Limitation of Henkin sematics in second order logic

Question

I am reading Enderton's A Mathematical Introduction to Logic.

1. The book explains that compactness fails in second order logic by a counter example $\Sigma = \{\neg \lambda_{\infty} , \lambda_2, \lambda_3,\lambda_4,\lambda_5, \cdots \}$ where $ \lambda_{\infty} = \exists X[\forall u \forall v \forall w (Xuv \rightarrow Xvw \rightarrow Xuw) \land \forall u \neg Xuu \land \forall u \exists x Xuv ]$ (Existence of transitive irreflexive relation on the entire set having no last element) and $ \lambda_n = \exists x_1 \exists x_2 \cdots \exists x_n(x_1 \neq x_2 \land x_1 \neq x_3...\land x_{n-1} \neq x_n) $ (There exist at least n different points)

2. Then the book explains many sorted logic and a 'structure' which is a reduction of second order language to first order many sorted logic by adding sorts of function and relation universe and using appropriate definitions, (I assume it is the Henkin semantics) and shows enumerability, compactness and LST theorem.

3. My question is about satisfiability of $\Sigma $ on the henkin semantics. To me it looks that $\Sigma$ still is a counter example at first glance. Furthermore, what makes compactness(and other good propertices) possible in Henkin semantics but not in general(usual) semantics? What is the limitation of henkin semantics compared to general semantics?

Answer 1

In a Henkin model, the second-order quantifiers are not required to range over all possible properties/relations over the domain of objects. In other words, the domain of the second-order quantifiers can be just some of the properties/relations which could be in principle be defined over the domain of objects.

So take e.g. a Henkin model with (i) infinitely many objects in the domain of the first-order quantifiers, but where (ii) the only (two-place) relations in the domain of the second-order quantifiers are reflexive. Then $\lambda_\infty$ is false (because there doesn't exist a relation $X$ which is irreflexive etc.); so this model indeed satisfies $\Sigma$.

Think of it this way. In the Henkin semantics, we can have fewer relations in the domain of the second-order domain than in the full semantics: that's why it is easier for a second-order existential quantification to fail (easier for its negation to be satisfied) when interpreted Henkin-style.