ACKNOWLEDGEMENTS My thanks to the many people whose thoughts have contributed to this thesis and to my mathematical development, including In particular, my collaborations with Tyrrell McAllister and Herb Scarf have been tremendously invaluable. Many thanks to my doctoral committee, especially John Stembridge for his careful reading of this thesis. I am… (More)
It is an open problem in the area of computable ran-domness whether Kolmogorov-Loveland randomness coincides with Martin-Löf randomness. Joe Miller and André Nies suggested some variations of Kolmogorov-Loveland randomness to approach this problem and to provide a partial solution. We show that their proposed notion of partial permutation randomness is… (More)
We make progress toward solving a long-standing open problem in the area of computable linear orderings by showing that every computable η-like linear ordering without an infinite strongly η-like interval has a computable copy without nontrivial computable self-embedding. The precise characterization of those computable linear order-ings which have… (More)
If F ⊆ N N is an analytic family of pairwise eventually different functions then the following strong maximality condition fails: For any countable H ⊆ N N, no member of which is covered by finitely many functions from F , there is f ∈ F such that for all h ∈ H there are infinitely many integers k such that f (k) = h(k). However if V = L then there exists a… (More)
We consider the possible cardinalities of the following three cardinal invariants which are related to the permutation group on the set of natural numbers: a g := the least cardinal number of maximal cofinitary permutation groups; a p := the least cardinal number of maximal almost disjoint permutation families; c(Sym(N)) := the cofinality of the permutation… (More)
Hirschfeldt and Shore have introduced a notion of stability for infinite posets. We define an arguably more natural notion called weak stability, and we study the existence of infinite computable or low chains or antichains, and of infinite Π 0 1 chains and antichains, in infinite computable stable and weakly stable posets. For example, we extend a result… (More)
ACKNOWLEDGEMENTS This thesis is the result of research I have done while I was a PhD student at the University of Michigan. I would like to thank everyone who has played a part in my doing this. In particular I am grateful to Andreas Blass for being an excellent advisor, Peter Hinman for many nice discussions and much information about recursion theory, and… (More)
We formalize the usual proof that the group generated by the function k → k + 1 on the integers gives rise to a cofinitary group.