Lindenbaum algebras and partial conservativity

Abstract:

A partial Lindenbaum algebra ${\Gamma ^A}$, where $A$ is a theory extending Peano arithmetic and $\Gamma \in \{ {\Pi _n},{\Sigma _n}\}$, is the full Lindenbaum algebra for $A$ restricted to sentences in $A$ provably equivalent to ${\Gamma _n}$-sentences. Using a new result on pairs of partially conservative sentences, we show that $\Pi _n^A$ and $\Sigma _n^A$ are not isomorphic.