Wilkie's well known question asks whether $I\Delta\_{0}$ proves the unboundedness of primes. We know that by adding a sentence to $I\Delta\_{0}$ which says "the exponential function is total", it is possible to prove the unboundedness of primes. This sentence is $\Pi\_{2}$. Suppose $\Pi\_{1}\text{-Th}(\mathbb{N})$ denotes the set of all $\Pi\_{1}$ sentences that are true in $\mathbb{N}$. My question is:


Is it known that $I\Delta\_{0} 
+\Pi\_{1}\text{-Th}(\mathbb{N})$ proves the unboundedness of primes?



Unboundedness of primes in bounded arithmetic
Yes, because $\: I\Delta\_0 + \text{WPHP}\left(\Delta\_0\right) \:$ proves the unboundedness of primes (see [this answer](https://mathoverflow.net/questions/75995/formalizing-euclids-proof-of-the-infinitude-of-primes/76058#76058)),
  

since the assetion that a $\Delta\_0$-defined relation is an injection from $\:[0\hspace{.01 in},\hspace{-0.02 in}2\hspace{-0.05 in}\cdot\hspace{-0.04 in}x]\:$ to $\:[0,\hspace{-0.01 in}x]$
  

can be made itself $\Delta\_0$ by modifying the relation to also require that its output is in $\:[0,\hspace{-0.01 in}x]\;$.


