A note on uniform density in weak arithmetical theories

Abstract

Answering a question raised by Shavrukov and Visser (Notre Dame J Form Log 55(4):569–582, 2014), we show that the lattice of $\exists \Sigma ^\mathsf {b}_1$-sentences (in the language of Buss’ weak arithmetical system $\mathsf {S}^1_2$) over any computable enumerable consistent extension T of $\mathsf {S}^1_2$ is uniformly dense (in the sense of Definition 2). We also show that for every $\mathcal {C} \in \{\Phi _n: n\ge 3\} \cup \{\Theta _n: n \ge 2\}$ (where $\Phi$ and $\Theta$ refer to the known hierarchies of arithmetical formulas introduced by Burr for intuitionistic arithmetic) the lattices of $\mathcal {C}$-sentences over any c.e. consistent extension T of the intuitionistic version of Robinson Arithmetic ${{\,\mathrm{\mathsf {R}}\,}}$ are uniformly dense. As an immediate consequence of the proof, all these lattices are also locally universal (in the sense of Definition 3).