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Fading Transition

  1. Dec 26, 2003 #1
    By ZF set theory we know that {a,a,a,b,b,b,c,c,c} = {a,b,c}

    It means that concepts like redundancy and uncertainy are out of the scpoe of set's concet in its basic form.

    When we allow these concepts to be inherent properties of set's concept, then we enrich our abilities to use set's concept, for example:
    Code (Text):

    <-Redundancy->
        c   c   c  ^<----Uncertainty
        b   b   b  |    b   b
        a   a   a  |    a   a   c       a   b   c
        .   .   .  v    .   .   .       .   .   .
        |   |   |       |   |   |       |   |   |
        |   |   |       |___|_  |       |___|   |
        |   |   |       |       |       |       |
        |___|___|_      |_______|       |_______|
        |               |               |


    Where:

        c   c   c  
        b   b   b  
        a   a   a  
        .   .   .  
        |   |   |  
        |   |   |  = {a XOR b XOR c, a XOR b XOR c, a XOR b XOR c}  
        |   |   |  
        |___|___|_
        |          

        b   b
        a   a   c      
        .   .   .      
        |   |   |      
        |___|_  |  = {a XOR b, a XOR b, c}    
        |       |      
        |_______|      
        |                

        a   b   c
        .   .   .
        |   |   |
        |___|   |  = {a, b, c}
        |       |
        |_______|
        |
     
    I think that any iprovment in set's concept has to include redundancy and uncertainty as inherent proprties of set's concept.

    The above point of view leading me to what I call Complementary logic, which is a fading transition between Boolean logic (0 Xor 1) and non-boolean logic (0 And 1), for example:

    Number 4 is fading transition between multiplication 1*4 and
    addition ((((+1)+1)+1)+1) ,and vice versa.

    These fading can be represented as:
    Code (Text):

    (1*4)              ={1,1,1,1} <------------- Maximum symmetry-degree,
    ((1*2)+1*2)        ={{1,1},1,1}              Minimum information's clarity-degree
    (((+1)+1)+1*2)     ={{{1},1},1,1}            (no uniqueness)
    ((1*2)+(1*2))      ={{1,1},{1,1}}
    (((+1)+1)+(1*2))   ={{{1},1},{1,1}}
    (((+1)+1)+((+1)+1))={{{1},1},{{1},1}}
    ((1*3)+1)          ={{1,1,1},1}
    (((1*2)+1)+1)      ={{{1,1},1},1}
    ((((+1)+1)+1)+1)   ={{{{1},1},1},1} <------ Minimum symmetry-degree,
                                                Maximum information's clarity-degree
                                                (uniqueness)
    ============>>>

                    Uncertainty
      <-Redundancy->^
        3  3  3  3  |          3  3             3  3
        2  2  2  2  |          2  2             2  2
        1  1  1  1  |    1  1  1  1             1  1       1  1  1  1
       {0, 0, 0, 0} V   {0, 0, 0, 0}     {0, 1, 0, 0}     {0, 0, 0, 0}
        .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
        |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
        |  |  |  |       |__|_ |  |       |__|  |  |       |__|_ |__|_
        |  |  |  |       |     |  |       |     |  |       |     |
        |  |  |  |       |     |  |       |     |  |       |     |
        |  |  |  |       |     |  |       |     |  |       |     |
        |__|__|__|_      |_____|__|_      |_____|__|_      |_____|____
        |                |                |                |
        (1*4)            ((1*2)+1*2)      (((+1)+1)+1*2)   ((1*2)+(1*2))
     
     4 =                                  2  2  2
              1  1                        1  1  1          1  1
       {0, 1, 0, 0}     {0, 1, 0, 1}     {0, 0, 0, 3}     {0, 0, 2, 3}
        .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
        |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
        |__|  |__|_      |__|  |__|       |  |  |  |       |__|_ |  |
        |     |          |     |          |  |  |  |       |     |  |
        |     |          |     |          |__|__|_ |       |_____|  |
        |     |          |     |          |        |       |        |
        |_____|____      |_____|____      |________|       |________|
        |                |                |                |
    (((+1)+1)+(1*2)) (((+1)+1)+((+1)+1))  ((1*3)+1)        (((1*2)+1)+1)

       {0, 1, 2, 3}
        .  .  .  .
        |  |  |  |
        |__|  |  |
        |     |  |
        |_____|  |
        |        |
        |________|
        |    
        ((((+1)+1)+1)+1)
     
    Multiplication can be operated only among objects with structural identity .

