I've seen a construction of a sentence of first order logic that is consistent, but has no models with underlying set $\mathbb{N}$ and recursive functions and relations. Do there also exist consistent sentences with no model on $\mathbb{N}$ with arithmetically definable functions and relations? If no, at which levels of the arithmetical hierarchy are there consistent sentences with no model at that level? If yes, what about the same question for analytically definable models?



Consistent sentences with no arithmetically definable models
Tarski proved that there is no arithmetically definable truth predicate. But one can express what it means to be a truth predicate in a single assertion expressing the Tarskian recursion, in the language with a symbol for that predicate. So there is no arithmetic model of the sentence asserting that $T$ is a truth predicate, if the rest of the model is the standard arithmetic, even though this theory is consistent. 


But the requirement that the rest of the model is standard is required for this argument. Ultimately, the answer to your question is negative, because if $\sigma$ is a consistent sentence, then one can build the Henkin model of $\sigma$, and this model will be arithmetically definable, and indeed computable from $0'$. Every consistent sentence has an arithmetically definable model on $\mathbb{N}$ of low complexity. Indeed, the Henkin construction shows that every arithmetically definable consistent theory has a model whose elementary diagram has complexity at most one level higher in the arithmetic hierarchy. For some introductory background, see Reed Solomon's [Effective Classical Completeness Theorem](http://logic.uconn.edu/wp-content/uploads/sites/508/2014/01/effectivecompleteness.pdf).


