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 Apr 12 awarded Popular Question Jun 5 awarded Yearling Oct 15 awarded Notable Question Jul 2 awarded Curious Jul 2 awarded Inquisitive Jun 5 awarded Yearling Dec 11 accepted Is there any countable ordinal number which has a member undefinable? Nov 9 awarded Popular Question Aug 4 comment Is there any countable ordinal number which has a member undefinable? Helpful, thank you. Jul 16 comment Is there any countable ordinal number which has a member undefinable? @CarlMummert I was trying to understand you. Would you please show me more detail? Do these models of ZFC, which every set is definable over the model, themselves may have uncountable(outside) many subsets? Or the cardinalities of their powersets are all at most countable(outside), but some of which are uncountable(inside)? Besides, what about $(V,<)$? Jul 16 comment Is there any countable ordinal number which has a member undefinable? It's nice for me, thank you. Jul 16 revised Is there any countable ordinal number which has a member undefinable? highlight Jul 16 comment Is there any countable ordinal number which has a member undefinable? @CarlMummert "There are models of ZFC in which every set is definable over the model". Did you mean definable with parameters, or just definable(without parameters)? Jul 16 asked Is there any countable ordinal number which has a member undefinable? Jul 3 comment What's correspondence between the model theoric and the set theoric kernel of homomorphism? Okay, it is right now. Jul 2 comment What's correspondence between the model theoric and the set theoric kernel of homomorphism? Thank you for answering and sorry for late commenting as I'm in traveling yesterday. I have a question: do you mean $ker(h)$ the set-theoric kernel if $h$? If so, then $ker(Id_{\mathfrak{A}})=Id_{\mathfrak{A}} \ne \emptyset$ whenever $\mathfrak{A}$ is not empty, but $Id_{\mathfrak{A}}$ is an automorphism, hence an embedding. Thus $\ker_m(Id_{\mathfrak{A}})=diag(\mathfrak{A})$ but $ker(Id_{\mathfrak{A}}) \ne \emptyset$ in cases $\mathfrak{A}$ is not empty... Jun 30 revised What's correspondence between the model theoric and the set theoric kernel of homomorphism? clearify and retag Jun 23 comment What's correspondence between the model theoric and the set theoric kernel of homomorphism? @kahen Thank you for your suggestion. Jun 23 revised What's correspondence between the model theoric and the set theoric kernel of homomorphism? patch Jun 23 comment What's correspondence between the model theoric and the set theoric kernel of homomorphism? @tomasz Okay, I have clarified my post.