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?