# Boundedly axiomatizable theories meet (in)completeness

Ali Enayat^{1}

^{1}* University of Gothenburg*

Recently Albert Visser and I wrote a paper on the incompleteness of boundedly
axiomatizable theories [1], which arose from a question posed by
Steffen Lempp and Dino Rossegger in the context of calibrating the Borel
complexity of the family of models of a given first order theory. The main
result of our paper shows that any consistent extension of a sequential theory
that is axiomatizable by a set of axioms of bounded complexity is incomplete. In
the theorem “complexity” refers to the *depth-of-quantifiers-alternation*, which
in the context of arithmetical theories that contain
$\mathsf{I\Delta}_0+\mathsf{Exp}$, agrees with the usual
$\Sigma_n$-complexity measure (in which $\Sigma_0$ corresponds to formulae
all of whose quantifiers are bounded). This result has also had a number of
offshoots, one of which is the ongoing work of Visser, Mateusz Lelyk, and myself
on the construction of completions of the arithmetical theories ${\sf PA}^-$
and ${\sf IOpen}$ that, in contrast, lend themselves to axiomatizability by
sentences of bounded complexity as measured by the $\Sigma_n$-complexity
measure. This tutorial aims to provide an exposition of these recent
developments.