User blog comment:DontDrinkH20/H-Boogol-Boogol Bit by bit: PART 2: Higher Order Logic/@comment-35470197-20180904134342

> \((M;c) \models \forallx(x=C)\) if and only if for every \(x \in M\), \(x\) is \(C\).

The last \(C\) is a typo of \(c\).

> \(\mathcal{P}^2(M)\) is the set of all subsets of subsets of \(M\), and so on.

The set of all subsets of subsets of \(M\) is just the set of all subsets of \(M\). Maybe it is a typo of the set of all subsets of the set of all subsets of \(M\).

> "\(\forall_n v^{n-1}_m(t)(\psi)\)" where \(\psi\) is another \(n\)-th order \(\mathcal{L}\)-formula

Since the term \(t\) is not specified here, it is just a placeholder of a free variable occurring in \(v^{n-1}_m\), right? If so, it might be better to remove \((t)\) or distinguish it from a specific term symbol above.

Also, it might be better to distinguish the brackets for the substitution of \(t\) and the quantification of \(\psi\). For example, write \(v^{n-1}_m[t]\).

> \(2^{\mathbb{N}} \in P\)

Typo of \(2 \in P\).

> If \(\varphi\) is "\(\forall_n v^{a_0}_0(t)(\psi)\)" where \(\psi\) is another \(n\)-th order \(\mathcal{L}\)-formula, then \(\mathcal{M} \models \varphi[x_0,x_1,\ldots]\)

Maybe it is a typo of \(\varphi[x_1,x_2,\ldots]\), because \(\varphi\) has n free occurrence of \(v^{a_0}_0\).