Recently in chat, we investigated the explicit construction of $\omega_1$. Regardless on whether we use ZFC or hartogs number, we seemed to hit a roadblock.
Using either ZFC or hartogs number, we managed to came up a set of countable ordinals $S$ that corresponds to well orderings on $\omega$. However, other than its existence, there seemed to be no algorithmic way to describe that well ordering in more detail
(for example, some explicit but not necessary computable predicate P(x,y) such that given any pair (x,y) it unambiguously give either true or false to the question x < y, that is, $P(x,y)$ is not of the form "There exists a well ordering...")
therefore cannot show that $S$ is well ordered and hence $\omega_1$ cannot be shown to exist by construction.
- Is there exists a predicative well ordering of an uncountable set $S$ which the injection $\omega \to S$ can be constructed without first assuming $S$ can be well ordered without the axiom of choice (and possibly without excluded middle)?
- If there is, what is $P$, that is, the criteria on how exactly any given countable well orderings are well ordered within $S$?