Question 1:

In the same language, let $ X $ be a nonempty set, and let $ \{ (\forall x_{x(i)} f(i)) \ | \ i \in X \} $ be a set of formulas. We use $ x(i) $ to denote the index of the variable on which the universal quantifier acts in the formula $ f(i) $. Let $ \equiv $ denote that two assignments have the same value for the free variables of the formulas indexed by $ \equiv $. Let $ U $ be an ultrafilter on $ X $, and $ (M, \nu) $ be a model structure in the language.

The question is:

Statement 1: For all assignments $ \mu $ in the structure $ M $, if $ \{ i \ | \ \nu \equiv_{(\forall x_{x(i)} f(i))} \mu \} \in U, $ then $ \{ i \ | \ (M, \mu) \models f(i) \} \in U. $

And Statement 2: $ \{ i \ | \ \text{for all assignments } \mu \text{ in } M, \nu \equiv_{(\forall x_{x(i)} f(i))} \mu, \text{ then } (M, \mu) \models f(i) \} \in U. $

Are these two statements equivalent?

Question 2:

Is there a simpler formula that can equivalently replace this: $ \{i \mid \{j \mid \phi(i,j) \} \in U\} \in U \quad \text{and} \quad \{j \mid \{i \mid \phi(i,j) \} \in U\} \in U, $ where $U$ is an ultrafilter?

I understand that we might express it using some commutative statement of generalized quantifiers, such as $\forall^* i \forall^* j \phi(i,j)$, but I still hope to get some concrete content rather than just a different representation.

Additionally, I feel that the content of these two questions is quite different, but the difficulty I encounter when thinking about them feels very similar. What is it? I currently do not have the ability to abstract what this common difficulty is.

Question 1:

In the same language, let $ X $ be a nonempty set, and let $ \{ (\forall x_{x(i)} f(i)) \ | \ i \in X \} $ be a set of formulas. We use $ x(i) $ to denote the index of the variable on which the universal quantifier acts in the formula $\forall x_{x(i)} f(i) $. Let $ \equiv $ denote that two assignments have the same value for the free variables of the formula indexed by $ \equiv $ (In its lower right corner). Let $ U $ be an ultrafilter on $ X $, and $ (M, \nu) $ be a model structure in this language.

The question is:

Statement 1: For all assignments $ \mu $ in the structure $ M $, if $ \{ i \ | \ \nu \equiv_{(\forall x_{x(i)} f(i))} \mu \} \in U, $ then $ \{ i \ | \ (M, \mu) \models f(i) \} \in U. $

And Statement 2: $ \{ i \ | \ \text{for all assignments } \mu \text{ in } M, \nu \equiv_{(\forall x_{x(i)} f(i))} \mu, \text{ then } (M, \mu) \models f(i) \} \in U. $

Are these two statements equivalent?

Question 2:

Is there a simpler formula that can equivalently replace this: $ \{i \mid \{j \mid \phi(i,j) \} \in U\} \in U \quad \text{and} \quad \{j \mid \{i \mid \phi(i,j) \} \in U\} \in U, $ where $U$ is an ultrafilter?

I understand that we might express it using some commutative statement of generalized quantifiers, such as $\forall^* i \forall^* j \phi(i,j)$, but I still hope to get some concrete content rather than just a different representation.

Additionally, I feel that the content of these two questions is quite different, but the difficulty I encounter when thinking about them feels very similar. What is it? I currently do not have the ability to abstract what this common difficulty is.