Implications

We have the following indirect implication of form equivalence classes:

1 \(\Rightarrow\) 2
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
1 \(\Rightarrow\) 2

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
1:

\(C(\infty,\infty)\):  The Axiom of Choice: Every  set  of  non-empty sets has a choice function.

2:

Existence of successor cardinals: For every cardinal \(m\) there is a cardinal \(n\) such that \(m < n\) and \((\forall p < n)(p \le m)\).

Comment:

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