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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
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 ( 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 ( 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

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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In the specific case of the integers, every infinite subset either has no minimum or no maximum. However, this is certainly not true for all sets. There are infinitely many real numbers in the (closed) interval from 0 to 1, but there is a maximum, namely 1. So no, the concept of having a maximum element is not inherently tied to finite sets.

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