Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
2 \(\Rightarrow\) 2	Theorems on the existence of successors of cardinals and the axiom of choice, Tarski, A. 1954a, Indag. Math.

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

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

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