arXiv subjects

Bernard A. Anderson

Publications and source records attributed to Bernard A. Anderson.

6 recordsLinked to original sources

arXiv2014

Degrees that are not degrees of categoricity

A computable structure A is x-computably categorical for some Turing degree x, if for every computable structure B isomorphic to A there is an isomorphism f:B -> A with f computable in x. A degree x is a degree of categoricity if there is a computable A such that A is x-computably categorical, and for all y, if A is y-computably categorical then y computes x. We construct a Sigma_2 set whose degree is not a degree of categoricity. We also demonstrate a large class of degrees that are not degrees of categoricity by showing that every degree of a set which is 2-generic relative to some perfect tree is not a degree of categoricity. Finally, we prove that every noncomputable hyperimmune-free degree is not a degree of categoricity.

math.LO↗

arXiv2011

A bounded jump for the bounded Turing degrees

We define the bounded jump of A by A^b = {x | Exists i <= x [phi_i (x) converges and Phi_x^[A|phi_i(x)](x) converges} and let A^[nb] denote the n-th bounded jump. We demonstrate several properties of the bounded jump, including that it is strictly increasing and order preserving on the bounded Turing (bT) degrees (also known as the weak truth-table degrees). We show that the bounded jump is related to the Ershov hierarchy. Indeed, for n > 1 we have X <=_[bT] 0^[nb] iff X is omega^n-c.e. iff X <=_1 0^[nb], extending the classical result that X <=_[bT] 0' iff X is omega-c.e. Finally, we prove that the analogue of Shoenfield inversion holds for the bounded jump on the bounded Turing degrees. That is, for every X such that 0^b <=_[bT] X <=_[bT] 0^[2b], there is a Y <=_[bT] 0^b such that Y^b =_[bT] X.

math.LO↗

arXiv2010

Relatively computably enumerable reals

A real X is defined to be relatively c.e. if there is a real Y such that X is c.e.(Y) and Y does not compute X. A real X is relatively simple and above if there is a real Y <_T X such that X is c.e.(Y) and there is no infinite subset Z of the complement of X such that Z is c.e.(Y). We prove that every nonempty Pi^0_1 class contains a member which is not relatively c.e. and that every 1-generic real is relatively simple and above.

math.LO↗

arXiv2009

Partitions of trees and ACA'

We show that a version of Ramsey's theorem for trees for arbitrary exponents is equivalent to the subsystem ACA' of reverse mathematics.

math.LO↗

arXiv2008

Automorphisms of the truth-table degrees are fixed on some cone

Let Dtt denote the set of truth-table degrees. A bijection p from Dtt to Dtt is an automorphism if for all truth-table degrees x and y we have x <=tt y if and only if p(x) <=tt p(y). We say an automorphism p is fixed on some cone if there is a degree b such that for all x >=tt b we have p(x) = x. We first prove that for every 2-generic real X we have X' is not tt below X + 0'. We next prove that for every real X >=tt 0' there is a real Y such that Y + 0' =tt Y' =tt X. Finally, we use this to demonstrate that every automorphism of the truth-table degrees is fixed on some cone.

math.LO↗

arXiv2007

Reals n-generic relative to some perfect tree

We say that a real X is n-generic relative to a perfect tree T if X is a path through T and for all Sigma^0_n (T) sets S, there exists a number k such that either X|k is in S or for all tau in T extending X|k we have tau is not in S. A real X is n-generic relative to some perfect tree if there exists such a T. We first show that for every number n all but countably many reals are n-generic relative to some perfect tree. Second, we show that proving this statement requires ZFC^- + ``There exist infinitely many iterates of the power set of the natural numbers''. Third, we prove that every finite iterate of the hyperjump, O^(n), is not 2-generic relative to any perfect tree and for every ordinal alpha below the least lambda such that sup_{beta < lambda} (beta th admissible) = lambda, the iterated hyperjump O^(alpha) is not 5-generic relative to any perfect tree. Finally, we demonstrate some necessary conditions for reals to be 1-generic relative to some perfect tree.

math.LO↗