Good morning,
I usually really struggle with logical complexity, when I see a statement it's very difficult for me to understand if it should be $\Delta_0$ or $\Sigma_1$ or something else (in the Lévy hierarchy). Hence, I was trying to look at some usual definitions in set theory to check if my understanding of this complexity is correct. It's also pretty much impossible to find online if an answer to a question of this kind is correct, so I'm going to ask here hoping for maybe also some reference to a textbook where I can learn more?
So, for example, the statement "$f$ is a filter on $\omega$". I would write it formally this way: $$(f\subseteq p(\omega)) \land (\varnothing\not\in f) \land (\forall a \in f,\ \forall b\in p(\omega): a\subseteq b\to b\in f)\land(\forall a\in f,\ \forall b\in f: a\cap b\in f).$$
All the quantifiers look bounded to me, hence this is a $\Delta_0$ statement with no parameters? Am I right? And if I want to say "there is a filter on $\omega$" it would become a $\Sigma_1$ statement with no parameters(?)
Second question: now about ultrafilters. "$u$ is an ultrafilter on $\omega$" clearly means: $$(u\ \text{is a filter on}\ \omega)\land (\forall a\in p(\omega): a\not\in u \to (\omega-a)\in u).$$ This again looks $\Delta_0$ with no parameters, to me. And "there is an ultrafilter on $\omega$" is $\Sigma_1$.
Hence again, "$u$ is a nonprincipal ultrafilter on $\omega$" means $$(u\ \text{is an ultrafilter on}\ \omega)\land (\forall g\in \omega\ \exists b\in u: g\not\in b).$$ And this AGAIN looks $\Delta_0$ to me! Maybe I'm doing something wrong because clearly not every statement can be $\Delta_0$...
Now, given a set $h$, we can define $[h]^2$ as the set of all subsets of $h$ with cardinality $2$, that is $$\{s\subseteq h:\ \exists a\in s,\ \exists b\in s: a\neq b \land (\forall c\in s:\ a=c\lor b=c)\}.$$ This is again a $\Delta_0$ definition, besides the fact that now I am using $h$ as a parameter.
But then if I want to say that "$u$ is a Ramsey ultrafilter" I just say that $$(u\ \text{is a nonprincipal ultrafilter on}\ \omega) \land (\forall s \subseteq[\omega]^2\ \exists h\in u:(s\subseteq[h]^2\lor s\cap[h]^2=\varnothing)).$$ So I would again say that this is $\Delta_0$, and "there is a Ramsey ultrafilter" is a $\Sigma_1$ statement without parameters.
Is this reasoning true? To me it looks like it shouldn't be... too many "easy complexity" formulas...