A corrected proof that all numbers are the same.

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Discussion Overview

The discussion revolves around a proof claiming that all numbers are the same, using induction on countable sets and drawing parallels to a well-known fallacy regarding horses. Participants explore the validity of the proof, its implications, and the logical structure behind it, including critiques and clarifications of specific statements made in the proof.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant presents a proof claiming that all numbers in any countable set are the same, using induction and concluding that all numbers are identical.
  • Another participant challenges the proof by stating that one cannot compare a number to itself, questioning the validity of the argument regarding singleton sets.
  • A subsequent reply attempts to clarify the previous challenge, arguing that the statement about singleton sets is vacuously true and asking for further clarification on the criticism.
  • Another participant acknowledges a flaw in the horse color analogy, suggesting that the proof incorrectly compares the first horse to itself, which does not support the argument.
  • One participant proposes using the term "corral" instead of "set" when discussing horses, suggesting that the analogy could be made clearer by changing terminology.
  • A later contribution emphasizes the need to verify that the sets used in the proof have at least one common element, indicating that the proof fails for the case of two elements.
  • Another participant agrees with the critique regarding the need for verification of the common element in the proof.

Areas of Agreement / Disagreement

Participants express disagreement regarding the validity of the original proof and the horse analogy. There is no consensus on the correctness of the proof, with multiple critiques and clarifications presented without resolution.

Contextual Notes

The discussion highlights limitations in the original proof, particularly regarding the assumptions made about the sets involved and the implications of comparing elements within those sets. The need for additional verification in the induction process is also noted.

Jimmy Snyder
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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.
 
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You can't compare it to it self.
 
Bright Wang said:
You can't compare it to it self.
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?
 
jimmysnyder said:
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?

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? :rolleyes: Maybe sleep posting.
 
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.
 
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.
 
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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