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Dec 30, 2013 at 18:16 comment added Anonymous Thank you! Not only have you provided a great answer, you even read between the lines the true question I was trying (so unsuccessfully) to get at. By the way, this doesn't actually hinge on $\Sigma^0_1$, does it? It would work for any $\Sigma^0_k$ as long as the $k$ is constant (and doesn't depend on which slice of $S$ is being defined at the moment), right? (For example, throw in a new disjunct "or W_n is nonempty" to each right hand side, say.)
Dec 30, 2013 at 17:47 history answered Bjørn Kjos-Hanssen CC BY-SA 3.0