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Countable list

  1. Dec 6, 2003 #1
    Let us check these lists.


    P(2) = {{},{0},{1},{0,1}} = 2^2 = 4

    and also can be represented as:

    00
    01
    10
    11


    P(3) = {{},{0},{1},{2},{0,1},{0,2},{1,2},{0,1,2}} = 2^3 = 8

    and also can be represented as:

    000
    001
    010
    011
    100
    101
    110
    111


    ...


    P(aleph0) = 2^aleph0 = |R|

    and also can be represented as:

    aleph0
    ^
    |
    |
    0(...--> aleph0)0
    0(...--> aleph0)1
    0(...--> aleph0)0
    0(...--> aleph0)1
    1(...--> aleph0)0
    1(...--> aleph0)1
    1(...--> aleph0)0
    1(...--> aleph0)1
    |
    |
    v
    aleph0

    We can find a bijection between N and R by this way:

    Code (Text):

    aleph0
    ^
    |
    |
    7 <--> 0.0(...--> aleph0)0
    5 <--> 0.0(...--> aleph0)1
    3 <--> 0.0(...--> aleph0)0
    1 <--> 0.0(...--> aleph0)1
    2 <--> 0.1(...--> aleph0)0
    4 <--> 0.1(...--> aleph0)1
    6 <--> 0.1(...--> aleph0)0
    8 <--> 0.1(...--> aleph0)1
    |
    |
    v
    aleph0
     
    Therefore 2^aleph0 = aleph0
     
    Last edited: Dec 6, 2003
  2. jcsd
  3. Dec 6, 2003 #2
    But there is another thing that i have found.

    We still be able to use Cantor's function and get some number which is not in the list.

    For example:

    aleph0
    ^
    |
    |
    0.0,x23,x33,x43,... ,0
    0.0,x21,x31,x41,... ,1
    0.1,x22,x32,x42,... ,0
    0.1,x24,x34,x44,... ,1
    |
    |
    v
    aleph0

    Our new result, which is not in the list, is the opposite of 0.0,x22,x33,x44,...

    So in this point we are maybe in a logical disaster.

    I think the sulotion is to use the idea of the open interval on a single number.

    For example:

    [0.x1,x2,x3,x4,... ,1)

    More information you can find here:

    http://www.geocities.com/complementarytheory/RiemannsBall.pdf
     
    Last edited: Dec 7, 2003
  4. Dec 7, 2003 #3

    MathematicalPhysicist

    User Avatar
    Gold Member

    im not an expert in set theory but doesnt {}={0}=0 and so it would be useless to write it in different ways?
     
  5. Dec 7, 2003 #4
    Hi loop quantum gravity,

    {{}} = {0}
     
  6. Dec 7, 2003 #5

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    [itex]\{ \} \neq \{ 0 \}[/itex]. In general, [itex]0 \neq \{ \}[/itex], but some models (including the one typically used in set theory) do make that identification.


    Organic: you're missing a very important fact about the ordering structure of the integers:

    If a sequence of integers has a first element and a last element, then the sequence is finite.

    By definition, the digits in a decimal expansion are indexed by integers...

    This means that:

    (which I'm assuming is supposed to be some real number written in b-ary1 notation) cannot have an infinite number of digits.

    Similarly

    cannot be a sequence2 of binary digits.

    (by this notation I'm assuming you mean that there are countably infinite numbers between the first 0 and the last 0)

    Also,

    cannot be a binary expansion of a real number.

    (again I'm assuming that this notation means there are countably infinite numbers between the first and last digit)

    Furthermore

    The notation [itex]P(A)[/itex] requires [itex]A[/itex] to be a set. You should be saying [itex]P(\{ 0, 1\})[/itex] instead of [itex]P(2)[/itex], and you should definitely be saying [itex]P(\mathbb{N})[/itex] instead of [itex]P(\aleph_0)[/itex]. (That is, assuming I understand correctly what you mean)

    Moreover

    the way you are listing the elements of the powerset of a finite set does not generalize to an infinite set. In particular, every element of the list:

    Code (Text):

    ...0000
    ...0001
    ...0010
    ...0011
    ...0100
    ...
     
    has a finite number of ones. This is only a representation of the finite subsets of [itex]\mathbb{N}[/itex].


    Additionally

    This is NOT the idea of an open interval. You should tell us what this means or stop using it.


    footnotes:

    1: b-ary means base-b representation. e.g. binary is 2-ary, decimal is 10-ary

    2: Unless otherwise specified, a sequence is indexed by some segment the natural numbers.
     
    Last edited: Dec 7, 2003
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