Abstract
This paper formalizes Gödel’s 1946 conjecture that every set-theoretic sentence is decidable from the present axioms plus some true axioms of infinity; and we prove a weak variant of this conjecture to be true for every ZF {\text {ZF}} universe. We then make precise the extent to which unbound quantifiers can be taken to range only over ordinals in ZF {\text {ZF}} , obtaining a sort of normal-form theorem. The last section relates these results to the problem of how wide the class of all sets is.
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