Maximum Number in an Infinite set

Question

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?

Answer 1

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).