    Also multiplication is noncommutative, for example:

    2*3 = ( (1,1),(1,1),(1,1) ) or ( ((1),1),((1),1),((1),1) )

    3*2 = ( (1,1,1),(1,1,1) ) or ( ((1,1),1),((1,1),1) ) or ( (((1),1),1),(((1),1),1) )

    More about the above you can find here (the first 9 lines defined by Hurkyl):

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

    More about Complementary logic, you can find here:

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

    http://www.geocities.com/complementarytheory/4BPM.pdf



    Organic
     
    Last edited: Dec 28, 2003
  2. jcsd
  3. Dec 26, 2003 #2
    Adding redundancy into the basic definition of a set doesn't give us any new abilities though. Using sets without redundancy, we can construct sets with redundancy.

    Whether redundancy is an intrinsic property, or one that we add afterwards doesn't have any relevance.
     
  4. Dec 26, 2003 #3
    I think it is relevant because when we use concepts like redundancy, uncertainy, symmetry-degree, information's clarity-degree and complementarity, as fundamental set's properties, it immediately effcts on all things that using set concept as their building-block.
     
    Last edited: Dec 26, 2003
  5. Dec 26, 2003 #4

    Hurkyl

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    Only in the sense that you have to rebuild all of them from scratch.


    And, incidentally, the things you seem to be focused on studying are things that don't use sets as their building block. In particular, the theories of natural numbers and real numbers.
     
  6. Dec 26, 2003 #5
    But using your "enhanced" sets we can also construct the more "traditional" sets. So your enhancments don't affect objects that are usually constructed with the more traditional concept of a set.
     
  7. Dec 26, 2003 #6
    If "traditional" sets are private case of "enhanced" sets , then we can still use use "traditional" sets if we want to, but our abilites of using sets are now beyond "traditional" abilities.
     
  8. Dec 26, 2003 #7

    Hurkyl

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    And we can build enhanced sets using just traditional sets, so we have gained no ability we didn't have before.
     
  9. Dec 26, 2003 #8
    Hi Hurkyl,

    This is one of the new abilities of the new set theory, to conncet between set and number concepts in a very simple way, which its result can be very complex (but not complicated).

    Please read 1 to 7 for more details:

    1) http://www.geocities.com/complementarytheory/GIF.pdf

    2) http://www.geocities.com/complementarytheory/LIM.pdf

    3) http://www.geocities.com/complementarytheory/RiemannsBall.pdf

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

    5) http://www.geocities.com/complementarytheory/CL-CH.pdf

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

    7)http://www.geocities.com/complementarytheory/4BPM.pdf



    Organic
     
    Last edited: Dec 26, 2003
  10. Dec 26, 2003 #9
    I should point out, that traditional constructions such as constructing [itex]\mathbb{N}[/itex] using [itex]0=\varnothing[/itex] and [itex]a+1=a\cup\lbrace a\rbrace[/itex] do not tell us anything about the intrinsic properties of numbers.

    For example, real numbers are not in fact Cauchy sequences of rational numbers. And zero is not in fact the empty set.

    These proofs are only relative consistency proofs.


    So studying set theory doesn't tell us anything about natural numbers. Replacing ZFC set theory with another theory won't change the properties of other number systems.
     
  11. Dec 26, 2003 #10
    Dear master_coda,


    Please read all of my previous post.

    I'll be glad to get your insights and remarks on it.


    Thank you.


    Organic
     
  12. Dec 26, 2003 #11
    I recognize that you've made some effort to include the more rigorous concepts that've been discussed here. But you have to stop explaining your ideas by throwing more and more terminology at them, and making more and more pdfs.

    Look at the posts from pheonixtooth. His ideas are somewhat grandiose and vague, when he states them in words. But he then when he tries to state his ideas mathematically, he does it using standard mathematical terminology.
     
  13. Dec 27, 2003 #12
    what is zero, then? are you saying there is a definitive definition of 0? i know for a fact that the empty set isn't the only definition...

    in other words, you should define what your terms mean.

    for example, 0 can be defined as Ø vs 0 can be defined as the qualia of emptiness. the second definition is more like how you define things like uncertainty and information clarity degree, iow, not mathematically rigorous though intuitively clear. the goal is to obtain a rigorous framework for such ideas and then dispense with the foundation and actually put it to use. to satisfy the mathematician, the foundation needs to exist.

    using words i can find in www.mathworld.com, define things such as redundancy (which i see as being some class function whose domain is the class of all sets and range is the natural numbers), uncertainty, etc. i'd recommend defining one word at a time and posting it to be revised and critiqued by us.
     
