User blog comment:Emlightened/Sasquatch (Big Bigeddon)/@comment-35470197-20181017094736

> If \((\overline{\in},R,F) \models t \textrm{ is an ordinal}\)

I could not understand why you can redefine \(R\) under the assumption above referring to \(R\). What are \((\overline{\in},R,F)\) and \(t\) here? A relation and functions (not just symbols) on a set \(M \in V\), and a term \(t \in M\)? If so, is there any axioms on \(M\)? Otherwise, \(\cup^{\overline{in}}\) does not make sense.

Also, what does \((\overline{\in},R,F) \models \phi(a)\) mean for an \(a \in V\) which does not necessarily lie in the underlying set \(M\)?

At last, is your definition valid in \((V,\in,\overline{\in},<)\)? Or is it just valid in \((V,\in,<)\) for each undefinable triad \((\overline{\in},R,F) \in V\) satisfying certain axioms?