A proof and a challenge

Here is a mathematical proof I came over during my eleventh grade in the year 1996.

Let x be the biggest real number below 1:

    x < 1
=> 1-x > 0
=> 1/(1-x) > 0
=> 2/(1-x) > 0
=> 0 < (1-x)/2
=> x < x + (1-x)/2

since x is the biggest real number below 1, we may conclude

(x < ) 1 <= x + (1-x)/2
(between x and 1 there is no real number)
=> 1-x < = (1-x)/2
=> 1/(1-x) >= 2/(1-x)
=> 1 >= 2
=> 1 > 2 or 1 = 2

Since 1 is surely not any larger than 2 we may follow that 1 must be equal to 2. That said, the whole mathematics need to be renewed. Since 1+1=2 we can now follow that 2+2=1, since both are interchangeable. Therefore we just have to deal with ones for all positive numbers. Everything is equal to one, just as I proofed.

Now, you can either stick to believe me on this proof and help me in re-defining the mathematics field, or you can challenge the proof that I just showed you. So, either you have to forget everything you learned about calculations in the past 20 (or so) years, or you can start to investigate the flaws of this proof.

It’s the same with other, not so obvious things in our lives. You can either start to believe the evidence you get presented, or start to challenge it, or to challenge it, or maybe to challenge it, or just to challenge it, or even stick with challenging it. Take your pick.

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7 thoughts on “A proof and a challenge”

  1. Markus,
    You caught my mathematical curiosity…

    The starting point “Let x be the biggest real number below 1” is wrong. For any x you pick I can find a bigger number that is still less than 1.

    This results in the line (x < ) 1 <= x + (1-x)/2 not holding true (due to the invalid starting premiss) – hence the rest breaks down.

    Oh, I agree with the challenging part though! :)

  2. It’s been a while since I’ve messed with calculus, but this doesn’t sit well with me.

    (x < ) 1 <= x + (1-x)/2
    (between x and 1 there is no real number)

    Basically for the same reason. If you assert there is no number between x and 1, then you're also saying there is no 'difference' between them which nullifies the existence of 1-x. Since we know 1-x exists, the statement, "between x and 1 there is no real number" can't be true. 1 < x results in an infinite set. You cannot define infinity as a value as it has to be expressed as a function (right?)

    Regardless, my brain hurts anyway which I assume was your evil, evil intent :) Thanks for bringing back a very dark point in my history.

    1. No. If you take the discussion to the natural numbers, you can say

      1 < a 2 <= a

      Since there is no valid number between 1 and 2, a must be greater than 2 or equal to 2 if a is greater than 1.

  3. Uh, no.

    Let’s call the difference between X an 1 “Y.” Thus, (1-X)/2 is equal to 1/2Y.

    So in the second claim, you are basically saying this:


    Since Y is the difference between X and 1, you can’t ADD Y to X and get something greater than or equal to one. This holds for all positive real numbers < 1; it's a short form of proof by induction.

    I think you're splitting the unsplittable atom.

    X is something like 0.999999 with a line over it, right? It's as far out as you can take the .99's.

    Say that's a billion nines. It is infinite.

    Then you do this:

    (1-x)/2 == well, what is that?

    That's be like 0.00000000 with a 5 at the end – in the one billionth and one place.

    So basically, you are saying "X is the highest order of precision possible in the universe. Now, add an element one-half as big as that order of precision to it, and you've got to get one, right?"

    No you don't. You get cats and dogs living together and the universe exploding. You can't claim a real number is the highest order of precision, then add something smaller to it; that's embracing a contradiction.

    You did make me think though, and that was fun! My favorite proofs are induction and limits problems, :-)

  4. Typo. The second claim holds they I wrote it. I should have said “for the 2nd claim, you are bascially saying this”:


    It’s also not the second claim, it’s the 2nd major claim, I guess. Sorry.

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