Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Can |R| be uncountable but not a Continuum?

  1. Aug 5, 2003 #1
    If we take any R member (with an infinitely many digits) which is being used as a set's boundary, we find this:

    A member with an infinitely many digits never reaches 0 by definition.

    It means that there is always an unclosed interval
    for some line's segment {0_____R member with infinitely many digits}.

    More than that, any interval from X to ~X can be opend or closed
    only by a quantum-like leap, and no number which is not X, can close it.

    No infinitely many points can close this interval because any point has exactly 0 size, so the interval can be closed only by an element ~=0, and this element can't be but a quantum-like leaps of continuous smooth lines.

    In the middle of any qauntum leap there are exactly 0 points.

    It means that there is a XOR ratio between LINES to POINTS.

    XOR ratio between LINES to POINTS
    -------------------------------------
    0(LINE) 0(POINT) -> 0-(No information) -> no conclusion.
    0(LINE) 1(POINT) -> 1-(Clear Particle-like information) -> conclusions on points.
    1(LINE) 0(POINT) -> 1-(Clear Wave-like information) -> conclusions on lines.
    1(LINE) 1(POINT) -> 0-(No clear information) -> no conclusion.

    There is exactly a one and only one way to cover a ___ by a .

    We have to stretch the point until it overpals the line.

    But than you have no longer a point and a line, but two lines.

    Through topology there can be another solution.

    If a point and a stretched_point are the same through topology eyes, than,
    by definition, there cannot be any points in the middle of a
    stretched_point, and we have a quantum leap element.

    More than that, every number can exists(=set's centent ~=0) only by its relation to {}=0, so through this point of view, no number is a point but some distance from zero, and any distance is a quantum leap between 0(=set's content does not exist) to ~0(=set's content exists).

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

    An axiom:

    In the middle of any qauntum leap there are exectly 0 points.


    Theorem:

    In the middle of a continuous line there are exactly 0 points.


    Proof:

    {} = 0 = Set's content does not exist.

    ~0 = Set's content exists.

    Any transition from 0(= Set's content does not exist)
    to ~0(= Set's content exists) can not be but a quantum leap.

    Anything that cen exactly overlap(close) a quantum leap = quantum leap.

    A quantum leap can be exactly overlapped(closed) only by
    a continuous line.

    Now we can conclude that some continuous line = some quantum leap.

    Therefore, In the middle of a continuous line there are exactly 0 points.

    QED.

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


    It means that there are exactly 0 points in the continuum, and we can define a new set's content, which is {__} = C = Continuum.


    Now we have 3 structural types of set's contents:

    {} = The Emptiness = 0 = Content does not exist.

    Let power 0 be the simplest level of existence of some set's content.

    {__} = The Continuum = An indivisible element = 0^0 = Content exists (from the structural point of view, there are exactly 0 elements in the Continuum, so its base value = 0, But because it exists (unlike the emptiness) its full notation = 0^0).

    (You can break an infinitely long continuum infinitely many times, but always you will find an invariant structural state of {.___.} which is a connector between any two break points.)

    {...} = The Discreteness = Infinitely many elements = [oo]^0 = Content exists.

    Any transformation from {} to {__} or {...} is based on phase transition, because we have 0(=does not exist) to 1(=exists) transition.

    So, from a structural point of view, we have a quantum-like leap.


    Now, let us explore the two basic structural types that exist.

    0^0 = [oo]^0 = 1 and we can see that we can't distinguish between
    Continuum and Discreteness by their Quantity property.

    But by their Structural property {__} ~= {...} .

    From the above we can learn that the Structure concept have
    more information than the Quantity concept in Math language.

    Any element that is under a definition like "infinitely many ..." can not be but a member of {...}, which is the structure of the Discreteness concept.

    Through my point of view, the Continuum is not a container but a connector between any two points {.___.} and you can find this state in any scale that you choose.

    Through this approach you don't have any contradiction between the Discreteness and the Continuum concepts, because any point is not in the Continuum, but an event that breaks the Continuum.

