Is it possible the define an effectively generated first order axiomatic system that is complete for pure arithmetical sentences defined in its language?
A pure arithmetical sentences is a sentence using the traditional arithmetical operators (including $\neq, <, > $) but that doesn't use any logical connective!
In fact we can do much better: Robinson's $\mathsf{Q}$ proves every true $\Sigma^0_1$ sentence (this is called $\Sigma^0_1$-completeness - see the discussion here) and so a fortiori decides every quantifier-free sentence. It seems to me that your "purely arithmetical" sentences are in particular all quantifier-free, so this is more than enough.
If we try to go much beyond the quantifier-free sentences, however, things quickly break down: by the internal MRDP theorem (see Hajek/Pudlak section I.3(d)), no consistent computably axiomatizable extension of $I\Sigma_1$ can decide every sentence of the form "Such-and-such Diophantine equation has no solutions." Note that we don't even need any Booleans here - universally quantified equations are enough to be problematic.