Juan Aguilera (University of Vienna)

Non-linearities in the analytical hierarchy

It is commonly known that there exist true $\Pi^0_1$ sentences which are mutually independent over theories such as PA. The analogue of this fact for $\Pi^1_1$ fails: there do not exist true $\Pi^1_1$ sentences which are mutually independent over theories such as $PA$ + all true $\Sigma^1_1$ sentences. We study the corresponding situation for $\Sigma^1_n$ and $\Pi^1_n$ under assumptions such as $V = L$ or large cardinals. This is joint work with F. Pakhomov.