Is there a minimal model for ZC that includes Gödel constructible elements?

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Garrulo
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Is there a meta-mathematical model of ZC (Zermelo Set Theory with specification scheme and axiom of choice but not remplacement and) that includes a minimun quantity of Gödel constructible elements of L hierarchy with all the ordinals until ω⋅2 and that it "lives" in all the meta-mathematical models of ZC?
 

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I don't know the answer, but just a suggestion: perhaps you might get more replies by posting it in the "Set Theory, Logic..." section.