Let us say that *all* N members are marked along an infinitely long and non-composed (SOLID) line, where this line has a start but it has no end, for example: |_|_|_|__ ... As can be seen, there must be always an unmarked and non-composed line beyond the domain of N members, if N is a non-finite set. In that case, do we can conclude that the set of *all* N members does not exist, and N must be incomplete (its last member does not exist) in order to be a non-finite collection? Be aware that since _________ ... is a non-composed element, it is not the R set. What to you think?
Do you know what the enlightenment philosophy is? A cynical summary might be that nonsensical crackpots and idiots are debunked and not allowed to foist their completely stupid views on the world. Charlatans should not be tolerated, people who claim to be doing the logical and obivous shuld be able to explain it and not just offer half excuses and rely on ignorance.... You do a perfectly good job of debunking your own theory yourself by writing unintelligible garbage that a child can see through. I believe they're more tolerant round here so they won't ban you like physicsforums or scienceforums did, where you were adequately shown to not be talking about anything sensible. start here: http://www.wsu.edu:8080/~brians/hum_303/enlightenment.html
Locality and Non-Locality in pure Mathematics We reexamine the common notion of the Non-finite and discover a new insight of it. Keywords: Non-finite, Membership, Non-local, Local, Point-like, Line-like. “If set A can be put in a 1-1 correspondence with its proper subset, then set A is a non-finite set” (George Cantor) This definition holds only if the Set concept is equivalent to the Collection concept. Is there a way to define the Non-finite not in terms of a collection/sequence? The answer is yes if we define the non-finite by using an element that its membership cannot be determined by a XOR connective. The definition of locality: Let a be local only if it is in ({a}) XOR out ({}a) of some set. The definition of non-locality: Let a be non-local only if it is in AND out of some set (_{_}). By these definitions we get two different categories of the non-finite, where one category is based on locality and the other category is based on non-locality, and they can be researched by using Riemann's Ball: Please Register or Log in to view the hidden image! As can be seen, no infinitely many local elements (where a local element is equivalent to an intersection in this representation) can be a non-local element (where a non-local element cannot be understood as ‘a one of many …’ element). Only Z* members are represented here, but between each R pair (which are local elements) that exists between each Z* pair, there exists a non-local element. On the other hand, when we are in ∞ non-local state, no R members can be found. When totally ordered along the Real-line (which now it is understood as a non-local element), each R member has an accurate location along it, and when any two arbitrary a,b R members are chosen, there is at least one x member that is greater than the first arbitrary a member and smaller than the second arbitrary b member, notated as a < x < b, and this is a permanent state no matter what scale level is examined. We suggest a new insight about the a < x < b permanent state along the Real-line, which says that between any pair of local a,b R members, there exists a non-composed and non-local element that totally fills the gap between a and b and since it is a non-composed element, no x can be found within its domain. In other words, instead of a divisible domain between any R pairs, there exists a non-local and non-composed element, which fills the gap between them, and if each R member is represented as a point, then this non-local and non-composed element is represented as a segment. If we examine a point and a segment from a logical point of view that determine the Membership concept itself, then a point belongs {.} XOR does not belong {}. to some set, where a segment has the ability to simultaneously exist in AND out of some set _{_}. Furthermore, we get a new insight of the Non-finite, when we understand the Real-line not as a totally ordered collection of points-like (local) elements, but as a non-composed and non-local infinitely long line-like element. The immediate benefit of this approach, is our ability to distinguish between two categories of elements along the Real-line, where one category is local (the points-like R members), and the other category is a collection of non-local segments-like elements. Let us reexamine the elements that exist along the Real-line by using the Place Value Numeration System, and in this case let us reexamine 0.111…[base 2] and 0.222…[base 3]: Please Register or Log in to view the hidden image! Each vertical line in this diagram represents a local element, where each horizontal line represents a non-local element. From this point of view we realize that 0.111…[base 2] and 0.222…[base 3] do not have an accurate place on the Real-line, and they are not equal to 1. Furthermore, 0.111…[base 2] and 0.222…[base 3] are not sequences of numerals that represent the fully local number 1, but they are the two non-local numbers 0.111…[base 2] and 0.222…[base 3], where 0.111…[base 2] < 0.222…[base 3] < 1 if they have common non-finite scale levels. A local number along the Real-line has a non-finite sequence