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A corrected proof that all numbers are the same.

  1. Dec 18, 2009 #1
    All numbers are the same

    Pf: First I will show that for any countable set of numbers Z, all the numbers in Z are the same. Let A = {a, b, c, ..., d} be any nonempty subset of Z. Then a = b= c= ... = d as I shall show by induction on the number of elements in A. Notice that for all singletons, A = {a}, the statement is vacuously true. Now assume that the statement is true for any subset that contains n elements {a1, a2, ..., an} and let A = {a1, a2, ..., a(n+1)} with n + 1 elements. Then the set B= {a1, a2, ..., an} has n elements and so by assumption a1 = a2 = ... = an. Likewise the set C = {a2, a3, ..., a(n+1)} has n elements and so by assumption a2 = a3 = ... = a(n+1). Combining these two sets of equalities, a1 = a2 = a3 = ... = a(n+1). So all the elements of A are the same. Hence, by induction, all numbers in Z are the same.

    Next I will show that all numbers are the same. By the first part, all integers are the same. Suppose there is a number which is not the same as the integers, call it w. Then the set W = {w} union the integers is a countable set of numbers and so by the first part they are all the same, contradicting the statement that w is different. Therefore, all numbers are the same.

    This same argument can be used to prove that all horses are the same color. And since Washington rode a white horse, we know that all horses are white. Since horses have two hindlegs in back and forelegs in front, that makes 6 legs so they have an even number of legs. However, for a mammal, 6 is an odd number of legs and so they have a number of legs which is both even and odd. Since no number is both even and odd, they must have infinitely many legs. We know that in the Triple Crown of 1973, Secretariat raced on all three legs, but that's a horse of another color.
     
    Last edited: Dec 18, 2009
  2. jcsd
  3. Dec 18, 2009 #2
    You can't compare it to it self.
     
  4. Dec 18, 2009 #3
    I'm not sure what is meant by this. However, if it is a criticism of the statement that all the elements of a singleton set are equal to each other, then you are not correct. To see that all the elements of a singleton set are equal to each other, consider the contrapositive statement: There is no pair of elements of the set that are different. This is vacuously true since there is no pair of elements. On the other hand if you had some other meaning in mind, then can you please clarify?
     
  5. Dec 19, 2009 #4
    Sorry, I think in the proof of the that all horses are the same color, by induction. (Which is clearly not true.) The error is that it compared the 1st horse to itself, so its the same color as it self. It was 5am when i posted? :uhh: Maybe sleep posting.
     
  6. Dec 19, 2009 #5
    Perhaps it would be easier to use the word corral instead of the word set when it comes to horses. In every corral that contains a single horse, all the horses in that corral are the same color. For surely there are not two horses of differing colors in a corral that contains but a single horse. If you agree to that, then just change the word horse to number, corral to set, and strike the word color.
     
  7. Dec 19, 2009 #6
    To validate your proof you need to assure set B and C have at least one common element, that means simply verifying n=1 is not enough, you also need to verify n=2 case, and it will fail.
     
  8. Dec 19, 2009 #7
    Correct kof9595995.
    Please put your answer in spoiler tags like this, so those who are still puzzled won't accidentally read the answer, and those who just want to know the answer can still read it.
     
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