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===Computable sets=== If ''S'' is a [[Computable function#Computable sets and relations|Turing computable set]], then both ''S'' and its [[Complement (set theory)|complement]] are recursively enumerable (if ''T'' is a Turing machine giving 1 for inputs in ''S'' and 0 otherwise, we may build a Turing machine halting only on the former, and another halting only on the latter). By [[Post's theorem]], both ''S'' and its complement are in <math>\Sigma^0_1</math>. This means that ''S'' is both in <math>\Sigma^0_1</math> and in <math>\Pi^0_1</math>, and hence it is in <math>\Delta^0_1</math>. Similarly, for every set ''S'' in <math>\Delta^0_1</math>, both ''S'' and its complement are in <math>\Sigma^0_1</math> and are therefore (by [[Post's theorem]]) recursively enumerable by some Turing machines ''T''<sub>1</sub> and ''T''<sub>2</sub>, respectively. For every number ''n'', exactly one of these halts. We may therefore construct a Turing machine ''T'' that alternates between ''T''<sub>1</sub> and ''T''<sub>2</sub>, halting and returning 1 when the former halts or halting and returning 0 when the latter halts. Thus ''T'' halts on every ''n'' and returns whether it is in ''S''; so ''S'' is computable.
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