Controlling The Skips of Arithmetic Sets

Andrews, Ganchev, Kuyper, Lempp, Miller, A.\ Soskova, and M.\ Soskova conjectured that skip-cototality coincides with weak cototality. We show that the conjecture is false by constructing a ${\mathrm{\Pi }}_2^0$ set whose skip does not have a cototal degree. We further adapt this construction to obtain additional results. We prove that for each odd $n$, there is a ${\mathrm{\Pi }}_{n+1}^0$ degree whose $n$-iterated skip does not have a cototal degree. We show that there is a ${\mathrm{\Pi }}_2^0$ degree which does not contain an introenumerable set.