    The Continuum does not exist in this event (because of a XOR ratio beween any line to any point), but any two events can be connected by a Continuum, for example the end of a line is an event that breaks the line and it turns to a nothingness, so from one side we have the Continuum, from the other side we have the nothingness, and between them we have a break point, that can be connected to another break point that may exist on the other side of the continuous line.

    Another way to look at these concepts is:

    Let a Continuum be an infinitely long X-axis.

    Let a point be any Y(=0)-axis on the X-axis.

    So what we get is an indivisible X-axis and infinitely many Y(=0)-axises points on (not in) the X-axis.

    Through this point of view we get an indivisible X-axis, as a connector (not a container) between any two Y(=0)-axises events.



    Therefore we can conclude that |R| < C = Continuum .



    There are 4 important conclusions from the above:

    Let n = 3 = 1+1+1

    A) 0^0 = Contiunuous 1

    B) 1^0 = 1 Connector

    C) n/1^0 = n Connectors (.__.__.__. = 1 1 1)

    D) n/0^0 = Continuous n ( ________ = 3)

    Through this approach, each natural number is the associations (AND ratio) between its continuous side (Continuous n) to its discrete side (n Connectors).


    More detailed information about the structures that you can get from
    those associations, you can find here:

    http://www.geocities.com/complementarytheory/CATpage.html



    Please tell me what do you think ?



    Yours,

    Doron
     
    Last edited by a moderator: Aug 11, 2003
  2. jcsd
  3. Aug 5, 2003 #2

    selfAdjoint

    User Avatar
    Staff Emeritus
    Gold Member
    Dearly Missed

    We don't have to "know the exact value" of pi to use it. Likewise square root of 2. We can get either of them to any degree of accuracy we require.

    We understand limits, and convergence, and density, and completness without being able to fully specify every single term.

    Note that we can't specify all the integers either. There isn't any largest one, and when you think you have named them all, there are more.

    So math and continuum don't depend on human intuition but on proofs.
     
  4. Aug 6, 2003 #3
    Hi selfAdjoint,

    You ignored the second part of my argument which is:

    If we take any R member which is being used as a set's boundary, and has an infinite magnitude, we find this:

    A member with an infinite magnitude never reaches 0 by definition.

    It means that there is always an unclosed gap
    for some line's segment {0_____R member}.

    Therefore we can conclude that |R| < C .


    I'll be glad to now what do you think.
     
    Last edited by a moderator: Aug 6, 2003
  5. Aug 6, 2003 #4

    Integral

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    I hope you dont mind if I jump in!


    Such a number does not exist. All real numbers are finite in magnetude. Only when the Real number system is extended to include infinity is there a non finite element.

    I am not sure what your point is. Of course 0<> infinity.

    What has infinity got to do with an interval containing 0.
    How have you proven your claim? I think you are a bit confused about what the the term "magnetude" means. Are you refering, not to the magnetude but to the length of the decimal expansion? If so then consider this. Let us consider the number .1n. Now while you would be correct in saying that it is never = 0 (which may be your point). Consider that for any number you name,say &delta; greater then 0 I can find an n such that .1n < &delta;

    so now you give me a end point of you " unclosed gap" I I will find a number inside your gap, therefore it is not much of a gap is it? There is no gap around zero or any other number for that matter. If such a gap existed there would be a smalles number greater then zero such a number does not exist.

    There is no way you can draw that conclusion from the information you have provided.
    You have my thoughts. I will add that if you are truly interested in the Real number system you need to find a good text on Real Analysis. I have studied from books by Rudin, Royden and Kolgormoff (sp)

    Rudin if very good.
     
    Last edited: Aug 6, 2003
  6. Aug 6, 2003 #5
    Hi Integral,


    Please read my non-standard point of view:

    You can not explore a continuous line by a point, because there is a XOR ratio between tham.

    A point can not be but an exact place on the continuous line, but any continuous line is the opposite of being in an exact place (like wave/particle ratio in Quantum Mechanics).

    So when you speak in terms of an exact place, you can't conclude something on a continuous line, which has (by definition) no exect place.

    Any distance is a ratio between points, and there always exists a not exect place (a continuous line) between any R member and 0.

    A not exect place = the smallest number does not exist.

    Because no R member can reaches 0, we can conclude that |R| < C .

