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Thread: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

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    Default Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

    Different levels of infinity, another Cantor axiom to argue to death...

    Cheers
    L-zr

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    Default Re: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

    Quote Originally Posted by Lazer View Post
    Different levels of infinity, another Cantor axiom to argue to death...

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    Axiom means "self-evident" or "worthy". There is nothing self-evident about any of Cantor's idiotic ideas, much less worthy.
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    Default Re: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

    Quote Originally Posted by Lazer View Post
    Different levels of infinity, another Cantor axiom to argue to death...
    Pre-dates Cantor, not really an axiom
    Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?
    Seems pretty obvious that n --> 2n-1 is an exact pairing of the first set to the second set, every member of the first set is paired with exactly one member of the second set, no duplications, no elements left out.

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    Default Re: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?
















    koeddyrbh:



    invisitesimal c. postulate



    invisibility action by






    because



    and

    0.999... = 9/10 + 9/100 + 9/1000 + ... =



    0.999...9 (n 9s) = 0.999...
    Look:



    Comprehend ?











    2.7
    2.97
    2.997

    2.999...997

    Let Sn denote the nth partial sum. Clearly Sn = 1 - 1/10ⁿ
    Then for every ε > 0 we have an N = log10 (1/ε) such that:
    |Sn - 1| = |1/10ⁿ| < ε for all n > N.
    Therefore 0.999... = 1

    Let Sn denote the nth partial sum. Clearly Sn = 1 - 1/10ⁿ
    Then for every ε > 0 we have an N = log10 (1/ε) such that:
    |Sn - 1| = |1/10ⁿ| < ε for n=∞ (all n > N)
    Therefore 0.999... = 1

    Let S(∞) denote the ∞th partial sum. Clearly S(∞) = 1 - 1/10^∞
    Then for every ε > 0 we have an N = log10 (1/ε) such that:
    |S(∞) - 1| = |1/10^∞| < ε for n=∞ (all n > N)
    Therefore 0.999... = 1

    Let S(∞) denote the ∞th partial sum. Clearly S(∞) = 1 - 1/10^∞
    Then for ε=inf > 0 we have an Z = log10 (1/ε) also 10^Z = 1/inf such that:
    |S(∞) - 1| = |1/10^∞| < ε for n=∞ (all n > Z)
    Therefore 0.999... = 1
    Last edited by 7777777; 10-04-2016 at 02:02 AM.

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    Senior Member john_gabriel's Avatar
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    Default Re: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

    Quote Originally Posted by grapes View Post
    Seems pretty obvious that n --> 2n-1 is an exact pairing of the first set to the second set, every member of the first set is paired with exactly one member of the second set, no duplications, no elements left out.
    Um, no. Bijective cardinality does not mean every member of a set is paired with exactly one member of another set, because not all the "members" are numbers.

    So it's not only false, but stupid to say that no elements are left out. After all, most of those mythical objects called "irrational numbers" are not elements of any set. Chuckle.
    Last edited by john_gabriel; 04-19-2015 at 12:19 PM.
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    Default Re: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

    A set is a well defined collection of distinct objects.

    Since infinity is undefined 123...and 246.. are not sets.
    Lies have the stench of death and defeat.

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    Default Re: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

    Quote Originally Posted by astrotech View Post
    A set is a well defined collection of distinct objects.
    Since infinity is undefined 123...and 246.. are not sets.
    Huh...

    The set of all natural numbers {1,2,3...} is so well defined and important
    that it has its own symbol N. as has other similar sets. It can not be more
    well defined then that.

    What is at question here and in the .999... thread is how to interpret infiniinty and infinitesimality.
    If the set of natural number is finite or not. If line has a finite number of points or not. The answer
    to this is not crystal clear and both viewpoints exist. It has however been proven (I can not rember
    who did this) that it is not important for other mathematic in general. But it is an intersting subject.

    Both viewpoints are attractive for different reasons.

    Cheers
    L-zr
    Last edited by Lazer; 04-22-2015 at 08:30 AM.

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    Default Re: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

    Quote Originally Posted by Lazer View Post
    If the set of natural number is finite or not.
    Thanks! That makes things easy.

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    Default Re: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

    Quote Originally Posted by Lazer View Post
    Huh...

    The set of all natural numbers {1,2,3...} is so well defined and important
    that it has its own symbol N. as has other similar sets. It can not be more
    well defined then that.

    What is at question here and in the .999... thread is how to interpret infiniinty and infinitesimality.
    If the set of natural number is finite or not. If line has a finite number of points or not. The answer
    to this is not crystal clear and both viewpoints exist. It has however been proven (I can not rember
    who did this) that it is not important for other mathematic in general. But it is an intersting subject.

    Both viewpoints are attractive for different reasons.

    Cheers
    L-zr
    But if there are infinite many natural numbers then all natural numbers cannot be defined. So I think the point Astrotech was making is that a "set" is complete, all accounted for. The symbol N is used to describe a "set" that cannot be all accounted for, which really makes no sense.

    I liked your post though by the way.

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    Default Re: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

    Quote Originally Posted by grapes View Post
    Thanks! That makes things easy.
    im glad you think so

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