After Godel's groundbreaking results, a plethora of $\Pi\_1^0$ undecidable arithmetical sentences have been found by many authors.


But what about $\Pi\_n^0$ for $n=2,3,.....$ ? 


There are, to my knowledge (but I am no expert) a few examples on the low end of the ladder, for instance Svejdar, a student of Petr Hajek, in the early eighties initiated the study of [interpretability logic](http://en.wikipedia.org/wiki/Interpretability_logic), and in the process identified one such sentence. 


Now, my question:


**Is there some systematic procedure/strategy for building a list of progressively higher (in the sense of the arithmetical hierarchy) undecidable sentences?** 


NOTE: I would think that one option could be to follow Svejdar's steps, and identify sentences which express higher (and/or looser) forms of auto-referentiality. 


ADDENDUM: As Joel Hamkins has immediately pointed out in the comment below, the question, as formulated above, is entirely trivial (you simply join a pi\_0 godelian sentence with a known to be true pi\_n sentence and the game is over). I guess it should be emended by ruling out such cases, and stipulating that the rungs of the ladder should be $\Pi\_0^n$ sentences which are *undecomposable*, meaning that they cannot be boolean-broken in smaller pieces where a $\Pi\_0^k$ with k less than n undecidable is found. The idea is that the new rung should express a genuine new (higher) form of undecidability. 



Godel 's Ladder: Undecidable  PI_N sentences for N =2, 3, ......
The halting set relative to an oracle deciding $\Pi\_{n-1}$ sentences, provides such a sentence. This set is $m$-complete for $\Pi\_n$.


