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The difficulty is not simply that the vocabulary is not perfectly well defined; it's that "English" really means, in practice, something like "whatever someone considered an English speaker finds themselves able to understand as English" and yes, that's circular. Languages evolve, writers deliberately push the boundaries, etc. Is the property of being English even computable? Probably (though I wouldn't want my life or my job to depend on being able to provide even a sketchy proof) for finite sentences; for the infinite "sentences" you want to allow, though, not so much. It's not even clear what "computable" would mean there; it certainly can't be anything that involves feeding the "sentence" to a machine that does a finite amount of computation and then returns a verdict.
English is computable. You can describe, letter by letter any Python program you want. Letter by letter.
Not only its listing, but also its execution, step by step. So yes, English is computable.