about irrelevant examples of non-ordered sets because we are talking here about N, which is a well-ordered set. Now, when you and I sticking only to these initial terms, please show why my |N| - Epsilon does not hold (and my argument is written in a perfect French, unless you prove me wrong in this case). If you are going to use the Axiom of Choice on an unordered N, then it is not relevant, because N was born ordered by Peano's axioms or ZF axiom of infinity. Thank you. Yours, Doron.

Not exactly dear Bloodhound, I can do that if the new definition is still consistent with the rest of my framework. And it is consistent with the rest of my framework, and also gives a solution to Matt Grime's example. This is a good example of how I can correct my framework through a fruitful dialog with professional mathematicians like Matt Grime. I do not expose my ideas to show that I am "the smartest person in the world" but exactly the opposite. I expose my ideas in order to be corrected by other persons, because, in my opinion, this kind of framework can be developed only through a teamwork. If you find that my correction is inconsistent with my framework, then please reply exactly why this correction is inconsistent with my framework. Thank you. Yours, Doron

Matt thank you for this, so I have to do some corrections in my definitions, so: A set is complete only if it does not include infinitely many elements. On the contrary' date=' in my system Uncertainty and Redundancy are first order properties, so ZF or Peano's Axioms define only the 0_Rundancy_AND_0_Uncertainty part of it. You have to understand Matt, that no one can stop fundamental changes in our civilization, and Math as part of it, is not protected from deep and comprehensive changes, specially in our time, where updated knowledge is available to anyone in the modern world. The academic institutions are no longer the one and only one alternative to understand and develop our cognitive and intellectual skills, and you can find scholars around the world that can contribute their talents to the society in unaccepted and non-conventional ways that can lead traditional frameworks to new frontiers. This is something which is inherent to the evolution of complex systems like us, and if some developments are useful to us, then they will be spread and will be used by us. Sometimes, there are ideas which come before their time, and most people do not understand them, but if they are fruitful ideas, they will be understood and be developed by our civilization. One of the most important properties of complex systems like us is our abilities to go beyond our current limitations in order to create/discover now frontiers that will be used by us in the near and maybe far future. This ability to act in both tactical and strategic levels are essential to our own existence as complex, yet simple living creatures. If to speak more to the point, I think that I have new insights about fundamental concepts, which are not exclusively belong to what is agreed as the Language of Mathematics, and even today these concepts are understood differently by different schools of thoughts. My goal is to check the possibility to integrate these points of view to a one comprehensive framework, which is based on organic and non-destructive associations between different points of view. By this goal we maybe can re-define and develop the deep sources of life phenomena itself, where the Language of Mathematics and it logical reasoning is one of the most important tools for this goal. This is a wishful thinking of a lot of people, but nothing is totally protected from fundamental changes in our non-trivial reality. I do not use words like rubbish about anyone's work, but I can explain why I think that Cantor made, in my opinion, conceptual mistakes about basic concepts like Infinity, for example. And I am not just say it, but also explain it in my non-conventional way and also show alternatives to these conceptual mistakes. I do not force anyone to agree with me, but I definitely air my view clearly and do the best I can in order to share my non-conventional ideas with others. The nature of fundamental non-conventional ideas is not to be expressed by the conventional ways, but this is exactly the very nature of fundamental non-conventional ideas, and this is a very hard work to develop a new terminology in order to share these ideas with others. And it is very important to share them with others, because the heart of this thing is to be able to develop a teamwork, which is based on these ideas, and work together in order to develop them, by using an opened mind.