Abstract
This chapter provides a note on arithmetical sets of indiscernibles. It has been proved that there is a recursive partition (of the set N of natural numbers) that possesses no infinite recursively enumerable (Σ 0 1 ) set of indiscernibles; the existence of some infinite set of indiscernibles is the familiar theorem of Ramsey. The chapter presents an entirely different proof based on the theory of retraceable sets. A set is called Σ 0 n , Π 0 n , Δ 0 n if it is definable in prenex normal form with n alternating quantifiers where the first quantifier can be chosen to be respectively existential, universal or both. This general procedure is referred to as the priority method, which could be more accurately replaced by the approximation method or the trial-and error method.
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