Constructibility in the Simpson Set Theory Without the Local Countability Axiom

Following some ideas and constructions of Simpson, we introduce a set theory whichextends the Zermelo theory taken without the Power Set axiom by the three axioms postulating that, (1) for any set, there exists a set containing all finite subsets of the given set, (2) every set has a transitive superset, and (3) every well-founded binary relation admits a collapse map onto a transitive set. We prove that the theory introduced is strong enough to adequately define the class of Gödel-constructible sets, and that this class still satisfies the same axioms and admits a definable well-ordering of all sets. Metamathematical consequences are discussed as well.