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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?

deft_code
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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 – RougeSegwayUser 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 00:05
  • The maximum of an infinite number of elements in a poset (http://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 (http://en.wikipedia.org/wiki/Total_order) always exists. – Qiaochu Yuan Jun 17 '13 at 00:12
  • 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 00:12

2 Answers2

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Possibly, but not necessarily. Any such set of integers 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).

Cameron Buie
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  • what is we say that the integers are finite ? – Our Nov 11 '18 at 11:59
  • The set of all integers is infinite. A finite subset of the integers will have a maximum element, unless the subset is empty. – Cameron Buie Nov 11 '18 at 13:11
  • For future readers: note that the question is about integers. In the case of the real numbers, "contains maximum iff bounded above" and "infinite and bounded above implies not bounded below" are both false; consider $[0,1)$. – jskattt797 May 18 '20 at 03:58
  • @jskattt797: Fair point! I've adjusted my wording to that end. :-) – Cameron Buie May 18 '20 at 10:44
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In the specific case of the integers, an infinite subset has no minimum or has no maximum (possibly neither). 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.

Cameron Buie
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Ryan Lafferty
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