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Shortest Long Number (Posted on 2004-04-15) Difficulty: 3 of 5
What is the smallest positive integer that cannot be defined in less than twenty-five syllables?

See The Solution Submitted by Sam    
Rating: 3.0833 (12 votes)

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But can any set of positive integers exist without a least member? | Comment 20 of 49 |
(In reply to re(6): Tentative solution REVISED (Paradoxes) by Sam)

Sam asks: "But can any set of positive integers exist without a least member?"

No. The proof is as follows.

Let S be a set of positive integers. S will either be finite or infinite. If S is finite, it is easy to show that S has a least member. Mathematical induction will prove that.

Suppose S is infinite. Arbitrarily select integer Z>1 in S. Now define set T as {all positive integers less than Z}. T is obviously finite, so T has a least element.

Let X be the least element of T that is also contained in S.

If there is no such X, then Z is the least element of S.

If there is such an X, then X is the least element of S.

 

 

 

 

Edited on April 15, 2004, 8:30 pm
  Posted by Penny on 2004-04-15 20:27:28

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