Is the 'greatest negative number' indeterminate, not defined or doesn't exist?

Question

Let's take S as a set of negative real numbers. Intuitively it seems that there's no greatest element in S because the term 'greatest' only makes sense if the set S is finite. But I don't know the exact term to describe this situation. How to describe the 'greatest negative number'?

Answer 1

The notion of a "greatest element" does not only make sense for finite sets, without defining posets, lets just focus on the real numbers:

Given a set $A\subseteq\mathbb{R}$, a real number $x$ is said to be the maximum element of $A$ if \begin{align} &x\in A\\ &\mbox{and} \\ &\forall_{y\in A}[y\leq x] \end{align}

(I hope you are familiar with quantifiers?)

Anyway, the set

$$ (-\infty,0) \cup \{1\} $$

has a maximum element, even though it has infinitely many elements.

Now to answer your question, the set of negative real numbers, $S$ does not have a maximum element, since if $a\in S$ were a maximum, then $\frac{a}{2}\in A$ but $a< \frac{a}{2}$ contradicting our assumption that $a$ was a maximum.

If you want to gain some more understanding, try to prove that if a subset of the real numbers has a maximum, then it is unique (this was implicitly assumed in the definition with the wording "the maximum"). For generalisations, that might help you answer questions like this in the future, look into the concept of a poset. (Beware that there is a difference between a maximal element, and an element which is the maximum, although the maximum is necessarily maximal, but the converse is not true in general)

Answer 2

Your intuition is broadly correct here and raises an interesting point.

What the set $S$ is (letting $S$ be the set of all negative real numbers), is a set that is open on one side, more precisely there is no greatest element that you could write down that is a part of that set because I can always give you a slightly bigger number that is always in the set.

This also applies for any set of real numbers that is open on one side, say the set $J=[0,1)$. The square bracket indicates that $0\in J$ and the round bracket indicates that $1\notin J$. So here we have the same problem when it comes to defining the greatest element of $J$.

It is clear that it doesn't really make sense to define the largest number of $S$ or $J$ but it does make sense to define a smallest upper bound as follows. The least upper bound or supremum of any non-empty set $K$ that is bounded above is defined as the smallest real number $x$ such that $x\geq a$ for all $a\in K$ and, for all $\epsilon >0,\; x-\epsilon$ is not an upper bound.

For the set $S$, this would be $0$ and for $J$, it would be $1$.