I'm interested in methods for proving a set is at some level $\Sigma^0_n$ (or $\Pi^0_n$) in the arithmetical hierarchy, and in particular, proving it is at the level with the smallest $n$ possible.
I know that a set is in $\Sigma_n^0$ if it can be expressed as a sequence of $n$ quantifiers (starting with $\exists$ and then switching between $\exists$ and $\forall$) on variables, followed by a decidable formula on those variables. I also know membership in $\Pi_n^0$ can be proved in a similar way, just starting with $\forall$ instead of $\exists$.
These slides use that approach to place several sets at different places in the arithmetic heirarchy (on slides 26,31,33, and 35, for example).
Here's where things start to get tricky for me. If a language is in $\Sigma^0_n$, it seems like it's also trivially in $\Sigma^0_{n+1}$, $\Sigma^0_{n+2}$, and so on -- because you can take the formula you used to show that the language was in $\Sigma^0_n$, and tack on "dummy" quantifiers that don't actually do anything.
So, let's say I want to proving that a language is in $\Sigma^0_n$, but not $\Sigma^0_{n-1}$, i.e. I want to put the language in its lowest spot on the arithmetical heirarchy. Now, it seems like just showing that the language corresponds to a a $\exists,\forall,...$ sequence followed by a decidable formula won't cut it, because there's a chance that I was adding useless quantifiers to artificially inflate my language's difficulty. What techniques exist to get around this issue -- How do I prove that my quantifiers are "essential", so to speak?
I'm definitely interested in being as formal as possible with such a proof, so any links to textbooks or papers would be much appreciated.