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ABS Oghbatalab(Latin)

ABSTRACT:

  A Paris model M is a model of set theory all of whose ordinal are first order definable in M. Jeffrey Paris (1973) initiate the study these models and showed that (1) every consistent extension T of ZF has a Paris model, and (2) for complete extentions T, T has a unique Paris model uo to isomorphism iff T proves V=OD. In this thesis we study Paris models,  

including the following results.

(1) If T is a consistent completion of   ZF + V = OD then T has continuum-many countable nonisomorphic Paris models.

(2) Every countable models of ZFC has a paris generic extention.

(3) If there is an uncountable well-founded model of ZFC, then for every infinit cardinal k there is a Paris model of ZF of cardinality k which has a nontrivial automorphism.

(4) For a model M of ZF, If M is a prime model then M is a paris model and satisfies AC, and If M is a paris model and satisfies AC then M is a minimal model. Moreover, Niether implication reverses assuming Con(ZF).

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