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...they say it's very difficult to show what sqrt2 is...

  1. Jul 9, 2015 #1
    Hello,
    Since nobody ever showed which x is such that x2=2
    I confidently fumble on and say:
    α=sup{x∈R: x2<2}
    Show ∝∈R; ∝2=2

    I also say:
    ∝≠max

    Also, I restrict:
    2not>2
    2not<2,

    ....and then,
    I wake up and scream: "Can anyone go further?!"

    Thank you.
     
  2. jcsd
  3. Jul 9, 2015 #2

    mathman

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    What is the problem?
     
  4. Jul 9, 2015 #3
    Given that the rationals are dense in the reals, you give me an epsilon and I'll give you a rational number whose distance is less than epsilon from the sqrt(2). Existential crises are unnecessary.
     
  5. Jul 9, 2015 #4

    mathwonk

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    it just depends on what version of real numbers you are using. If you realize they are just lengths of line egments, then take a unit square and use the diagonal. the difficulty comes when trying to reduce everything to rational numbers.
     
  6. Jul 9, 2015 #5
    Thank you, everyone!

    1) The surreals contain the reals.
    2) A Cauchy sequence comes as close as needed to the epsilon chosen.
    3) The construction of reals from Cauchy sequences determines that every rational sequence that converges to x is a representation of x.
    4) No construction of R seems to be definite. Therefore, how can its completeness be?

    My problem is:
    1) Could rigor even be considered as being respected in the attempt above (especially in view of the notes below)?
    2) Since the reals are contained in the surreals, could there be a surreal number that is sqrt2, not just close to it - since completeness of R is questionable?
    3) Is rigor impossible because no R construction is convincing?
    4) I also considered the construction of R by Dedekind cuts, and amazingly one example was exactly sqrt2. All went well until they had to show x2=2.
    Other constructions seem to wobble also.

    I incline to believe - particularly bolstered by the paper quoted at the bottom - that rigor is not possible due to the fact that there has not yet been an irrefutable construction of R.

    Any thoughts?
    Thank you.
    -------------------------------------------------------------------------------------------

    "Few mathematical structures have undergone as many revisions or
    have been presented in as many guises as the real numbers. Every
    generation re-examines the reals in the light of its values and mathematical
    objectives.
    It is often deplored that the field of real numbers is not constructive
    in any of the currently accepted meanings of the word. How then
    do we propose to adhere to the seemingly impossible objective of
    making the real numbers conform to the credo of constructivity ?"

    ( "The Real Numbers as a Wreath Product"
    F. FALTIN, Cornell University, Ithaca, New York 14850
    N. METROPOLIS, Los Alamos Scientijic Laboratory, Los Alamos, New Mexico 87544
    B. Ross AND G.-C. ROTA, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 )
     
    Last edited: Jul 10, 2015
  7. Jul 10, 2015 #6

    mathwonk

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    to repeat, the reals are not the rationals. there IS a real sqrt of 2, but not a rational one. it is hard as you have observed to go from the rationals to the reals, since it takes an infinite number of rationals to specify a real, but it does seem possible and rigorous. E.g. it is not clear to me you understand the technical meaning of "constructive" as opposed to "construction", in the article cited.

    the rest of your post seems to be philosophy, and may belong elsewhere.
     
    Last edited: Jul 10, 2015
  8. Jul 10, 2015 #7
    You realize that mathwonk is a professional mathematician (well, now retired) right? He knows much more mathematics than many of us. If you are trying to learn, you would be wise to listen to him.
     
  9. Jul 10, 2015 #8

    mathwonk

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    i soon felt my post was impolite, and tried to delete it, but not quickly enough it seems. it is not easy to instruct some people, especially angry people.
     
  10. Jul 10, 2015 #9

    Evo

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    This thread is over, offending posts removed.
     
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