1. Standard memberroyalchicken
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    29 Aug '03 21:51
    A paradox is essentially something that cannot be handled by logic from within your axioms. A simple, standard one is:

    The barber shaves all men and only those men who do not shave themselves. Who shaves the barber?

    If the barber shaves himself, he shouldn't. If he doesn't, he should. bad explanation?
  2. Hendersonville, NC
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    29 Aug '03 22:06
    Originally posted by royalchicken
    A paradox is essentially something that cannot be handled by logic from within your axioms. A simple, standard one is:

    The barber shaves all men and only those men who do not shave themselves. Who shaves the barber?

    If the barber shaves himself, he shouldn't. If he doesn't, he should. bad explanation?
    Now, if you would be so kind, apply that to my original post so that I can understand what you mean.
  3. Standard memberroyalchicken
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    29 Aug '03 22:18
    Your conclusion, "1=2", is a paradox within the standard axioms, resulting from uncritical use of the function f(x)=sqrt(x).
  4. Standard memberFiathahel
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    29 Aug '03 23:50
    Look at the function f(x):=1 for all x

    If you use a simmilar reasoning as in the problem '1=2' you'll get that because f(a)=1=f(b) for all a and b, a = b.

    The problem is that the inverse of g(x)=x^2 is a metafunction, meaning that g^-1(x) is not one point, but a set of points. g^-1(x)={sqrt(x), -sqrt(x)}. So is sin^-1(0)={n*pi | n an integer}. Because metafunctions are more difficultto handle (they don't have a derivitive, for example), and contain more information than you need, it is easier to define only the inverse of a funtion on an interval: sqrt(x) is the inverse of x^2, x in [0,inf], arcsin(x) is the inverse of sin(x), x in [-1/2*pi, 1/2*pi].
  5. Standard memberroyalchicken
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    29 Aug '03 23:591 edit
    Exactly right. Your explanation is absolutely more complete, and absolutely worse pedagogically for CC 😉. Good job!
  6. Hendersonville, NC
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    30 Aug '03 04:24
    Originally posted by royalchicken
    Exactly right. Your explanation is absolutely more complete, and absolutely worse pedagogically for CC 😉. Good job!
    You are correct on this one. I am much more confused now....🙄
  7. Standard memberFiathahel
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    30 Aug '03 10:43
    Originally posted by Cheshire Cat
    You are correct on this one. I am much more confused now....🙄
    sorry 😉
    the idea was only that if f(a)=f(b) that this doesn't mean that a = b.
  8. Joined
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    18 Sep '03 20:24
    Isn't it that

    1/3 is 0.3333333333333333...
    and 3 x 0.333333333333... = 0.999999999999999...

    but 3 x 1/3 is 1 ?

    so... that's what our maths teacher told us :-)
  9. Standard memberTheMaster37
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    19 Sep '03 08:24
    Originally posted by seewinkler
    Isn't it that

    1/3 is 0.3333333333333333...
    and 3 x 0.333333333333... = 0.999999999999999...

    but 3 x 1/3 is 1 ?

    so... that's what our maths teacher told us :-)
    Yes, that is one of many ways of showing that it's true
  10. Standard memberFiathahel
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    19 Sep '03 17:31
    They are the same because there difference equals zero
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