Recently I have learned that the Completeness Axiom is equivalent to the Monotone Convergence Theorem. I also saw here (without reading the proof admittedly) that the intermediate value theorem is also equivalent.
Let $C = \{c_1,c_2,\ldots\}$ where $c_i$ is a logical statement equivalent to the completeness axiom. Is it true that $|C| \leq k$ for some $k \in \mathbb{N}$. If this is not true, can we say anything about the cardinality of $C$? Is it countable?
Thanks.