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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.

1729
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    The set of all statements is countable, It is easy to find infinitely many theorems equivalent to another, so there are infinitely many such statements, hence countably many. – Thomas Andrews Jan 27 '18 at 21:11
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    What do you mean by a "statement", and by "equivalent"? If you allow arbitrary parameters there is a proper class of such statements, since for any set $S$, you could take the statement "the completeness axiom and $S=S$". – Eric Wofsey Jan 27 '18 at 21:16
  • Unfortunately, I haven't taken any courses in logic so I can't really clarify "statement" or "equivalent" rigorously. I think I was hoping to somehow avoid trivial cases like the one you gave but I am not sure how that would be done. – 1729 Jan 27 '18 at 21:22
  • I never get tired of sending people links to https://arxiv.org/abs/1204.4483 . – Patrick Stevens Jan 27 '18 at 21:30

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