Implications

We have the following indirect implication of form equivalence classes:

81-n \(\Rightarrow\) 81-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
81-n \(\Rightarrow\) 81-n

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

Howard-Rubin Number Statement
81-n:

(For \(n\in\omega\)) \(K(n)\): For every set \(S\) there is an ordinal \(\alpha\) and a one to one function \(f: S \rightarrow {\cal P}^{n}(\alpha)\). (\({\cal P}^{0}(X) = X\) and \({\cal P}^{n+1}(X) = {\cal P}({\cal P}^{n}(X))\). (\(K(0)\) is equivalent to [1 AC]  and \(K(1)\) is equivalent to the selection principle (Form 15)).

81-n:

(For \(n\in\omega\)) \(K(n)\): For every set \(S\) there is an ordinal \(\alpha\) and a one to one function \(f: S \rightarrow {\cal P}^{n}(\alpha)\). (\({\cal P}^{0}(X) = X\) and \({\cal P}^{n+1}(X) = {\cal P}({\cal P}^{n}(X))\). (\(K(0)\) is equivalent to [1 AC]  and \(K(1)\) is equivalent to the selection principle (Form 15)).

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