  14. Dec 27, 2003 #13
    Dear master_coda,


    Please let me put it this way:

    1) I am not a professional mathematician, therefore I don't know how to express my ideas in the standard mathematical terminology.

    2) Moreover, maybe this is the reason why I can see things differently form well educated professional mathematicians, which, in general, is not bad (in my opinion).

    3) Because I see things not in their common way, I may develop tools, which are not standard, to explore and research my personal point of view on what is called accepted mathematical definitions.

    So, there is some paradox, because sometimes to see things in a non-traditional way, is easier to a non well-educated person, but then it is very hard to put his new point of view in an understandable format.

    Maybe this original point of view is so fundamental, that we have no choice but to change our paradigm about what we call well-defined terms.

    I'll appreciate any cooperation that will result in rigorous mathematical definitions.

    Yours,


    Organic
     
    Last edited: Dec 27, 2003
  15. Dec 27, 2003 #14
    Dear phoenixthoth,


    I think this is a beautiful question, because we can learn a lot of thinks out of nothing.

    I think to give deferent definitions to the same object is a legal mathematical attitude.

    If you don't agree with me, then please tell me why it can't be a legal mathematical attitude.

    Some examples:

    1) |{}| = 0 is a quantitative point of view on the emptiness concept.

    2) x - x = 0 is the "balanced" point of view of some system.

    3) 0^0 = 1 is a structural point of view of the continuum concept
    (Please see: http://www.geocities.com/complementarytheory/CATheory.pdf )

    4) x=0,y=0,z=0,w=0,... is the absolute coordinate of n-dimensional system.

    5) 0 is the south pole of Reimann's ball (the opposite of oo XOR -oo and the middle of -oo AND oo).

    6) (0,oo) is a quantitative point of view of any one of our positive real numbers.

    7) Also (0,oo) can be the open interval of any segment existing between 0 and oo .

    8) 0 is an even number.

    9) 0=false 1=true

    10) '0' = jump on this place in '0.0010...' but remember its power.

    11) By Complementary Logic ( http://www.geocities.com/complementarytheory/CompLogic.pdf )
    0 = |{oo ... nor <--> Emptiness <--> nor <--> Emptiness <--> nor ... oo}|



    Please add more ...
     
    Last edited: Dec 27, 2003
  16. Dec 27, 2003 #15
    Organic, if you want to talk math you have to use at least some standard terminology. That doesn't mean you have to use terms from higher mathematics. Indeed, I wish you would stop using terms from higher math, because you aren't using them correctly and it only makes your ideas more confusing.

    For example, when non-Euclidean geometry was first being explored, it wasn't a popular idea. It went against the philosophy of math that was accepted at the time.

    But that didn't mean that the proponents of non-Euclidean geometry did things in a non-mathematical way. It was still a mathematical way, just a way that no one had ever thought of before.


    What I'm trying to say is that if your ideas are so different from math that you can't use anything from contemporary math to describe them at all, then your ideas probably aren't mathematical. That isn't necessarily a bad thing...there are a lot of good things that aren't math. But there isn't much that mathematicians can do to help you develop your ideas.
     
  17. Dec 27, 2003 #16

    Hurkyl

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    Which of those examples is supposed to be a definition of zero?
     
  18. Dec 27, 2003 #17
    Dear master_coda,

    Please be more specific, take for example: http://www.geocities.com/complementarytheory/GIF.pdf

    Please read all of it and then return and find any thing that you can't understand mathematically, and please write your remarks on it.


    Thank you.


    Organic
     
  19. Dec 27, 2003 #18
    Dear Hukyl,

    This is the all point, in my opinion there cannot be one definition for Zero, and any definition of it depends on the way it is being used through some system.
     
  20. Dec 27, 2003 #19

    Hurkyl

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    As far as I can tell, none of those statements could be construed to be a definition of zero.

    I'm not even sure what point you're trying to make, though; of course the same entity can have multiple definitions. The point is, though, that the definitions are equivalent; so that no matter which definition you use, you can prove all of the others are still true.

    If you have inequivalent definitions, then you're talking about different concepts (such as zero the natural number and the zero vector in R^2)
     
  21. Dec 27, 2003 #20
    Ok Hurkyl,

    Can you please give your definition to Zero?
     
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