Low sets without subsets of higher many-one degree

Given a reducibility $\leq_\mathrm{r}$, we say that an infinite set $A$ is $r$-introimmune if $A$ is not $r$-reducible to any of its subsets $B$ with $|A\backslash B|=\infty$. We consider the many-one reducibility $\leq_\mathrm{m}$ and we prove the existence of a low$_1$ $m$-introimmune set in $\Pi^0_1$ and the existence of a low$_1$ bi-$m$-introimmune set

Coding information into all infinite subsets of a dense set

Suppose you have an uncomputable set $X$ and you want to find a set $A$, all of whose infinite subsets compute $X$. There are several ways to do this, but all of them seem to produce a set $A$ which is fairly sparse. We show that this is necessary in the following technical sense: if $X$ is uncomputable and $A$ is a set of positive lower density then $A$ has an infinite subset which does not compute $X$. We will show that this theorem is sharp in certain senses and also prove a quantitative version formulated in terms of Kolmogorov complexity. Our results use a modified version of Mathias forcing and build on work by Seetapun and others on the reverse math of Ramsey's theorem for pairs.Comment: 30 page

Relativization of the theory of computational complexity.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1972.Vita.Bibliography: leaves 123-124.Ph.D

Generalized Domination.

This thesis develops the theory of the everywhere domination relation between functions from one infinite cardinal to another. When the domain of the functions is the cardinal of the continuum and the range is the set of natural numbers, we may restrict our attention to nicely definable functions from R to N. When we consider a class of such functions which contains all Baire class one functions, it becomes possible to encode information into these functions which can be decoded from any dominator. Specifically, we show that there is a generalized Galois-Tukey connection from the appropriate domination relation to a classical ordering studied in recursion theory. The proof techniques are developed to prove new implications regarding the distributivity of complete Boolean algebras. Next, we investigate a more technical relation relevant to the study of Borel equivalence relations on R with countable equivalence classes. We show than an analogous generalized Galois-Tukey connection exists between this relation and another ordering studied in recursion theory.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113539/1/danhath_1.pd