Re: Paper and slides on indefiniteness of CH

The basic issue has been raised as to how the axiom of choice is to follow from “maximality”. This has been particularly well understood for a long time, e.g., in the following way.

THEOREM. In ZF, the following are equivalent. i. Every graph has a maximal clique. ii. The axiom of choice.

It is most convenient to define a graph as a pair (V,E), where E is an irreflexive symmetric binary relation on V. A clique is a set where any two distinct elements are related by E. Maximal means inclusion maximal.

Alternatively, one can use digraphs in the sense of paris (V,E), where E is a binary relation on V. A clique is a set where any two elements are related by E. (You can also use: any two distinct elements are related by E).

Harvey

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