    (By the way, you can take the digits of any infinitaly long R member, and build a set
    of an infinite magnitude)
     
    Last edited by a moderator: Aug 6, 2003
  7. Aug 6, 2003 #6

    Integral

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    What do you mean by XOR, to me this says eXclusive OR, which is an OR function that is false if both inputs are true. Not sure what this has to do with anything.

    How does a continus line disallow the existance of a fixed point? The real numbers are dense, this means that every point can be assigned a decimal representation.
    Sure you can, every point is a precise point. nothing moves, how could it be anything else.
    No, intervals are a difference between points, no ratio involved. The interval (-(.1n),.1n contains only 0 for all n, there is no ratio involved.
    You have neither proven or even demonstrated that fact. If you leave out all of the transentental number the cardinality of R is still C.
    The existance of a hole or even an infinite number of holes does not effect the cardinality of R. Perhaps you need to get a better understanding of this concept.
     
  8. Aug 6, 2003 #7
    By XOR ratio I mean that you can not use term A concluse something on term B if there is a
    XOR ratio between them.

    If you speak in term A, which is, for example, an exect place (a point), you can not conclude somthing on term B, which is, for example, a continuous line (no exec place).

    (By the way, you can take the digits of any infinitaly long R member, and build a set
    of an infinite magnitude)



    [l]A more general point of view of mine[/l]

    There is no way to associate between a discrete set {…} and a Continuous set {___} by means of the Quantity concept, without forcing the Continuum concept to be expressed in terms of the Discreteness concept, and what the Common Math does is:

    {.<-- . -->.} = Extrapolation over scales = elements with finite magnitude = N, Z.

    {.--> . <--.} = Interpolation over scales = elements with finite or infinite magnitude, where those with an infinite magnitude are built on repetitions over scales = Q.

    {. --> . <-- .} = Interpolation over scales = elements with infinite magnitude without repetitions over scales = R.

    But the infinite { . --> . <-- . } magnitude never reaches the {___} state, and this is an axiomatic fact that no mathematical manipulation (which is based on the quantity concept) can change.

    For example, please show me how we can use the bijection method between {...} and {__} ?

    We find that |R| > |Q| by using the bijection method, and for this, the strucrute of each compared elemant in both sides
    MUST BE {. <-- . --> .} or {. --> . <-- .}, so we are closed under {...} and can't conclude that |R| = {__} = Continuum.

    All we can conclude is that the magnitude of the infinitely many elements of |R| is bigger than the magnitude of the infinitely many elements of |Q|.


    Here are again my non standard basic definitions:

    If we use the idea of sets and look at their contents from
    a structural point of view, we can find this:

    {} = The Emptiness = 0 = Content does not exist.

    Let power 0 be the simplest level of existence of some set's content.

    {__} = The Continuum = An infinitely long indivisible element = 0^0 = Content exists (from the structural point of view, there are exactly 0 elements in the Continuum, so its base value = 0, But because it exists (unlike the emptiness) its full notation = 0^0).

    (You can break an infinitely long continuum infinitely many times, but always you will find an invariant structural state of {.___.} which is a connector between any two break points.)

    {...} = The Discreteness = Infinitely many elements = infi^0 = Content exists.

    Any transformation from {} to {__} or {...} is based on phase transition, because we have 0(=does not exist) to 1(=exists) transition.

    So, from a structural point of view, we have a quantum-like leap.


    Now, let us explore the two basic structural types that exist.

    0^0 = infi^0 = 1 and we can see that we can't distinguish between
    Continuum and Discreteness by their Quantity property.

    But by their Structural property {__} ~= {...} .

    From the above we can learn that the Structure concept have
    more information than the Quantity concept in Math language.

    Any element (with finite or infinite magnitude) that is under a definition like "infinitely many ..." can not be but a member of {...}, which is the structure of the Discreteness concept.
     
    Last edited by a moderator: Aug 6, 2003
  9. Aug 6, 2003 #8

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    "Because we can't know the exact value of any R member, we can't use any R member as a common boundary between any two sets."


