TITLE: Unboundedness of primes in bounded arithmetic
QUESTION [14 upvotes]: 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?

REPLY [8 votes]: Yes, for trivial reasons: $\forall x>0\,\exists y\le2x\,(y>x\land\mathrm{Prime}(y))$ is a true $\Pi^0_1$ sentence which implies the unboundedness of primes. (Of course, weaker bounds than Bertrand’s postulate would also do, such as $y\le x^2$.)