Suppose we take MK and add a new unary function symbol α, with a countable set of extra axioms α(1) ∈ α(0)
α(2) ∈ α(1)
α(3) ∈ α(2)
(i) Show that the resulting theory is consistent.For all n ∈ N $\vdash$ α(n + 1) ∈ α(n)
(ii) Show that $\nvdash$ (∀n∈N)α(n+1)∈α(n).
I see a Compactness theory application but how?
This is most interesting if we interpret the task to mean that it must be allowed to use the new $\alpha$ in class comprehensions. Some hints:
(i) is then still a straightforward compactness argument. Given any finite set of axioms, take a (presumed existing) model of MK and extend it with an appropriate interpretation of $\alpha$ that satisfies the finitely many new axioms you have to satisfy. This can easily be done such that your extended model's $\alpha$ is definable in terms of $\in$, which allows you to argue that all instances of the class comprehension axiom must still be satisfied.
For (ii), note that $\forall n\in\omega: \alpha(n+1)\in\alpha(n)$ implies that $\{\alpha(n)\mid n\in\omega\}$ (which must exist because we can use $\alpha$ to define classes) would be a counterexample to Foundation.