    Perhaps it would be better to say "I" rather than "we". I suspect that most people know the exact value of "1", "2", "1/3". In any reasonable sense of "know the exact value", many people know the exact value of "pi" and "square root of 2". Are you confusing "know the exact value" with "write out the complete decimal expansion"?

    "Please read my non-standard point of view:"

    The moment you say that, you are no longer talking about mathematics and, unless you explain how your "non-standard point of view" is connected to the "standard view", you are not saying anything about mathematics.

    "You can not explore a continuous line by a point, because there is a XOR ratio between tham."

    Okay, so you are using "XOR" is a "non-standard" way, again without defining it. Do you EXPECT other people to read your mind?

    "A point can not be but an exact place on the continuous line, but any continuous line is the opposite of being in an exact place (like wave/particle ratio in Quantum Mechanics)."

    Okay, but since a "point" is NOT a "continuous line" this doesn't say a point can't be ON a continuous line.

    "So when you speak in terms of an exact place, you can't conclude something on a continuous line, which has (by definition) no exect place."

    Whose definition? A good secondary school course in geometry ought to clear that up for you.
     
  10. Aug 6, 2003 #9
    By XOR ratio I mean that you can not use term A to conclude something on term B if there is a XOR ratio between them.

    If you speak in term A, which is, for example, an exect place (a point), you can not conclude somthing on term B, which is, for example, a continuous line (no exect place).

    (By the way, you can take the digits of any infinitaly long R member, and build a set of an infinite magnitude)


    Through my point of view, the Continuum is not a container but a connector between any two points {.___.} and you can find this state in any scale that you choose.

    Through this approach you don't have any contradiction between the Discreteness and the Continuum concepts, because any point is not in the Continuum, but an event that breaks the Continuum.

    The Continuum does not exist in this event, but any two events can be connected by a Continuum, for example the end of a line is an event that breaks the line and it turns to a nothingness, so from one side we have the Continuum, from the other side we have the nothingness, and between them we have a break point, that can be connected to another break point that may exist on the other side of the continuous line.

    Another way to look at these concepts is:

    Let a Continuum be an infinitely long X-axis.

    Let a point be any Y(=0)-axis on the X-axis.

    So what we get is an indivisible X-axis and infinitely many Y(=0)-axises points on (not in) the X-axis.

    Through this point of view we get an indivisible X-axis, as a connector (not a container) between any two Y(=0)-axises events.
     
    Last edited by a moderator: Aug 6, 2003
  11. Aug 6, 2003 #10

    Integral

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    AS long as you are speaking a language which only you understand there is no point in holding a discussion. I gave you the meaning of XOR but you insist on definining it for your own use.

    There used to be a fellow posting here who claimed that he had a rational definintion of Pi (who all remembers Donde?) We tried, with no sucess to teach him that some words had precisly defined meaning among scientists, if you use these words incorrectly you only create confusion.

    It is clear to me that you have adatpted much terminology from math but simply do not understand the formal meaning of these words. As long as you are misusing these terms there can be no possiblity of meaningful communication. You need to develope your OWN set of words to describe your concepts and not borrow from math.
     
  12. Aug 6, 2003 #11
    XOR
    ----
    00 -> 0
    01 -> 1
    10 -> 1
    11 -> 0

    XOR
    ----
    0(LINE) 0(POINT) -> 0-(No information)
    0(LINE) 1(POINT) -> 1-(Exact_place information)
    1(LINE) 0(POINT) -> 1-(No_exact_place information)
    1(LINE) 1(POINT) -> 0-(No clear information)
     
    Last edited by a moderator: Aug 6, 2003
  13. Aug 6, 2003 #12

    selfAdjoint

    User Avatar
    Staff Emeritus
    Gold Member
    Dearly Missed

    Since you are depending on boolean logic, XOR or Exclusive OR being a boolean operator, you should know that there is an axiomatic development of the real number system from set theory. How do you deal with that?

    Note that boolean operators can't generate new facts outside of logic. Your argument won't be good because you use XOR, but because your hypotheses are good (if they are).
     
  14. Aug 6, 2003 #13
    Hi Selfadjoint,

    The ratio between A LINE and A POINT is a XOR boolean ratio, therefore any exploration of some line by points can find information on points, and can't conclude anything about the line.

