User blog comment:Emlightened/Sasquatch (Big Bigeddon)/@comment-1605058-20170327204644

Since it's late and I don't have too much time, I'll just leave a few quick comments/questions.

\(F\) and \(R\) appear to be unary functions (functional predicate), not predicates, so I would appreciate if you've clarified that.

What do you mean by "adjunction" in the definition of \(R(\alpha)\)? And, what does the \(\overline\in\) in the superscript indicate?

What are the \(a\) in the definition of \(\llcorner a \lrcorner\)? Are they arbitrary natural numbers? If so, and you include infinitely many terms in your language, there might be problems with this number being well-defined (what about the infinitely many formulas \(\phi(x) = (x=a)\)?).

If they aren't natural numbers, then what does \(\langle 5,a\rangle\) mean? Do they have an intended interpretation? If not, then how can terms represent unique objects, so that a statement \(t\text{ is an ordinal}\) is a well-defined statement which your structure can satisfy?