of zeros that is "perpendicular" to the Real-line (as can be seen in the diagram), where a non-local number does not have this property. Each non-local [base n>1] number is based on a unique invariant proportion, which gives us a new insight about the non-finite collection that is: Any non-finite collection is incomplete, when compared to the non-composed and non-local Real-line, because no non-finite collection can reach the completeness of a one infinitely long and non-composed line-like element. A finite collection is not effected by this incompleteness, because it does not have the tendency to reach the non-composed Real-line’s completeness. In other words, a finite collection has an accurate cardinal, where a non-finite collection does not have an accurate cardinal because it is an incomplete mathematical element. From the new insight of the Non-finite, Cantor's second diagonal argument is understood as a proof of the incompleteness of a non-finite collection. For example, let us say that we have a non-finite collection which is composed of unique non-finite collections, where each non-finite collection has a unique order of empty and non-empty sets: Code: { { [B][COLOR="Blue"]{ }[/COLOR][/B],{ },{ },{ },{ },... } { {#},[B][COLOR="blue"]{ }[/COLOR][/B],{ },{#},{ },... } { { },{#},[B][COLOR="blue"]{#}[/COLOR][/B],{ },{ },... } { {#},{#},{ },[B][COLOR="blue"]{#}[/COLOR][/B],{#},... } { { },{ },{#},{ },[B][COLOR="blue"]{ }[/COLOR][/B],... } ... } We can define another unique non-finite collection, which is actually the non-finite diagonal opposite collection { {#},{#},{ },{ },{#},... } that has to be added to our non-finite collection, etc., etc. … ad infinitum. From this point of view the identity-map of a non-finite collection, which is composed by unique non-finite collections, does not exist because its accurate cardinality cannot be satisfied. Let us take for example the non-finite collection of the Natural numbers, but instead of the standard way, where each member is a unique element, we shall use a non-finite multiset {1,1,1,1,1,…}, where +1 (the successor) is the permanent next element {1,1,1,1,1,…} +1, the existence of which was shown by Cantor’s second diagonal method. Since the successor is permanently out of the multiset, the right notation of the Natural numbers' cardinality is |N|-successor. Let @ be |N|-successor. If A = @ and B = @-2^@, then A > B by 2^@, where both A and B are collections of infinitely many elements. By using the new insight of the Non-finite, we have both non-finite collections and a unique arithmetic between non-finite collections, which its results are always non-finite collections. These results are finer than the results of the Cantorian transfinite universe, for example: By Cantor aleph0+1 = aleph0 , by the new insight @+1 > @. By Cantor aleph0 < 2^aleph0, by the new insight @ < 2^@. By Cantor aleph0-2^aleph0 < aleph0 is undefined, by the new insight @-2^@ < @. By Cantor 3^aleph0 = 2^aleph0 > aleph0 and aleph0-1 is problematic. By the new insight 3^@ > 2^@ > @ > @-1 etc. We know that the Cantorian transfinite universe is considered as an actual infinity, where the limiting "process" is considered as a potential infinity, but by the new insight of the Non-finite we realize that since no non-finite collection can reach the completeness of a one non-composed and non-local line-like element, then only the non-composed and non-local line-like element can be considered as an actual infinity, where any non-finite collection is no more than a potential infinity. These collections can have infinitely many degrees of potential infinities, which no one of them reaches the non-composed and non-local line-like actual infinity. This new approach of the Non-finite is not counter intuitive as the Cantorian transfinite universe, because it clearly separates between the Continuum (which is not less than a non-composed and non-local finitely/infinitely long line-like element that is not considered as a "one of many" element) and the Discreteness (which is no more than a collection of finitely/infinitely many mixed or unmixed local points-like and/or non-local lines-like elements). From a Cantorian point of view, cardinals are commutative (1+aleph0 = aleph0+1) and ordinals are not (1+ω not= ω+1). By using the new insight of the Non-finite, both cardinals and ordinals are commutative because of the inherent incompleteness of any non-finite collection. In other words, @ is used for both ordered/unordered non-finite collections and x+@ = @+x in both cases. By the common notion of the Limit concept elements get closer to the limit, but this is not the case when we use the proportion concept, for example, let us say that we have a sports car (where the name of the back wheels is "epsilon" and the name of the front wheels is "delta") and our mission is to cross the zero point of X,Y-axis with both "delta" and "epsilon" wheels: Please Register or Log in to view the hidden image! We are seated in the car and trying to reach point zero. We realize that we are not getting any closer to the zero point, and the reason is: the faster we drive, the smaller we become (as can be seen in the picture above) and we have here a Lorentz-like transformation that has an invariant proportion along non-finite scale levels. According to this invariant proportion, nothing gets closer to the Zero point. Strictly speaking, our mission cannot be completed, because