Set of sentences of FO+LFP

The set of wffs of first-order logic is recursive. Taking that checking if a wff is a sentence is straightforward I take that the set of sentences of FO is also recursive.

If we add FO the least fixed point operator, does the set of sentences stay recursive? I guess yes, but I haven't seen this proved anywhere. Where can I find a (rigorous) proof?

What about other logics, say MSO?

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