# Collapsing the Linear Time Hierarchy and finite axiomatizability of bounded arithmetic

It is well known that if ${\bf T_2}$ (or $I\Delta_0+\Omega_1$) is finitely axiomatizable, then the Polynomial Hierarchy collapses.

Q. Is there any similar relation between $I\Delta_0$ and Linear Time Hierarchy?

Edit: related questions

Q'. Why standard proof (like corollary 4.39 in chapter V of Metamathematics of first-order arithmetic) of

Finite axiomatizability of ${\bf T_2}$ $\Rightarrow$ PH collapse.

does not work for proving

Finite axiomatizability of ${\bf {\bf I}\Delta_0}$ $\Rightarrow$ LiH collapse ?

Relativized Case

Q''. Is there any formula $\phi(x)$ in the language of arithmetic with following properties?

I. $\mathbb{N} \models\forall x(\phi(x)\leftrightarrow \alpha(x))\to$ "LiH($\alpha$) does not colapse",

II. there exists $i\in \mathbb{N}$ such that, ${\bf I}E_i(\alpha)+\forall x(\phi(x)\leftrightarrow \alpha(x))\vdash {\bf I}\Delta_0(\alpha)$.

• ... An exception is Buss’s paper math.ucsd.edu/~sbuss/ResearchWeb/collapseBAPH/index.html , which uses a more light-weight argument to get a weaker collapse. On a quick look in the paper, I didn’t notice anything that would require genuinely polynomial growth, so there is a nontrivial chance it may actually be adapted to $I\Delta_0$ and the linear hierarchy, but the devil is in the detail. – Emil Jeřábek Dec 2 '16 at 14:49