Reflection schema for PA I have some questions concerning reflection principle and Peano arithmetic:
1) PA + reflection for $\Pi_{1}$ sentences is equivalent to PA + CON(PA), I saw the proof but I dont quite get why $\bot$ is $\Pi_{1}$ sentence?
2) What about PA + reflection for $\Sigma_{1}$ sentences, is this theory sound, conservative over PA ?
3) What about adding reflection schema for $\Pi_{n}$ sentences for arbitrary $n$? I mean we will get stronger and stronger theories or is there a fixed point?

For 1): The usual definition is that the class of $\Pi_1$ formulas of arithmetic is the smallest class containing the atomic formulas and closed under conjunction, disjunction, bounded existential quantification, and universal quantification.  Depending on one's conventions, $\bot$ is either atomic or the disjunction of 0 disjuncts; either way, it is $\Pi_1$.
For 2): Since $\bot$ is also a $\Sigma_1$ sentence, reflection for $\Sigma_1$ sentences lets you prove the consistency of PA and is therefore not conservative over PA.  In fact, it's equivalent $\omega$-consistency of PA.  Of course PA plus any amount of reflection over PA is sound, because it's true.
For 3): Yes, as you increase $n$, you get stronger and stronger theories by adding reflection for $\Pi_n$ sentences to PA.  Each theory in this infinite sequence (as $n$ increases) proves the consistency of the preceding one.
