Can an infinite well ordered set $A \subseteq \mathbb{R}$ have a greatest element? I've read it a couple of times that the answer is yes, but I just couldn't agree with that.
If set $A$ is well ordered, then surely it has least element. Okey, but then if it have a greatest element, as well, the set will be finite.
If I'm wrong, could you give me example of such set? Thanks!
Maximal element of infinite well ordered set
well-orders
Best Answer
One idea is to find a well-ordered set of real numbers that fits nicely into a fixed interval, then to pick some individual value bigger than all of them and add it in as the maximum element. For example, you could pick
$$S = \{ 1 - \frac{1}{n} \ | \ n \in \mathbb{N} \} \cup \{ 137 \}.$$
Here, the first half of the definition of $S$ gives you an infinite well-ordered set, and adding in $137$ preserves the well-orderedness while giving you a maximum element.