I have recently been thinking about whether it is possible to construct a free algebra on a signature containing an operation of nonfinite arity. This lead me to propose a question to myself:
- For any set S, is it possible to find a well ordered set $ \mathbb T:= (T, <)$, such that for all functions $g: S \to T$ their exists an element $\alpha \in T$ such that $\alpha$ dominates $g[S]$ (i.e. $\forall x \in g[S]: \alpha > x$)?
It is a quick remark to note that any such $T$ must satisfy $|T| >|S|$ otherwise we have a surjective function $ f: S\twoheadrightarrow T $ of which we can find no such dominating $\alpha_f$ for!
I have been attempting to no avail to prove that such $\mathbb T$ exists for any set $S$ for a while now to no avail, and I am not certain that one necessarily does.
I have spotted an analogy between question (1) and having to take the existence of an infinite carnality set as an axiom in set theory. Note that the existence of $\omega$ is equivalent to asking if such a $\mathbb T$ exists when $S$ is a singleton set. This makes me wonder weather the totally ordered set I dream of is too big to exist if $S$ is of infinite cardinality, without making more assumptions than one usually does ($ZFC$), when doing set theory.
Therefore, I open the question (1) up to the floor.