Statement:
(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)).
Howard_Rubin_Number: 81-n
Parameter(s): This form depends on the following parameter(s): \(k\), \(k\): non-negative integer
This form's transferability is: Unknown
This form's negation transferability is: Negation Transferable
Article Citations:
Monro-1972: Models of \(ZF\) with the same sets of ordinals
Monro-1973b: Models of ZF with the same sets of sets of ordinals
Book references
Note connections:
Note 90
For all \(n\in\omega\), \(\neg (\hbox{ZF} \vdash K(n+1) \rightarrow K(n) )\)