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clique principle – ZF with Class-choice? Answer

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clique principle – ZF with Class-choice?

LASS ZF can breathe expanded with class and clique predicates and increase the idea with a category abstraction strategy in such a route that for all formulation $ phi $:

$$ forall vec {z} exists Y (Cls (Y) wedge forall x (x in Y leftrightarrow phi ( vec {z}, x))) $$

Let’s convene the idea $ ZFC ^ {Cls} $.

We know there are numerous courses in $ ZFC ^ {Cls} $ are actual courses, love Russell’s class $ R = x notin x $.

What if, as an alternative of the habitual axiom of alternative, now we have axioms that, for any clique s, of non-empty, non-overlapping units, merely a. posit Great Y that has precisely one member from every member of s?

For instance, it appears limpid that the Vitali clique can’t breathe outlined in $ ZFC ^ {Cls} $.

Does $ ZFC ^ {Cls} $ increase $ ZF $ kindly of attention-grabbing?

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