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Bijectional, Generic and Permutational models of ZF

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The consistency assumption of ZF implies the existence of a model (K,(epsilon)) for ZF by virtue of Goedel's completeness theorem. But then von Neumann's hierarchy (V(,a))(,a)(epsilon)(,R), where R is the class of all ordinals of K, implies the existence of a model (V,(epsilon)) for ZFG, where G is the Axiom of Regularity. This shows that the consistency of G with ZF can be proven by means of an inner model with methods which can be formalized in ZF. Similarly, Goedel's model (L,(epsilon)) of constructible sets and Cohen's minimal model (M,(epsilon)) of strongly constructible sets are inner models of (V,(epsilon)) satisfying ZFG + (V = L) and consequently satisfying ZFG + GCH, and thus ZFG + GCH + AC. As in the case of inner models, a natural tendency for construcing various ZF models is to consider the class of sets of (K,(epsilon)) that satisfy a set-theoretical formula R(x). But then, because of the minimality of (M,(epsilon)), no such inner model can be constructed for ZFG + (NOT) (V = L) and hence for ZFG + (NOT) AC or for ZFG + (NOT) CH;Thus, to construct ZFG models in which V = L is not valid, one must use methods which go beyond ZF, i.e. which cannot be formalized in ZF. Examples of such models are the Bijectional, Generic and Permutational models which we consider in this dissertation;First, we construct Bijectional and Dyadic Sequential ZF models with sets having k atoms for any cardinal k and ZF models with the negation of the Extensionality Axiom;We also construct standard models MG of ZF where G is not an element of M. To this end we introduce the notion of an elementhood relation with respect to a suitable generic subset G of a partially ordered set P of M and consider the corresponding partially-order-valued class of progenitors. We then find the quotient class of progenitors with respect to G to obtain MG. Moreover, we obtain some results concerning the cardinalities of the dense subsets of partially ordered sets. Furthermore, we give necessary and sufficient conditions for the existence of generic subsets of partially ordered sets by means of the notion of molecule;Using some of the above results, we consider Permutational models based on decreasing chains of equivalence classes of permutations and construct a model for ZF and (NOT) AC.