Abstract
We exposit two previously unpublished theorems of Leo Harrington. The first theorem says that there exist arithmetical singletons which are arithmetically incomparable. The second theorem says that there exists a ranked point which is not an arithmetical singleton. Unlike Harrington's proofs of these theorems, our proofs do not use the finite- or infinite-injury priority method. Instead they use an oracle construction adapted from the standard proof of the Friedberg Jump Theorem.
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