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Given an infinite set of random integers, is there a largest element?
In other words is maximum as a concept inherently tied to finite sets?

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Is the set random or infinte? If the set is finite, the answer is yes, there's a largest element. If it's infinite, than there might be (i.e it could be bounded above). But, if you're working just with the natural numbers-rather than all of the integers, it would always never have a maximum element –  Chris Dugale Jun 16 '13 at 23:53
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Note that "infinite set of random integers" is not a well-defined statement -- you need to state where these integers come from to give it meaning. –  usul Jun 16 '13 at 23:59
    
Think about $\{0,-1,-2,\dots\}$. –  Zeyu Jun 17 '13 at 0:05
    
The maximum of an infinite number of elements in a poset (en.wikipedia.org/wiki/Partially_ordered_set) is well-defined and unique if it exists, but doesn't exist in general. The maximum of a finite number of elements in a totally ordered set (en.wikipedia.org/wiki/Total_order) always exists. –  Qiaochu Yuan Jun 17 '13 at 0:12
    
You need to clarify what you mean by "random" here. The integers don't admit a uniform probability distribution, so you need to pick one. –  Qiaochu Yuan Jun 17 '13 at 0:12
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Possibly, but not necessarily. Any such set will have a maximum if and only if it is bounded above (necessarily not bounded below, since infinite). For example, consider the negative integers (has a maximum) and the positive integers (has no maximum).

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