between the "delta" front wheels and the zero point, there is a non-local and non-composed line-like element that cannot be eliminated by a non-finite collection/sequence of local elements. In the same manner R set is an incomplete collection. Actually we reach point zero, if and only if we don’t have a car anymore but a single point, which is a phase transition that cuts the infinitely many smaller states (smaller cars), and we don’t have an incomplete collection over infinitely many scales (infinitely many cars), but a finite collection of many scales (a finite collection of sports cars). Summery: By clearly distinguishing between the actual infinity (which is not less than a non-composed and non-local line-like element) and the potential infinity (which is no more than a collection/sequence of infinitely many elements), we define a new arithmetic of the Non-finite, which its results are potential infinities that are not counter intuitive, because of the simple insight that no non-finite collection/sequence can reach the completeness of a non-composed and non-local infinity. Let us return once again to Cantor’s definition of the Non-finite: “If set A can be put in a 1-1 correspondence with its proper subset, then set A is a non-finite set” . By clearly distinguishing between the Local and the Non-local, we can add that if set A is a non-finite collection/sequence, then it is an incomplete mathematical element, by definition. ------------------------------------------------------------ References: Tall David: Natural and Formal Infinities, Mathematics Education Research Centre, Institute of Education, University of Warwick ( http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001p-esm-infinity.pdf ). Tall David , Tirosh Dina: Infinity - the never ending struggle, published in Educational Studies in Mathematics 48 (2&3), 199-238 ( http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001o-esm-intro-tirosh.pdf ). Joseph W. Dauben: George Cantor and the battle for transfinite set theory, Department of history, University of New-York (http://www.acmsonline.org/Dauben-Cantor.pdf ). ------------------------------------------------------------ If a 1-1 correspondence is measured between two sets, their sizes are related to each other by a variant or an invariant proportion. For example, let S be a non-finite collection of circles, where each diameter has a unique length. In that case Pi is a constant proportion between circle's circumference and diameter, no matter how many circles are in the collection. In other words, circumference/diameter does not depend on ordinality or cardinality. The same invariant proportion can be found between some non-finite set and its proper subset, for example: Let us check the 1-1 mapping between A={1,2,3,4,5,6,…} and B={ 2,4,6,…} non-finite sets. We know that they have the same cardinality even if B is a proper subset of A, but now let us check if there is some proportion between A and B that does not depend on their ordinality or cardinality, which is equivalent to the circumference/diameter case, for example: By this mapping we can clearly see that B is a proper subset of A: 1 <--> 2 <--> 2 3 <--> 4 <--> 4 5 <--> 6 <--> 6 … But by this mapping we realize that B and A have the same cardinality: 1 <--> 2 2 <--> 4 3 <--> 6 4 <--> 8 5 <--> 10 6 <--> 12 … Now let us use the invariant proportion that exists between A and B (which is equivalent to the circumference/diameter case): Let |Aip| be the invariant proportion value of A. Let |Bip| be the invariant proportion value of B. 1 <--> 2 2 <--> 4 3 <--> 6 4 <--> … 5 <--> … 6 <--> … … In that case: |Aip| > |Bip| |Bip|*2 = |Aip| |Aip|/2 = |Bip| In other words, if some member is written in B set side, then it must be written in A set side, because if a 1-1 correspondence is measured between two sets, their sizes are related to each other by a variant or an invariant proportion, and since 12 was written in B side in this particular example, then it must be written also in A set side, and as a result we get: 1 <--> 2 2 <--> 4 3 <--> 6 4 <--> 8 5 <--> 10 6 <--> 12 7 <--> … 8 <--> … 9 <--> … 10 <--> … 11 <--> … 12 <--> … … etc. … etc. ad infinitum. and we can clearly see that: |Aip| > |Bip| |Bip|*2 = |Aip| |Aip|/2 = |Bip| even if we deal with non-finite sets that their last members do not exist (as they do exist in finite sets). From this new notion we suddenly understand that |A| = |B| standard result is based on a non-accurate understanding of the Non-finite collection because |A| = |B| result is based on a partial picture of the 1-1 mapping, where this partial picture is: 1 <--> 2 2 <--> 4 3 <--> 6 4 <--> 8 5 <--> 10 6 <--> 12 … which gives us the illusion that |A| = |B|, and the full picture of the 1-1 mapping is actually: 1 <--> 2 2 <--> 4 3 <--> 6 4 <--> 8 5 <--> 10 6 <--> 12 7 <--> … 8 <--> … 9 <--> … 10 <--> … 11 <--> … 12 <--> … … etc. … etc. ad infinitum. It is not some artificial extension of the finite case, because A and B are non-finite sets as well (no one of them has a last element), and now we can conclude that the proportion concept is deeper and finer than the standard notion about the 1-1 mapping technique. Furthermore, the proportion's new notion is consistent with the notion which says that any non-finite collection/sequence is no more than a potential-infinity, when we distinguish between locality and non-locality as two fundamental mathematical definitions.