    Some analogy is the XOR ratio between a particle(point) and a wave(line) in Quantum Mechanics.


    |R|=2^aleph0 > |Q| and we can conclude that |R| is uncoutable, but the jubject in this thread is about C and |R|.
     
    Last edited by a moderator: Aug 6, 2003
  15. Aug 6, 2003 #14

    selfAdjoint

    User Avatar
    Staff Emeritus
    Gold Member
    Dearly Missed

    So you are just asserting this? Baire theory and measure theory are false just because you have this XOR idea?

    You have to know that the intuition of others does not match yours. For example the Dedekind cut has seemed intuitive to many:

    Let A and B be two sets of points, with every point of A to the right of every point of B. Then there is a point with these properties.
    1) No point of B is to the right of it.
    2) No point of A is to the left of it.

    For example let B be the set of positive numbers whose square is less than 2. And let A be the numbers whose square is greater or equal to 2. Clearly these to sets of numbers (regarded as points on the line) satisfy the hypothesis. And the point, or number guaranteed by the conclusion is precisely the square root of 2.
     
  16. Aug 6, 2003 #15

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    I presume you meant to say "transcendental"?

    Actually, if you leave out all of the transcendental numbers, the cardinality of what's left is no longer c, it is aleph_0.

    Every algebraic number (aka nontranscendental real number) is a root of a polynomial with integral coefficients. There are countably many such polynomials, and each polynomial only has countably many roots, so there can be at most countable * countable = countable algebraic numbers.
     
  17. Aug 7, 2003 #16
    Hi selfAdjoint,

    I updated my original message, which is in the top of this thread.

    Please read it and tell me what do you think.

    Thank you.

    Doron
     
  18. Aug 7, 2003 #17

    Integral

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Hurkyl,
    Opps..Thanks for the correction. You can delete any countable set from the reals and not lose the cardinality, so there must be a uncoundable number of transcendental's. My bad!


    Doron,
    By distinquishing between an infinite number of digits and an infinite magnetude you have taken the first step on the path of understanding the real number system. Now as you have been told none of your claims constitutes a proof. You have shown that the difference between any real number and zero is greater then zero. What you have failed to prove is that the gap between the number and zero is empty, and that is what you need to prove to get to your conculusion. Read, again, my argument that I can find a number in the middle of any "gap" you can find. Not much of a gap, now is it.
     
  19. Aug 7, 2003 #18
    Hi Intergal,

    For every point that you give me, I give you a gap that you have to close.

    You will find this {X____~X} as a fractalic invariant state (a quantum leap) in any scale that you choose forever.

    Therefore between any two members in set R there is {.__.} a quantum leap (with exectly 0 elemets in it) and we can conclude that |R| < C.

    More than that, if R={___} and Q={...} than please show me how do you
    check if there is a 1 to 1 correspondence (a bijection) between them,
    without forcing R {___} to be expressed in terms of {...} ?

    If it can't be done, I think it means that any mathematical research
    that is based on a bijection, is closed under the discreteness concept.
     
    Last edited by a moderator: Aug 7, 2003
  20. Aug 7, 2003 #19

    Integral

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    You have not proven that, I can show by counter example, (ie point in the middle of said gap) that the gap does not exist. Simple because the difference between 2 numbers is greater then zero does not imply a gap in the real number line. That is your only argument and it does not hold water.
     
  21. Aug 7, 2003 #20
    I talking about an axiomatic state.


    How do you come to the conclusion that you have 2 numbers, if there is no gap between them ?

    And if so, why just 2 numbers in a point ? Why not infinitely many numbers in a point ?

    If R={___} and Q={...} than please show me how do you check if there is a 1 to 1 correspondence (a bijection) between them, without forcing R {___} to be expressed in terms of {...} ?

    If it can't be done, I think it means that any mathematical research
    that is based on a bijection, is closed under the discreteness concept.
     
    Last edited by a moderator: Aug 7, 2003
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Can |R| be uncountable but not a Continuum?
  1. Continuum Hypothesis (Replies: 2)

  2. Uncountable Sum (Replies: 12)

Loading...