Can a Transfinite set of any given size possess a top element?

Question

Can an infinite set have a Top element? Can a set of any given transfinite Cardinality possess a Top element? Are there any examples of known sets that have top elements and are more than finite? I am pretty certain that this is possible but I have only heard finite cases talked about.

Answer 1

Yes. Every set can be linearly ordered, assuming the axiom of choice, and given any linear order $(X,\leq)$, we can pick one $x\in X$ and define $\leq_x$ which is $\leq$ on $X\setminus\{x\}$, and $y\leq_x x$ for all $y\in X$.

Since assuming choice gives you that every set can also be well-ordered, you can require that $\leq$ (and consequently $\leq_x$) is a well-order as well, which means that every non-empty set has a minimum too.

Another approach here is a model theoretic approach: let $T$ be the theory in the language of $\leq$ stating that $\leq$ is a linear ordering with a maximum. It is certainly consistent, and it has models of every finite size, therefore, by the compactness theorem, it has an infinite model, and therefore, by the Löwenheim–Skolem theorem, an infinite model of any given size.